Excitations in Quantum Liquids.

Reports on Progress in Physics

ACCEPTED MANUSCRIPT

Excitations in Quantum Liquids To cite this article before publication: Henry R Glyde et al 2017 Rep. Prog. Phys. in press https://doi.org/10.1088/1361-6633/aa7f90

Manuscript version: Accepted Manuscript Accepted Manuscript is “the version of the article accepted for publication including all changes made as a result of the peer review process, and which may also include the addition to the article by IOP Publishing of a header, an article ID, a cover sheet and/or an ‘Accepted Manuscript’ watermark, but excluding any other editing, typesetting or other changes made by IOP Publishing and/or its licensors” This Accepted Manuscript is © 2017 IOP Publishing Ltd.

During the embargo period (the 12 month period from the publication of the Version of Record of this article), the Accepted Manuscript is fully protected by copyright and cannot be reused or reposted elsewhere. As the Version of Record of this article is going to be / has been published on a subscription basis, this Accepted Manuscript is available for reuse under a CC BY-NC-ND 3.0 licence after the 12 month embargo period. After the embargo period, everyone is permitted to use copy and redistribute this article for non-commercial purposes only, provided that they adhere to all the terms of the licence https://creativecommons.org/licences/by-nc-nd/3.0 Although reasonable endeavours have been taken to obtain all necessary permissions from third parties to include their copyrighted content within this article, their full citation and copyright line may not be present in this Accepted Manuscript version. Before using any content from this article, please refer to the Version of Record on IOPscience once published for full citation and copyright details, as permissions will likely be required. All third party content is fully copyright protected, unless specifically stated otherwise in the figure caption in the Version of Record. View the article online for updates and enhancements.

This content was downloaded from IP address 128.192.114.19 on 10/09/2017 at 12:55

Page 1 of 48

Excitations and structure of liquid 4 He: A review H. R. Glyde1 1

Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716-2593, USA

(Dated: June 5, 2017)

Contents

4 4 4 6 6 7 8 9 9 10 10 11 11 11 13 15 15

dM

2. Collective excitations A. Low temperature 1. One and multiphonon scattering 2. Phonon-roton mode energies 3. S(Q) and sum rules 4. Anomalous dispersion 5. Multphonon contributions B. Pressure Dependence 1. Phonon-roton energies 2. Structure of S(Q, ω) vs pressure 3. Higher pressure C. Temperature Dependence 1. Broad features of S(Q, ω) 2. Mode energies and lifetimes, low temperatures 3. Models of S(Q, ω), higher temperatures 4. Mode energies and lifetimes, higher temperatures D. Normal Liquid 4 He

1 1 2 3

17 17 18 19 19 21 21 22 23 24 25 26 27

4. Single particle excitations A. Introduction B. Impulse approximation C. One body density matrix D. Final state effects 1. Additive Approach E. Model OBDM and momentum distribution

28 28 30 30 31 32 33

pte

3. Interpretation: Collective Excitations A. Landau Theory B. Bijl Feynman Theory C. Feynman and Cohen Theory D. Correlated Basis Function Theory E. Shadow Wave Functions F. Diffusion Monte Carlo G. Moments of S(Q, ω) H. Field Theory Methods I. Dielectric Formulation of χ(Q, ω) J. Exact results from field theory K. Temperature dependence of S(Q, ω) 1. Role of the condensate

ce

F. Measurements of the condensate, the momentum distribution and Final State effects 1. PIMC calculations of Si (Q, ω) 2. Kinetic energy and the condensate fraction G. Kinetic energies in condensed Matter H. Superfluidity, modes and BEC 1. Landau and Superfluidity 2. Path integrals, symmetrization and superfluidity 3. BEC and superfluidity

an

1. Introduction A. In the beginning and why liquid 4 He B. Historical sketch C. Neutron scattering and the dynamical structure factor

us

cri pt

Progress made in measuring and interpreting the elementary excitations of superfluid and normal liquid 4 He in the past 25 years is reviewed. The goal is to bring up to date the data, calculations and our understanding of the excitations since the books and reviews of the early 1990s. Only bulk liquid 4 He is considered. Reference to liquid 3 He, mixtures, reduced dimensions (films and confined helium) is made where useful to enhance interpretation. The focus is on the excitations as measured by inelastic neutron scattering methods. The review covers the dynamic response of liquid 4 He from the collective excitations at low energy and long wavelength (i.e. phononroton modes) to the single particle excitations at high energy from which the atomic momentum distribution and Bose-Einstein condensate fraction are determined. A goal is to show the interplay of these excitations with other spectacular properties such as superfluidity and the test of fundamental calculations of quantum liquids that is possible. The role of Bose-Einstein condensation in determining the nature of the P-R mode and particularly it’s temperature dependence is emphasized. The similarity of normal liquid 4 He with other quantum and classical liquids is discussed.

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

AUTHOR SUBMITTED MANUSCRIPT - ROPP-100857.R1

35 36 38 39 40 40 41 41

5. Concluding remarks and future opportunities A. Concluding remarks 1. The P-R mode 2. BEC, the momentum distribution and final state effects B. Some future opportunities

42 42 42

6. Acknowledgments

44

References

43 43

44

1. INTRODUCTION A. In the beginning and why liquid 4 He

Helium was first liquefied by H. Kamerlingh Onnes (1908) in Leiden. With this breakthrough, liquid 4 He became the most accessible quantum fluid in nature for the study of the fundamental properties of Bose quantum fluids. Liquid 4 He and 3 He remain today the most accessible strongly interacting quantum fluids partnered with the trapped cold gases as the most accessible dilute quantum fluids. Access to liquid 4 He also opened low temperature physics. Using liquid 4 He as a cryogenic fluid, it became possible to cool materials to low temperatures. For example, Kamerlingh Onnes (1911) reported that the electrical resistance of Hg (and some other metals) dropped to practically zero below a critical temperature, the discovery of superconductivity. These two remarkable achievements have been colourfully re-


Page 2 of 48 2

cri pt

tion are obtained. We do not cover liquid 3 He nor 3 He−4 He mixtures. Recent articles are (Bryan et al., 2016; Diallo et al., 2006; Glyde et al., 2000a; Krotscheck and Lichtenegger, 2015). We do not cover 4 He films, (2D or 1D systems) nor helium in porous media (reduced dimensions and quantum liquids in disorder). Recent articles are (Bertaina et al., 2016; Bossy et al., 2012a, 2015; Diallo et al., 2014; Godfrin et al., 2012; Gordillo and Boronat, 2016; Prisk et al., 2013; Vranjeˇs Markić and Glyde, 2015). We do not consider helium droplets nor the cold Bose gases. We apologize for the brief treatment of many fascinating topics. We have, for example, aimed at integrating neutron scattering measurements with superfluidity and the development of theory of Bose liquids (e.g. section 4.H)

B. Historical sketch

us

Early investigation of liquid helium up to 1950 is superbly reviewed by London (1954). The measurements in the 1920s and 1930s soon showed that liquid 4 He was quite different above and below about 2.2 K (Keesom, 1942). The high and low temperature liquids were denoted He I and He II, respectively. Keesom and Clausius (1932) and Keesom and Keesom (1932) reported that the specific heat diverged at the transition from He I to He II. It was denoted the λ transition, at temperature Tλ , after the shape of the divergence. Earlier, Bose (1924) had proposed a quantum statistics for gases, denoted Bose statistics. Einstein (1924) noted that Bose statistics require that below a critical temperature, Tc , a macroscopic fraction of the gas atoms must occupy the lowest energy single particle state, now denoted Bose-Einstein condensation (BEC). This condensation was regarded as a curious peculiarity of a Bose gas. Kapitza (1938) reported that the viscosity of He II was many orders of magnitude lower that of He I, the first report that He II was a “superfluid”. Allen and Misener (1938), in a letter immediately following Kapitza’s, reported the same unmeasurably low viscosity in He II. Further fascinating properties of He II were reported that year by Allen and Jones (1938), Kikoin and Lasarew (1938) and Daunt and Mendelssohn (1938a; 1938b) in the same year. The discovery of superfluidity is superbly reviewed by Balibar (2007). Immediately, London (1938) proposed that the transition at Tλ was the onset of BEC in liquid 4 He and superflow in He II was a consequence of BEC. He showed how several observed properties of superfluid He II at T < Tλ could be explained if He II contained a macroscopic coherent component. Tisza (1938; 1940; 1947) proposed the famous two fluid model of superfluidity that we use today. He proposed that He II consisted of two interpenetrating but independent components, a normal liquid component that had normal friction with walls and a superfluid compo-

ce

pte

dM

an

viewed in Physics Today (van Delft, 2008; van Delft and Kes, 2010). Bose-Einstein condensation (BEC) and superfluidity were first observed in liquid 4 He. Understanding and interpreting these phenomena subsequently became an integral part of the development of quantum many-body theory. For example, the fundamental formulation of BEC in terms of the one-body density matrix (OBDM) (Penrose and Onsager, 1956) was motivated by the need to understand BEC in strongly interacting liquids as well as in Bose gases. Revealing and understanding superfluidity in liquid 4 He (and later in liquid 3 He) developed in parallel with that of superconductivity, as emphasized in the books by London (1950; 1954) and Tilley and Tilley (1990), Leggett (2006) and Ginsberg (2009). Today the nature of coherence in the liquid state arising from BEC and its consequence for superfluidity remain fascinating topics. Particularly, revealing and understanding the interplay between BEC, the collective phonon-roton (P-R) modes and superfluidity remains a key goal. As discussed particularly by Leggett (2006), a fundamental understanding of superfluidity in liquid 3 He and 4 He and superconductivity can be developed in parallel beginning with BEC and phase coherence. Liquid 4 He also serves as a unique laboratory to explore transport properties and other fundamental science, such as cosmology and low energy particle physics (Baym et al., 2013, 2015a,b; Dubbers and Schmidt, 2011; Grigoriev et al., 2016) using ultra cold neutrons (Golub et al., 1991) and the posible detection of dark matter (Schutz and Zurek, 2016). Success in the creation of ultra cold neutrons and detection of dark matter requires a precise knowledge of the dynamical response of liquid 4 He (Schmidt-Wellenburg et al., 2009; Schutz and Zurek, 2016). Both the collective modes in liquid 4 He and the single particle response from which BEC is determined are uniquely observed using neutron scattering methods. Our purpose is to review measurements made using neutrons and the understanding they bring. The focus is on measurements and their interpretation since the books by Nozières and Pines (1990), Silver and Sokol (1989), Griffin (1993), Glyde (1994a), and Griffin et al. (1995). A central aim is to bring the data and interpretation on P-R modes and BEC in these books up to date in this review surveying earlier work where useful. This review considers only bulk liquid 4 He. We open the Introduction below with a brief history of BEC, superfluidity and neutron scattering measurements in bulk liquid 4 He. We follow with a brief sketch of the dynamic structure factor, S(Q, ω), of liquid 4 He observed in the inelastic neutron scattering cross-section. In section 2 we review measurements of the collective P-R modes that are observed in S(Q, ω) at low momentum transfer, ~Q. In section 3 we review the theory of these modes and the interpretation of the data. In section 4 we review measurements of S(Q, ω) at higher ~Q from which the momentum distribution, n(k), and the condensate frac-

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 3 of 48

AUTHOR SUBMITTED MANUSCRIPT - ROPP-100857.R1 3

cri pt

To close this sketch, we recall that Penrose and Onsager (1956) defined BEC in terms of the OBDM, ρ1 (r) ≡ ˆ † (r)Ψ(0)⟩, ˆ ⟨Ψ a definition valid for a strongly interacting Bose fluid as well as a dilute gas (see section 4.C). The dimensionless OBDM n(r) ≡ ρ1 (r)/n, where n = N/V , is the Fourier transform of the atomic momentum distribution, n(k). This definition of n(k) and BEC offers a route to microscopic calculation of n(k) and the condensate fraction n0 = N0 /N using path integrals. This route provides the most accurate values of n(k) and n0 today, as discussed in section 4.H.2. The OBDM can also be readily modeled and fitted to data to obtained observed values of n(k)‘ and n0 , as discussed in section 4.E. In addition, the condensate state, χ0 (r), can be obtained by diagonalizing the OBDM. For a uniform 3D liquid these states are plane wave states with a phase. Superfluidity can be √ formulated√in terms of the condensate field, Ψ(r) = N0 χ0 (r) = N0 |χ0 (r)|eiϕ(r) and the condensate field velocity vs = (~/m)∇ϕ(r). Section 4 is devoted to reviewing recent measurements of n(k), of the condensate fraction and their interpretation in terms of superfluidity and other phenomena.

ce

pte

dM

an

nent that was frictionless. Tisza’s two fluid equations are reproduced in Eqs (4.33) of section 4.H.1. Conceptually, the physical origin of the superfluid component was BEC. However, the normal fluid density and velocity were defined empirically, as discussed in section 4.H.1, and the superfluid density and velocity were obtained from them as derived quantities, Landau (1941; 1947) in a tour de force of intuition developed the Tisza two fluid model into a complete and totally successful theory of superfluidity. The physical interpretation was quite different. Landau proposed that liquid 4 He, a dense, strongly interacting liquid supported only collective modes, denoted the phonon-roton mode (1947). Single particle excitations were excluded because the fluid was strongly interacting and because they would enable scattering of atoms between the normal fluid and superfluid components excluded in the two fluid model. At T = 0 K the liquid is in the ground state and all of the liquid is in the superfluid component. A normal component, defined empirically, is created at finite temperature by thermally exciting the characteristic P-R modes, as discussed more fully in section 4.H.1. The Landau theory does not draw on nor mention BEC. It was so successful that BEC was largely forgotten for many years. The P-R mode energy dispersion curve proposed by Landau 1947 is shown as an inset in Fig. 1.1. It is remarkably similar to the current, most accurate, observed curve. Early neutron scattering measurements in the 1950s verified the existence of the P-R mode, as sketched in section 2.A. Comments by Landau on the interpretation of the P-R mode are reproduced in section 3.A. The early microscopic calculations of the mode beginning with Feynman (1954) are reviewed in section 3.B as a basis for understanding current many-body theory calculations.

In a paper that remains a fundamental building block of many-body theory, Bogoliubov (1947) developed a theory of BEC and of superfluidiy arising from BEC for a weakly interacting Bose gas. As a consequence of BEC, the gas has only a single mode. The mode has sound dispersion at low Q, ωQ = cQ, as does the P-R mode, and has the character of both a density and quasiparticle mode. It is denoted quasiparticle sound. This formulation showed qualitatively, for a Bose gas, how a single mode having a phonon-roton dispersion (and no other modes) as well as superfluidity could arise if there is BEC. The theory is strictly valid for a Bose gas only. However, it is the foundation for the field theory formulations of Bose fluids developed in the 1960s, 70s and 80s as reviewed in section 3.H. These formulations show how well defined collective modes and superfluidity follow from BEC.

us

FIG. 1.1 Phonon-roton mode energy dispersion curve, ωQ , of liquid 4 He at low temperature (T = 0.5 K) and saturated vapor pressure (SVP). Triangles are from Donnelly et al. (1981) and triangles with dots from Glyde et al. (1998). The inset shows the mode proposed by Landau (1 meV = 11.605 K). Delta is the energy at the roton minimum.

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

C. Neutron scattering and the dynamical structure factor

A full introduction to neutron scattering and the wide range of properties that can be investigated can be found in Marshall and Lovesey (1971), Squires (1978), Lovesey (1984)) and the volumes edited by Sköld and Price ((Price and Sköld, 1987)), Fernandez-Alonso and Price (2013) and Price and Fernandez-Alonso ((2015). In condensed 4 He, neutrons scatter from the 4 He nuclei. Since the nucleus is spinless and 4 He is isotopically pure, the scattering length b of all the nuclei is the same. The scattering is therefore entirely coherent (see p. 21 (Squires, 1978)) and the coherent scattering length is just bc = b. The dynamical structure factor (DSF), S(Q, ω), is defined in terms of the coherent differential scattering cross


Page 4 of 48 4

section as (Lovesey, S. W., 1984),

A. Low temperature

′

where S(Q, t) is the coherent intermediate scattering function (ISF), 1 ∑ −iQ·ˆrj (t) iQ·ˆrl (0) 1 ρ(Q, t)ˆ ρ(−Q, 0)⟩ = ⟨e e ⟩ S(Q, t) = ⟨ˆ N N j,l

(1.3) ∑ The ρˆ(Q, t) = j e−iQ·ˆrj (t) is the Fourier transform of ∑ the nuclear density ρˆ(r, t) = j (ˆ r − rˆj (t)), ∫ ρ(Q, t) = dre−iQ·r ρ(r, t). (1.4)

=

1 ∑ −iQ·(rj −rl ) ⟨e ⟩ N j,l

(1.5)

dM

The scattering from liquid 4 He is always coherent. However, at higher Q (Q ≥ 12 ˚ A−1 ) the j = l term in S(Q, ω) dominates and S(Q, ω) is well represented by the j = l term only, i.e. by the incoherent Si (Q, ω) defined as, 1 ∑ −iQ·rj (t) −iQ·rj (0) Si (Q, t) = ⟨e e ⟩ (1.6) N j The incoherent static structure factor is Si (Q) = Si (Q, t = 0) = 1.

pte

2. COLLECTIVE EXCITATIONS

In this section we discuss neutron scattering measurements of the dynamical structure factor, S(Q, ω), of liquid 4 He at low wave vector (Q . 4 ˚ A−1 ) and low energy (~ω . 2 meV) transfer. At low wave vector and energy transfer from the neutron to the liquid, (energies less than or comparable to the interatomic potential) the neutrons create collective excitations in the liquid. At low temperature in the Bose-condensed phase, these are the well known phonon-roton (P-R) modes. We begin with liquid 4 He at saturated vapor pressure (SVP) (p ≃ 0) (section 2.A) and follow with the pressure and temperature dependence of these modes, sections 2.B and 2.C, respectively. The broad response of normal liquid 4 He at higher Q is discussed in section 2.D. The theory and interpretation of the dynamics is presented in section 3.

ce

At low Q and low temperature, the dominant feature observed in S(Q, ω) is the collective phonon-roton (P-R) mode. The energy dispersion curve, ωQ , of the P-R mode at low temperature and SVP was shown in Fig. 1.1. The P-R mode, proposed originally by Landau (1947) and derived first by Feynman (1954) and Feynman and Cohen (1956), was first observed by Palevsky et al. (1957), Yarnell et al. (1958) and Henshaw (1958) in the late 1950s. In the 1960s, 1970s and 1980s, the pressure and temperature dependence of S(Q, ω) and the P-R mode energy, ωQ , were extensively measured. The measurements of this period are reviewed in detail in (Woods and Cowley, 1973), (Price, 1978), (Glyde, 1984), (Glyde and Svensson, 1987) and discussed in books (Griffin, 1993) (Glyde, 1994a). Since these excellent reviews, measurements at Q ≤ 2.3 ˚ A−1 by (Andersen et al., 1994a), (Andersen et al., 1994b), (Gibbs et al., 1999), Zsigmond (2007), (F˚ ak et al., 2012) and (Beauvois et al., 2016) have been made. At higher wave vectors “Beyond the roton” (BTR), at 2.5 < Q < 3.6 measurements by (F˚ ak and Andersen, 1991; F˚ ak and Bossy, 1998a,b; F˚ ak et al., 1992; Glyde et al., 1998; Pearce et al., 2001; Pistolesi, 1998) have appeared. The end point of the P-R mode is at Q = 3.6 ˚ A−1 where the intensity, ZQ , in the mode goes to zero. The measurements at higher energy resolution ((FWHM), W = 50 meV) (Glyde et al., 1998; Pearce et al., 2001) ωQ show that the P-R energy dispersion curve at Q > 2.8 ˚ A−1 becomes independent of Q at ωQ = 2∆. The dispersion curves at higher Q shown in Fig. 1.1 and in Fig. 2.1 below are taken from these data. The P-R mode energy, ωQ , and intensity, ZQ , are well determined at low temperature. The chief ambiguity remaining today is at higher temperature. This ambiguity is not in the data itself but rather in the analysis of data. It is in separating S(Q, ω) unambiguously into single PR mode and multiphonon-roton mode components,

an

From Eqs. 1.2 and 1.3 we see that the density response of liquid 4 He is observed in S(Q, ω). The static structure factor, S(Q), is ∫ S(Q) = S(Q, t = 0) = dωS(Q, ω)

1. One and multiphonon scattering

cri pt

d σ k = b2c S(Q, ω), (1.1) dΩdω k where k and k ′ are the wave vectors of the incoming and outgoing neutrons, respectively, Q = k − k ′ is the wave vector transfer and ~ω is the energy transfer in the inelastic scattering. S(Q, ω) is defined as ∫ 1 S(Q, ω) = dteiωt S(Q, t) (1.2) 2π

us

2

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

S(Q, ω) = S1 (Q, ω) + SM (Q, ω),

(2.1)

when the mode has a significant half width, ΓQ , at finite temperature. In Eq. (2.1), S1 (Q, ω) is defined as the sharp P-R mode component of the single P-R response. The P-R mode energy, ωQ , is defined as the energy of the sharp component. SM (Q, ω) is the remainder of S(Q, ω) at ω > ωQ , chiefly ω > 2∆, where ∆ is the P-R energy at the roton minimum. At low temperature (T < ∼ 1.3 K), S1 (Q, ω) can be readily separated from SM (Q, ω) in Eq. (2.1) since S1 (Q, ω) is well represented by delta functions, S1 (Q, ω) = [nB (ω) + 1]ZQ [δ(ω − ωQ ) − δ(ω + ωQ )]. (2.2) where nB (ω) is the Bose function. Using nB (−ω) + 1 = −nB (ω), the second term can be written as

Page 5 of 48


us

FIG. 2.2 S(Q, ω) = S1 (Q, ω) + SM (Q, ω) at Q = 0.9 and 1.1 ˚ A−1 vs energy transfer, ~ω, in liquid 4 He at SVP and T = 1.3 K (Andersen et al., 1994a). The sharp peak (dotted line) is the P-R mode in S1 (Q, ω). The multi-phonon component, SM (Q, ω) is at higher energy transfer, ω ≥ 1.4 meV, and has two peaks. The intensity in SM (Q, ω) is multiplied by a factor of 10.

an

FIG. 2.1 Phonon-roton mode energy dispersion curve, ωQ , at four pressures. At SVP and Q < 2.3 ˚ A−1 , the blue triangles are from (Donnelly et al., 1981) and the red open circles from (Andersen et al., 1994a). At SVP and Q > 2.3 ˚ A−1 , the red triangles are from (Glyde et al., 1998). At 20 bar and Q < 2.3 ˚ A−1 , the green open squares are from (Gibbs et al., 1999); at 20 bar and Q > 2.3 ˚ A−1 , the green open circles from (Pearce et al., 2001). The stars are ωQ at 31.2 bar (Bossy et al., 2008a). The black open circles at 0.4 ≤ Q ≤ 1.6 are ωQ at − 5.5 bar (Albergamo et al., 2004) .

ce

pte

dM

+nB (ωQ )ZQ δ(ω + ωQ ). In addition at low temperature where kB T ωQ , SIN T (Q, ω) is the interference terms in S(Q, ω) between the single and multiP-R components as are clearly identified in solid helium

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

cri pt

5

(Glyde, 1994a) and S2 (Q, ω) is the scattering that creates two P-R modes. There are also higher order terms. Since, the single P-R response at ω ≥ 2∆ decays predominantly to two P-R modes and the interference is chiefly between S1 (Q, ω) and S2 (Q, ω), the observed intensity at ω ≥ 2∆ is predominantly proportional to the two P-R density of states (Manousakis and Pandharipande, 1986; Stirling, 1983). At temperatures T ≤ 1.3 K the width of S1 (Q, ω), 2ΓQ , is small compared to the resolution width (W = 50 - 100 µeV) of a typical instrument used to measure S(Q, ω)(Andersen et al., 1994a; Beauvois et al., 2016; Gibbs et al., 1999). For example, at T = 1.3 K and Q = 1.9 ˚ A−1 , 2ΓQ ≃ 10 µeV (F˚ ak et al., 2012; Mezei, 1980) (see Fig. 2.12). Thus in analysis of data at T ≤ 1.3 K, S1 (Q, ω) can be represented by a delta function, S1 (Q, ω) = ZQ δ(ω − ωQ ), The width of S1 (Q, ω) shown in Fig. 2.2 arises entirely from the IN6 instrument resolution width. A clear separation of S1 (Q, ω) and SM (Q, ω) is needed in order to extract the P-R mode ωQ , ΓQ , and the intensity, ZQ , in the P-R mode. This is possible at T ≤ 1.3 K, and ωQ and ZQ are well determined. At higher temperature the separation is more difficult, as discussed below. Fig. 2.3 shows S(Q, ω) at T = 0.6 K at higher Q. At higher Q, the intensity in the single P-R peak, S1 (Q, ω),


Page 6 of 48 6

2∆

0.3

Q = 2.4 Å

-1

0.2

0.0

S(Q,ω) (arb. units)

0.06

Q = 2.8 Å

-1

Q = 3.2 Å

-1

0.04 0.02 0.00 0.03 0.02

us

0.01 0.00

-1

Q = 3.6 Å

0.03

0.00 -2

2 4 6 Energy (meV)

8

dM

0

an

0.02 0.01

FIG. 2.3 S(Q, ω) as Fig. 2.2 at higher wave vector transfer, Q, in liquid 4 He at 20 bars. The black dots show a sharp peak at T = 0.6 K and ω ≤ 2∆, the P-R mode in S1 (Q, ω). The multi-phonon, SM (Q, ω) is the broad intensity at higher ω (ω ≥ 2∆), At Q ≥ 2.8 ˚ A−1 , S1 (Q, ω) is small compared to the broad SM (Q, ω). The weight ZQ in S1 (Q, ω) goes to zero at Q = 3.6 ˚ A−1 . The open circles show the broad S(Q, ω) at T = 2.10 K (Pearce et al., 2001)) .

pte

is small and that in the broad component SM (Q, ω) at higher energy (ω ≥ 2∆) is large. As Q increases to the end point of the dispersion curve (Q = 3.6 ˚ A−1 ), the weight, ZQ , in S1 (Q, ω) decreases until ZQ goes to zero at Q = 3.6 ˚ A−1 . While in Fig. 2.2, there is clear structure in SM (Q, ω) at low temperature,(Andersen et al., 1994a; Manousakis and Pandharipande, 1986; Stirling, 1983) at higher Q in Fig. 2.3, SM (Q, ω) is dominated by a broad, largely featureless, temperature independent scattering that extends up to high ω.

ce

ωQ shown are in good agreement with earlier data. Measurements (Glyde et al., 1998; Pearce et al., 2001) at Q > ∼ 2.6 ˚ A−1 show that the single P-R energy ωQ comes up to twice the roton energy, 2∆, but does not exceed 2∆. Earlier measurements (Cowley and Woods, 1971; Smith et al., 1997) suggested the ωQ might exceed 2∆. If ωQ > 2∆, the single P-R mode has enough energy to decay spontaneously to two P-R modes, two modes in the roton region, as noted first by Pitaevskii (Pitaevskii, 1959) and formulated in detail by Ruvalds and Zawadowskii (1970), Zawadowskii et al. (1972) and Bedell et al. (1984), Pistolesi (1998) and others. Specifically, the response in S(Q, ω) will be sharp for ω < 2∆ but will be broad for ω > 2∆ in the two P-R mode band. We return to S(Q, ω) in the wave vector range 2 ≤ Q ≤ 4 ˚ A−1 below. At T = 1.0 K, the half width, ΓQ , of the P-R mode at the roton Q (0.5 µeV) is one thousand times smaller than that of a longitudinal phonon mode at intermediate Q in solid helium or of the zero sound mode in liquid 3 He. As T decreases, ΓQ of the P-R mode continues to decrease exponentially while that in the solid and liquid 3 He is approximately independent of T . A unique and remarkable feature of the P-R mode in the Bose condensed phase is that the width 2ΓQ goes to zero at low temperature. This is because, when there is BEC, there are no single particle modes at low energy to which the P-R mode can decay. There is only the P-R mode, a combined density/single particle mode and so the mode can decay only to itself. Except at some low Q values (see section 2.A.4), this decay mechanism requires a thermal P-R mode and so the mode width 2Γ → 0 as T → 0. The absence of low lying single particle modes in liquid 4 He at T < Tλ (Tλ = 2.17K at SVP) is a consequence of Bose-Einstein condensation(Bogoliubov, 1947; Gavoret and Nozières, 1964; Glyde, 1994a; Glyde and Stirling, 1990; Griffin, 1993; Griffin and Cheung, 1973; Hohenberg and Martin, 1965; Szépfalusy and Kondor, 1974).

cri pt

T = 0.60 K T = 2.10 K

0.1

2. Phonon-roton mode energies

Fig. 2.1 shows the P-R mode energy dispersion curve at low temperature. For wave vectors Q ≤ 2.3 ˚ A−1 , the

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

3. S(Q) and sum rules

The weight, ZQ , in the P-R mode and the static structure factor, S(Q), are defined in Eqs. (2.2) and (1.5), respectively. S(Q) is the total integrated intensity in S(Q, ω). At Q → 0, ZQ → S(Q). At Q → 0, there is only the single P-R mode and SM (Q, ω) = 0. (Gavoret and Nozières, 1964). At Q ≃ 0.4 ˚ A−1 , ZQ ≃ S(Q). As Q increases, ZQ falls substantially below S(Q) at Q ≃ 1.1 ˚ A−1 and then rises sharply to a significant peak at Q ≃ 2 ˚ A−1 in the roton region (see Fig. 2.4). ZQ then falls rapidly to zero at Q values beyond the roton. As noted, ZQ = 0 defines the end point of the P-R dispersion curve. The interesting dip in ZQ at Q = 1.1 ˚ A−1 and the −1 ˚ peak at Q ≃ 2.0 A can be clarified using the f-sum

Page 7 of 48

AUTHOR SUBMITTED MANUSCRIPT - ROPP-100857.R1 7 1.6

1.2

Superfluid 4He 1.4

S(Q)

T = 0.5 K

H

1

1.2

0.6 0.4

0.6

0.2

0.4 SM(Q)

0

0.5

1

1.5

2 Q (Å)

0 2.5

3

3.5

4

FIG. 2.4 The static structure factor, S(Q), defined in Eq. (1.5), and the weight, ZQ , in the P-R mode S1 (Q, ω) = ZQ δ(ω − ωQ ) in liquid 4 He at T = 0.5 K and SVP. SM (Q) is the multi-mode component, SM (Q) = S(Q)− ZQ . S(Q) is from Svensson et al. (1980) and ZQ from Gibbs et al. (1999) (black triangles) and Azuah et al. (2013) (red circles)

rule,

(2.4)

dM

where ~ωR = (~Q)2 /2m is the recoil energy of a stationary free atom of mass m when a momentum ~Q is transferred to it. The f -sum rule is a number (of particles) conserving sum rule that is independent of interaction, phase, pressure and temperature. As in Cowley and Woods (1971), we introduce the normalized sum rule, ∫ 1 H = fsum /ωR = dωωS(Q, ω) = 1. (2.5) ωR Substituting S(Q, ω) = S1 (Q, ω) + SM (Q, ω) into Eq. (2.5), H can be separated into single and a multi-PR parts, H = H1 + HM = 1.

(2.6)

pte

At low temperature (e.g. T ≤ 1.3 K) where S1 (Q, ω) in Eq. (2.2) can be accurately written as S1 (Q, ω) = Z(Q)δ(ω − ωQ ), ∫ 1 H1 (Q) = dωωS1 (Q, ω) = Z(Q)ωQ /ωR . (2.7) ωR

lim S(Q, ω) = S1 (Q, ω) = S(Q)δ(ω − ωQ )

(2.8)

ce

At T = 0 K and in the limit Q → 0 (Gavoret and Nozières, 1964) (see section 3.J),

(2.9)

Q→0

and ωQ → cQ where c is the sound velocity. At Q → 0 there is only the P-R mode. Comparing Eq. (2.8) with Eq. (2.2), we have ZQ = S(Q) at Q → 0 . Using the Q → 0 value of S(Q), S(Q) = ~Q/(2mc), we have lim ZQ = S(Q) = ~Q/(2mc)

Q→0

0.5

1

1.5

2

2.5

3

3.5

4

−1

Q (Å )

FIG. 2.5 The f-sum rule normalized to unity, H, as defined in Eqs. (2.5) and (2.6) as a function of wave vector Q. H = H1 (Q) + HM (Q) is separated into contributions from the single P-R mode, H1 (Q), and multi-mode, HM (Q), intensity, respectively.

an

dωωS(Q, ω) = ~Q2 /2m = ωR ,

0

At Q → 0, Eq. (2.2) therefore satisfies the f -sum rule Eq. (2.4) and H1 = 1 so that there is no intensity outside the P-R mode at T = 0.

∫ fsum =

H1 (Q)

us

0.2

cri pt

H (Q)

S(Q)

0.8

0

HM (Q)

0.8

ZQ

1

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

4. Anomalous dispersion

At low Q (Q ≤ 0.8 ˚ A−1 ), ωQ can be formally expressed as a power series in Q, ωQ = cQ = c0 Q[1 + ω2 Q2 + ω3 Q3 + ....].

(2.10)

where c0 = c(Q = 0) is the sound velocity at Q = 0 and ω2 and ω3 are parameters. The term in Q2 is found to be small. During the 1960s-80s, much effort was devoted to determining ω2 and ω3 and the coefficients in similar expansions of S(Q) and ZQ , e.g. S(Q) = ~Q/(2mc0 )[1 + c2 Q2 + c3 Q3 + ....].

(2.11)

A goal was to predict the thermodynamic and sound properties from the P-R dispersion curve at low and higher Q (Donnelly et al., 1981; Maris, 1977; Sridhar, 1987). From analysis of sound measurements, Maris and Massey (1970) deduced that ω2 was positive; that ωQ and the sound velocity c(Q) have upward or anomalous dispersion with Q. At very low Q (Q ≤ 0.3 ˚ A−1 ), SM (Q, ω) = 0 and Eq. (2.9) holds. Substituting the expansion Eq. (2.10) into Eq. (2.9) and expanding the denominator we have, comparing with (2.11), s2 = −ω2 and s3 = −ω3 up to O(Q3 ) so that these expansions are related. The simple relations above hold only if SM (Q, ω) = 0. From Figs 2.4 and 2.5, we see that ZQ clearly falls below S(Q) at Q ≥ 0.5 and SM (Q, ω)̸= 0. Sophisticated formulations (e.g. (Lin-Liu and Woo, 1973; Pines and Woo, 1970; Rugar and Foster, 1984)) were developed to include SM (Q, ω), fulfil sum


Page 8 of 48 8

rules and develop consistent expansions of ωQ , ZQ and S(Q). These are fully reviewed by Sridhar (1987). Maris (1973) proposed and obtained excellent fits to sound properties at SVP using the low Q dispersion relation, (1 − Q /Q2A ) γQ2 ], (1 − Q2 /Q2B )

6

0.35 ’final.dat’ using 1:2:3

2

(2.12)

0.3

0.25

4

0.2

3

0.15

2

0.1

us

with parameters c0 = 238.3 m/sec, γ = 1.112 ˚ A2 , QA −1 −1 ˚ ˚ = 0.5418 A and QB = 0.3322 A . This relation was confirmed by Stirling (1983) who obtained a good fit to his accurate measurements of ωQ in the range 0.1 ˚ A−1 ≤ Q ≤ 0.8 ˚ A−1 using Eq. (2.12) with parameters 238.3, 0.977, 0.548, and 0.384, respectively. This measurement (Stirling, 1983) appears to remain the most precise at low Q to date and Eq. (2.12) the best representation of ωQ up to Q = 0.8 ˚ A−1 . Eq. (2.12) shows upward dispersion with c(Q) reaching a maximum of 4 % above c0 at Q ≃ QB ≃ 0.35 ˚ A−1 At Q ≤ QB , ω2 = γ. The upward dispersion is also confirmed in a measurement of the phonon width (Mezei and Stirling, 1983). In the region of upward dispersion the width increases since in that region temperature independent, three phonon decay processes become allowed.

cri pt

5

Energy (meV)

c = c0 [1 +

1

an

5. Multphonon contributions

ce

pte

dM

Fig. 2.5 shows H1 (Q) at low T and SVP obtained from Eq. (2.7) using the ωQ in Fig. 2.1 and ZQ in Fig. 2.4. As Q increases, H1 (Q) shows a significant dip at Q ≃ 1.3 ˚ A−1 and a rapid rise to nearly H1 (Q) = 0.4 at Q ≃ 2.0 ˚ A−1 . The interference term SIN T (Q) in SM (Q, ω) introduces precisely this behavior in solid 4 He (see sections 6.3 and 12.5 of (Glyde, 1994a). In the solid, SIN T (Q) transfers intensity from S1 (Q, ω) to SM (Q, ω) at Q ≃ 1,5 ˚ A−1 and from SM (Q, ω) to S1 (Q, ω) at Q ≃ 2.5 ˚ A−1 . −1 ˚ The small values of ZQ at Q ≃ 1.3 A and the large values of ZQ at Q ≃ 2.0 have been reproduced using shadow wave function (Moroni et al., 1998a) and correlated basis function (Campbell et al., 2015) methods. Interference terms are clearly included in the functions defining the modes in these formulations. Fig. 2.6 shows the intensity in the P-R mode (ZQ vs Q) and in multi-phonon component SM (Q, ω). The intensity (Z(Q) in the P-R mode has been divided by a factor of three relative to that in SM (Q, ω). Even with this factor, the intensity in the roton dominates all other intensities. Beauvois et al. (2016) have recently made highprecision measurements of S(Q, ω) in liquid 4 He at T < 0.1 K and SVP (ρ = 0.0219 ˚ A−3 ). In addition to a welldefined PR mode, they document precisely S(Q, ω) at energies above the PR mode. Specifically there is intensity which is a linear extension of the phonon mode (see Fig. 2.7). This intensity is denoted the “ghost phonon”. There is also intensity above the PR mode at higher Q. ˚−1 , there is intensity between Particularly at Q > ∼ 1.3 A

0.05

0

0

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1

2

0 3

4

Q (Å−1)

FIG. 2.6 Intensity in S(Q, ω) = S1 (Q, ω) + SM (Q, ω) vs Q and ω. The intensity in the P-R mode is ZQ from Fig. 2.4. SM (Q, ω) contributes at energies ω ≥ 2∆ (above the thin horizontal dashed line) (data from (Andersen et al., 1994b) and (Glyde et al., 1998)). The intensity in SM (Q, ω) is multiplied by a factor of 3. The short dashed line shows ~Q2 /2m. The solid line shows the peak, and the long dashed lines the half heights, in SM (Q, ω) vs Q at higher Q.

the PR mode and the energy 2∆. At Q ≥ 1.3 ˚ A−1 this intensity extends up to high energy. Intensity above the PR mode ∑ arises from the two PR mode DOS, G2 (Q, ω) ∝ l,m δ(ω − ωl − ωm )∆(Q − ql − qm ). This DOS appears in both the S1 (Q, ω) and S2 (Q, ω) terms in S(Q, ω). The G2 (ω) becomes significant at ω > 2∆ when both ωl and ωm are energies in the roton region where the single mode DOS is high. Indeed, there is an abrupt increase in intensity in both the observed and calculated S(Q, ω) in Fig. 2.7 when ω reaches 2∆. However, S(Q, ω) also shows some intensity at ω < 2∆ when one or both of ωl and ωm are phonons. The intensity in the ghost phonon arises largely from two phonons that have parallel wave vectors (B. F˚ ak, private communication). For this process G2 (Q, ω) has a significant matrix element in S1 (Q, ω) (see Eqs. (3.9) and (3.10)). The intensity at Q & 1.3 ˚ A−1 at ω < 2∆ arises largely from a phonon and a roton. Intensity at ω < 2∆

Page 9 of 48


cri pt

Wavevector transfer Q !!"1"

us

Energy transfer Ω !meV"

FIG. 2.8 P-R mode energy ωQ at Q = 1.1 ˚ A−1 (the maxon Q) versus pressure (Albergamo et al., 2004). The ωQ increases with increasing pressure at low pressure but becomes limited by twice the roton energy, 2∆, at higher pressure reaching a maximum ωQ = 2∆ at p ≃ 20 bar.

dM

FIG. 2.7 Upper: Intensity in S(Q, ω) = S1 (Q, ω) + SM (Q, ω) observed by Beauvois et al. (2016) in liquid 4 He at SVP. The heavy line is the P-R mode in S1 (Q, ω). The intensity above the P-R mode is SM (Q, ω). Lower: the corresponding S(Q, ω) calculated by Campbell et al. (2015). The intensity in SM (Q, ω) increases significantly at ω > ∼ 2∆.

pte

˚−1 is also seen in Andersen et al. (1994a). at Q ≃ 1.3 A At Q ≃ 1 ˚ A−1 , the observed intensity shows at peak at ω ≃ 1.8 meV, a peak that forms an apparent ‘upper branch’. This peak arises from a combination of maxons and rotons in G2 (Q, ω). This peak is seen in Fig. 2.6 at Q = 1.1 ˚ A−1 as one of two peaks above the PR mode. The new features in Beauvois et al. (2016) is the clear demonstration of multi-PR intensity at energies ω < 2∆ in the form of a ghost phonon and intensity at ω < 2∆ at higher Q. It is interesting that in spite of this intensity, a width to the P-R mode itself is not seen, even at higher Q where ωQ approaches 2∆.

ce

− 5 bar. The sound velocity in liquid 4 He at low temperature increases from c = 238.3 m/sec at SVP (Maris, 1977) to c = 343.5 m/sec at 20 bar (Stirling, 1991). As a result the P-R mode energy ωQ = cQ in the “phonon” region increases with increasing p as does ωQ at the “maxon” wave vector Q = 1.1 ˚ A−1 . In contrast, at wave vectors Q ≃ 2.0 ˚ A−1 in the “roton” region ωQ decreases with increasing pressure. Specifically, the “roton” energy, ∆, the energy at the roton minimum, decreases from ∆ = 0.742 meV at SVP (Andersen et al., 1994a; Woods et al., 1977) to ∆ = 0.652 meV at 20 bars (Pearce et al., 2001) This pressure dependence can be reproduced (Campbell et al., 2015) in first principles calculations of ωQ .

an

Wavevector transfer Q !!"1"

B. Pressure Dependence 1. Phonon-roton energies

Fig. 2.1 shows the P-R mode energy dispersion curve, ωQ , in liquid 4 He at pressures SVP, 20 bar, 31.2 bar and

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Energy transfer Ω !meV"

9

Since ωQ at the “maxon” increases with pressure while ∆ decreases, ωQ at the “maxon” eventually becomes limited by 2∆ at higher pressure. Fig. 2.8 shows ωQ at the “maxon” as a function of pressure from p = − 5 bar to the solidification pressure, p = 25.3 bar. At low p, ωQ increases with p but ceases to increase with p above roughly 20 bar as ωQ approaches 2∆. Above 20 bar, ωQ at the “maxon” decreases somewhat with increasing p. Similarly, ωQ at large Q values “beyond the roton” is limited 2∆. At Q ≥ 2.8 ˚ A−1 , ωQ decrease with increasing p, from 2∆ = 1.484 at SVP to 1.304 meV at 20 bar (Pearce et al., 2001). We expect the width of S1 (Q, ω) at energies at or below 2∆ to remain unobservably small at low temperature (i.e S1 (Q, ω) given by Eq. 3.2). However, at ω > 2∆, the response will be broad in ω since at ω ≥ 2∆ spontaneous decay to two other P-R modes is energetically possible. As ∆ decreases we anticipate a transfer of intensity from the sharp P-R mode, S1 (Q, ω), to the broad component, SM (Q, ω).


Page 10 of 48 10

us

an

The observed weight ZQ in the P-R mode, as defined in Eq. (2.2), is a function pressure. At wave vectors Q ≃ 1.0 ˚ A−1 , ZQ decreases substantially with pressure, by a factor of two between SVP and 20 bar (Gibbs et al., 1999). In contrast at Q ≃ 2.0 ˚ A−1 , ZQ increases with pressure. This change in ZQ with pressure signals a transfer of intensity between S1 (Q, ω) and SM (Q, ω) defined in Eq. (2.1). Specifically, since the f-sum rule is a number sum rule, H in Eq. (2.6) is unity independent of pressure. If ZQ decreases significantly with pressure (e.g. at Q = 1.0 ˚ A−1 ) and ωQ changes little, there must be a transfer of intensity from S1 (Q, ω) to SM (Q, ω) between SVP and 20 bar at Q = 1.0 ˚ A−1 and the reverse in the roton region. This behavior could be understood if interference terms that transfer intensity between S1 (Q, ω) and SM (Q, ω) are important and increase in magnitude with pressure. For example, in solid helium these terms reduce the S1 (Q, ω) at Q ≃ 1.5 ˚ A−1 , change sign at Q −1 ˚ ≃ 2.0 A , and increase S1 (Q, ω) at Q ≃ 2.5 ˚ A−1 . The similarity between this transfer of intensity in solid and liquid 4 He is discussed in sections 6.3 and 12.5 of (Glyde, 1994a).

cri pt

2. Structure of S(Q, ω) vs pressure

3. Higher pressure

ce

pte

dM

Returning to Fig. 2.1, we see that the P-R mode energy ωQ is a sensitive function of pressure. Specifically, the energy at the roton minimum, ωQ = ∆, decreases with pressure. At p = 20 bars, ωQ in the “maxon” region ˚−1 is limited by 2∆,. We expect the and at Q > ∼ 2.5 A single P-R mode to remain unbroadened if (and only if) its energy lies below 2∆. In a porous media, it is possible to extend the liquid phase to higher pressures above 25.3 bar where the bulk liquid solidifies. The P-R mode energy ωQ observed at 31.2 bar in liquid 4 He confined in gelsils and MCM-41 (Bossy et al., 2012a) is shown in Fig. 2.1. At 31.2 bar 2∆ is substantially reduced (∆ = 0.57 meV). A P-R mode was observed only in the wave vector range 1.7 ≤ Q ≤ 2.4 ˚ A−1 . A sharp P-R mode in the maxon region was not observed. The shape of ωQ at 31.2 bar suggests that at maxon wave vectors, 0.8 ≤ Q ≤ 1.7 ˚ A−1 , all of the intensity in P-R mode has been transferred to SM (Q, ω), specifically to S1B (Q, ω) in Eq. (2.3), at energies ω > 2∆. A sharp P-R mode may exist at the “maxon”, but it’s energy would have to be less than 2∆ = 1.14 ± 0.04. It is more likely that a sharp mode in the “maxon” region does not exist at 31.2 bar. At higher pressure (e.g. p = 35 bar) the P-R mode is expected (Bossy et al., 2008a; Pearce et al., 2004) to exist at low Q (Q . 0.5 ˚ A−1 ) and in a limited range (1.8 < Q . 2.5 ˚ A−1 ) at higher Q only. This pressure dependence clearly illustrates the limitations imposed on ωQ by 2∆. Calculations show that the roton energy, ∆, continues to decrease with increasing pressure but remains finite (∆ > ∼ 0.25 meV) up to 250

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

FIG. 2.9 Temperature dependence of S(Q, ω) at Q = 1.1 ˚ A−1 in liquid 4 He at 20 atm. At T < Tλ (upper frame), the intensity in the sharp P-R mode S1 (Q, ω) decreases with increasing temperature and goes to zero at Tλ . At T > Tλ (lower frame), S(Q, ω) is largely independent of temperature. Adapted from (Talbot et al., 1988)

bar. (Vranjes et al., 2005)

Page 11 of 48


ce

pte

dM

an

In the superfluid phase, between 0.5 K and Tλ , S(Q, ω) changes dramatically with temperature, . For example, at the maxon wave vector, Q = 1.1 ˚ A−1 , shown in Fig. 2.9, a sharp P-R mode is clearly observed in S(Q, ω) at low temperature, T < ∼ 1.3 K. When temperature is increased, the mode broadens (Andersen et al., 1994b; F˚ ak et al., 2012; Gibbs et al., 1999) and its intensity decreases (Andersen et al., 1994b; Gibbs et al., 1999; Stirling and Glyde, 1990; Talbot et al., 1988; Woods and Svensson, 1978). At T ≃ Tλ (Tλ = 1.928 K at p = 20 bar) and at temperatures above Tλ , Fig. 2.9 shows that there is no sharp P-R mode, only broad intensity in S(Q, ω). At T ≥ Tλ , S(Q, ω) changes little with temperature (see section 2.D). In contrast, the total integrated intensity in S(Q, ω) (the static structure factor, S(Q)) changes little with temperature (less than 5 % between 0.5 K and Tλ (Mineev, 1980)). The intensity in the P-R mode at low T (T ∼ Tλ . The sharp P-R mode observed at low T in the superfluid/Bose condensed phase is not observed in the normal phase. Similar behavior is observed at higher Q, up to the end point Q = 3.6 ˚ A−1 . Fig. 2.10 shows the observed spectral function, A(ϕ, ω), at constant scattering angle ϕ corresponding to Q values in the roton region, Q ≃ 1.95 ˚ A−1 . Since S(ϕ, ω) = (1/(2π))[nB (ω) + 1] A(ϕ, ω), A(ϕ, ω) is the DSF with the Bose thermal factor n(ω) removed. The nB (ω) is large at low ω and diverges at ω = 0. A(ϕ, ω) is an odd function of ω and vanishes at ω = 0. The sharp roton peak observed at low temperature in Fig. 2.10 decreases in height with increasing temperature (note the log scale). At T = 2.05 K there remains a clear but small, broadened P-R mode peak in A(ϕ, ω). This peak gives way to a broad, featureless A(ϕ, ω) at T ≥ Tλ which changes little with further increase in temperature. Fig. 2.11 shows S(Q, ω) at Q = 3.0 ˚ A−1 where the intensity ZQ in the P-R mode is small at low temperature (see ZQ vs Q in Fig. 2.4). As at lower Q, the intensity in the mode decreases and the mode broadens with increasing T . In the normal liquid phase, there is no sharp P-R mode. There has been much discussion on whether the intensity in the P-R mode goes to zero at T = Tλ or whether the mode simply broadens into very broad intensity at T ≥ Tλ . Either result can be obtained depending on the model used to analyze data, as we discuss below. How˚−1 , the P-R mode ever, whatever the model, at Q > ∼ 0.8 A has broadened by a factor of 1000 between 1 K and Tλ . Above Tλ there is only broad, largely temperature independent intensity. Data on liquid 4 He under pressure in porous media (Bossy et al., 2015), where the P-R mode vanishes at lower temperature (less broadening), suggest

cri pt

1. Broad features of S(Q, ω)

that the intensity in the mode can vanish at T ≃ Tλ with little broadening. As in other liquids, normal liquid 4 He supports a sound mode. At Q → 0 there remains a well-defined sound mode in normal liquid 4 He. Thus, at low Q (Q < ∼ 0.5 ˚ A−1 ) the extremely sharp P-R mode in superfluid 4 He broadens (Glyde, 1994a; Hohenberg and Martin, 1965; Stirling and Glyde, 1990; Woods, 1966) into a still relatively sharp sound mode in normal liquid 4 He. The analysis of data as a function of temperature may be said to have two goals. The first is to provide a general description of S(Q, ω) as it changes rapidly with T in the superfluid phase and evolves into a largely temperature independent S(Q, ω) in the normal liquid phase. The second is to determine the change in energy, ωQ , half width ΓQ and weight ZQ of the P-R mode with temperature in the superfluid phase. At lower temperature (T ≤ 1.5 K), S1 (Q, ω) remains sharply defined relative to the other components of S(Q, ω). In this case S1 (Q, ω) can be readily separated from the other components of S(Q, ω) such as the multi-P-R component, SM (Q, ω), in Eq. (2.1). However, at higher temperature as the P-R width increases, S1 (Q, ω) cannot be separated unambiguously from SM (Q, ω). Models of S(Q, ω) are needed and different models lead to different identifications of S1 (Q, ω) and therefore differing values of ωQ , ΓQ and ZQ . The differing values of ωQ , ΓQ and ZQ found in the literature arise chiefly from differing models of S(Q, ω) used. Some differences also arise from the use of different functions to fit S1 (Q, ω). In section 2.C.3 we illustrate these differences using two models; the WoodsSvensson (WS) and the Simple Subtraction (SS) models. We will also discuss a third model which has been used to analyze S(Q, ω) in porous media (Bossy et al., 2012a), a model which avoids some of the assumptions on the temperature dependence built into the WS and SS models. Below we first discuss lower temperatures where S1 (Q, ω) can be readily identified. We then discuss higher temperatures where models are required.

us

C. Temperature Dependence

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

2. Mode energies and lifetimes, low temperatures

The temperature dependence of the P-R mode energy, ωQ (T), and line width, γ(T ) = 2ΓQ , provides information on the nature of P-R mode interactions in liquid 4 He. At the roton wave vector (Q = 1.95 ˚ A−1 ), a theory of the shift in energy, δ(T) = ∆(T) - ∆(0), and the line width γ(T ) with temperature was first proposed by Landau and Khalatnikov (1965; 1949) (LK) with subsequent improvements by Bedell et al. (1982; 1984) (BPZ). In the LK theory, a roton lifetime (and energy shift) arises from the interaction and scattering of the roton with an existing thermally excited roton in the liquid. The two rotons are annhilated and two other P-R modes in the roton region are created, a four P-R mode process. Since a thermal roton is required in this process, the linewidth


Page 12 of 48

FIG. 2.10 Temperature dependence of the spectral function A(ϕ, ω) [S(ϕ, ω) = (2π)−1 (nB (ω)+1)A(ϕ, ω)] at constant scattering angle ϕ corresponding to Q values in the roton region in liquid 4 He at SVP (Zsigmond et al., 2007). The sharp peak in S(ϕ, ω) at T < Tλ = 2.17 K gives way to broad scattering at T > Tλ .

an

T = 1.29 K

dM

S(Q,ω) (arb. units)

T = 0.60 K

0.15

0.10

T = 1.70 K

0.05

T = 2.10 K

pte

0.00 0

2

4

6

Energy (meV)

ce

FIG. 2.11 Temperature dependence of S(Q, ω) at Q = 3.0 ˚ A−1 in liquid 4 He at 20 bar (Pearce et al., 2001). The solid line is a fit of Eq. (3.53) to data showing that the intensity in the P-R peak goes to zero as the condensate fraction n0 (T) goes to zero at Tλ = 1.93 K.

and energy shift is proportional to the number of thermally excited rotons, Nr (T ) =

√ √ T [1 + α µT ]e−∆(T )/T ) .

√ In Eq. (2.13), α µT is a small term arising from the deviation of ωQ from a parabola around the roton minimum (Bedell et al., 1982, 1984). In the LK theory, the roton energy shift and width are δ(T ) = −δR NR (T ) and γ(T ) = γR Nr (T ), respectively, where δR and γR are unknown magnitudes. Clearly δ(T ) and γ(T ) have the same temperature dependence. The LK and BPZ expressions have been used extensively to describe data (Andersen et al., 1994a; Gibbs et al., 1999; Glyde and Svensson, 1987; Mezei, 1980; Woods and Cowley, 1973). Early measurements were made on neutron spectrometers which had energy resolution widths W ≃ 100 µeV ≃ 1.2 K or larger. Line widths down to γ(T ) ≃ W/10 = 10 µeV = 0.1 K only could be resolved. The roton γ(T ) is 0.1 K at T ≃ 1.4 K (see Fig. 2.12). A factor of ten increase in precision was made using spin echo spectrometers (Mezei, 1980; Mezei and Stirling, 1983). Linewidths down to 1 µeV = 10−2 K = 10 mK could be resolved. The roton line width falls below 10 mK at T = 1.0 K and cannot be resolved at T ≤ 1.0 K. This limit of resolution to temperatures T ≥ 1.0 K and the dependence of γ(T ) and δ(T) on models used to analyse data at T ≥ 1.5 K made it difficult to make an unambiguous test of LK theory(Andersen et al., 1994a). The measurements and interpretation up to 1993 are reviewed in section 10.6 of (Glyde, 1994a). Subsequent improvements in instrument resolution and precision enabled more precise measurement of δ(T ) and γ(T ) to lower temperatures (0.8 K) (Andersen et al., 1996; Bossy et al., 2000; Farhi et al., 2001). These measurements showed (Andersen et al., 1996; Farhi et al., 2001) that while the line width, γ(T ), was well described by the LK theory, the observed temperature dependence of δ(T ) was much slower than predicted by LK or BPZ theory. There was a clear discrepancy between existing theory and experiment for δ(T ). More recently, instrument precision has been further increased using the TRISP spectrometer (F˚ ak et al., 2012). Line widths

us

-1

Q = 3.0 Å

0.20

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

cri pt

12

(2.13)

Page 13 of 48


us

an

FIG. 2.12 The temperature dependence of the roton (a) energy shift, δ(T) = ∆(T) − ∆(0), and (b) linewidth, γ(T ) = 2ΓQ (T ) (F˚ ak et al., 2012). The solid circles with error bars are the observed values. The solid line is the fit of Eqs. (2.14) and (2.15) to the observed values. The dashed and dotted lines are the terms in the fit arising from interaction of the roton with phonons and other rotons, respectively.

ce

pte

dM

down to 2 mK = 0.2 µeV (the roton line width in liquid 4 He at T = 0.8 K) could be resolved with a commensurate improved precision for δ(T). For T ≤ 1.5 K, we focus here on these recent measurements and theory (F˚ ak et al., 2012) Fig. 2.12 shows (a) the energy shift δ(T) = ∆(T) − ∆(0) and (b) the linewidth γ(T ) = 2ΓQ of the roton as a function of temperature up to 1.5 K. The γ(T ) is accurately determined between 0.8 K and 1.5 K. At T ≤ 0.5 K, δ(T) is zero within precision. The new precision reveals that δ(T) initially increases with T and then decreases with T . The LK theory provides only a decrease. There must be additional terms beyond the LK theory which increase δ at low temperature. Fak et al. (2012) identify two additional terms which have soley real parts and so contribute to δ(T) but not to γ(T ). These are a Hartree term, as in a Hartree-Fock approximation, involving phonons and a 3-P-R term involving phonons which is purely real and proportional to T 4 . A decay term involving phonons proportional T 7 was also identified. Including these new terms, the roton shift and linewidth are δ(T ) = δP T 4 − δR Nr (T )

and

7

γ(T ) = γP T + γR Nr (T ).

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

cri pt

13

FIG. 2.13 At T ≤ Tλ , the intensity ZQ in S1 (Q, ω), the single P-R mode component of S(Q, ω), versus temperature obtained fitting the Woods-Svensson (WS) and simple subtraction (SS) models to S(Q, ω). At T > Tλ , the intensity is the total integrated intensity in S(Q, ω). Adapted from (Andersen et al., 1994b).

The first terms, proportional to T 4 and T 7 , respectively, are the new terms involving phonons. The second terms are the original LK expressions proportional to Nr (T ). The δP , δR , γP and γR are unknown magnitudes. These expressions were fitted to the data with the magnitudes treated as free parameters. The fits are the solid blue lines in Fig. 2.12. Eq. (2.14) provides a good fit to the δ(T ) data with a positive value of δP . The γP is negligible and the LK theory alone continues to provide a good fit to γ(T ). In brief, the roton width remains well described by the LK theory alone. However, the roton energy shift with temperature is not well described by the LK theory alone. An additional term is needed to describe the shift correctly. This new term may be thought of as arising from a Hartree term and a second term, both involving phonons, which are purely real and contribute only to the energy shift.

(2.14)

3. Models of S(Q, ω), higher temperatures

(2.15)

In the superfluid phase at higher temperatures, T > ∼ 1.5 K, it is difficult separate the single P-R component


Page 14 of 48 14

1 [nB (ω) + 1]AD (Q, ω) 2π

(2.16)

where nB (ω) = [exp(~ω/kB T )−1]−1 is the Bose function and AD (Q, ω) is the DHO spectral function (Talbot et al., 1988) (see appendix C (Glyde, 1994a)), ] [ 2ΓQ 2ΓQ , AD (Q, ω) = ZQ − (ω − ωQ )2 + Γ2Q (ω + ωQ )2 + Γ2Q [ ] 8ωωQ ΓQ = ZQ . (2.17) 2 + Γ2 ])2 + 4ω 2 Γ2 (ω 2 − [ωQ Q Q

[SM (Q, ω)+ SB (Q, ω)] is then simply subtracted from the data. A single S1 (Q, ω) is fitted to the remaining data at all temperatures. In the SS model, the intensity ZQ in the P-R mode changes little with temperature up to Tλ . The fitted S1 (Q, ω) broadens into a flat, featureless intensity at Tλ but intensity remains at Tλ . To understand why intensity must remain at and above Tλ in the SS model, we note that the static structure factor of liquid 4 He, S(Q) given by Eq. (1.5), is roughly independent of temperature (within 5 %) (Mineev, 1980). Since, after subtraction of a constant background, the remaining integrated intensity over S1 (Q, ω) is proportional to S(Q), the integrated intensity in S1 (Q, ω) must remain roughly independent of temperature (Mineev, 1980; Talbot et al., 1988) and cannot go to zero at Tλ . Similarly, in the WS model, if we subtract a term proportional to (1 - ρS (T )/ρ) and S(Q) is constant, we expect the remainder of S(Q, ω) to be roughly proportional to ρS (T )/ρ. The drawbacks of these two models are (1) in the WS model the intensity in the P-R mode must go to zero at Tλ whereas (2) in the SS model intensity must remain in the P-R mode above Tλ . However, in either model there is no well-defined P-R mode at Tλ and above Tλ . To close this section we note that the weight or intensity in the P-R mode is most reliably given by the integral of S1 (Q, ω) over ω,

dM

an

The S1 (Q, ω) in Eqs. (2.16) and (2.17) is fitted to the observed S1 (Q, ω) with ωQ , ΓQ and ZQ treated as free fitting parameters. The best fit values of ωQ , ΓQ and ZQ are intepreted as the observed P-R mode energy, half width and intensity, respectively. In applications of 2 the DHO, the function E(Q) = [ωQ +Γ2Q ]1/2 is sometimes taken as the mode energy. The two models commonly used to describe the total S(Q, ω) are the Woods-Svensson (WS) (Talbot et al., 1988; Woods and Svensson, 1978) and SimpleSubtraction (SS) (Andersen et al., 1994b; Gibbs et al., 1999; Talbot et al., 1988) models. In the WS model, S(Q, ω) is expressed as the sum of a single P-R mode and a multi-P-R component, [S1 (Q, ω) + SM (Q, ω)], which is proportional to the superfluid fraction, ρS (T )/ρ, and a normal liquid component, SN (Q, ω), expected to be proportional to (1 - ρS (T )/ρ). This is written as,

SSS (Q, ω) = S1 (Q, ω) + [SM (Q, ω) + SB (Q, ω)]. (2.19)

cri pt

S1 (Q, ω) =

In the SS model, a multi-P-R mode and a background contribution, [SM (Q, ω)+ SB (Q, ω)] is first identified at low temperature and S(Q, ω) is written as,

us

from the remainder of S(Q, ω) unambiguously. In this case models of the total S(Q, ω) are needed to determine S1 (Q, ω) and the P-R mode energy, ωQ , mode inverse −1 lifetime τQ = ΓQ and mode intensity, ZQ from fits to data. We review two models below. In the models, S1 (Q, ω) is usually represented by a damped harmonic oscillator (DHO) function. Specifically,

pte

SW S (Q, ω) = [S1 (Q, ω) + (ρS /ρ)SM (Q, ω)] + (1 − ρS /ρ)SN (Q, ω). (2.18)

ce

where the intensity, ZQ , in the P-R mode in S1 (Q, ω) is expected to be proportional to ρS (T )/ρ and to go to zero at Tλ , The multi P-R term, SM (Q, ω), is obtained from a fit to data at low T where ρS (T )/ρ = 1. At Q < 2.3 ˚ A−1 , SM (Q, ω) is small compared to S1 (Q, ω). ∼ The SN (Q, ω) is determined from fits to S(Q, ω) at T > Tλ where S(Q, ω) = SN (Q, ω). The WS model describes the temperature dependence of S(Q, ω) quite well for wave vectors Q ≥ 0.8 ˚ A (Andersen et al., 1994a; Gibbs et al., 1999; Stirling and Glyde, 1990; Talbot et al., 1988; Woods and Svensson, 1978). For example, fits of Eq. (2.18) to data yield a parameter ZQ that is quite similar to ρS (T )/ρ, as shown in Fig. 2.13. At low Q there is a sound mode that survives into the normal phase, i.e. there is sound propagation in the normal liquid.

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

∫ S1 (Q) =

dωS1 (Q, ω).

(2.20)

If we use the DHO expression Eq. (2.17) for the spectral function, then in the limit of low temperature, kT Tλ , ωQ is the energy parameter in the DHO function fitted to the total S(Q, ω) and does not represent a mode energy. Adapted from (Andersen et al., 1994b).

4. Mode energies and lifetimes, higher temperatures

ce

pte

As discussed above, the P-R mode energies, ωQ , and inverse lifetimes, ΓQ , extracted from data at higher temperatures in the superfluid phase (1.75 < ∼ T ≤ Tλ ) depend on the model of S(Q, ω) used to interpret the data. For example, Figs. 2.14 and 2.15 show the ωQ and ΓQ obtained (Andersen and Stirling, 1994) by fitting the Woods-Svensson (WS) and Simple Subtraction (SS) models to the data of (Andersen et al., 1994a). Up to T ≃ 1.75 K the ωQ and ΓQ obtained using the two models are in reasonable agreement. However, for T > ∼ 1.75 K the ωQ and ΓQ clearly differ, by a factor of two or three for ΓQ . Thus unique and model independent values of ΓQ are still not available in bulk liquid 4 He at T > ∼ 1.75 K. The theory of the temperature dependence of ΓQ , (e.g. the Landau-Khalatnikov theory (1965; 1949)), is actually tested ((Andersen et al., 1994b; F˚ ak et al., 2012; Gibbs et al., 1999) over a limited temperature range only (0.5 < T < 1.75 K). ∼ ∼

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

cri pt

15

FIG. 2.15 The P-R mode width, 2ΓQ , versus temperature at T ≤ Tλ obtained fitting the Woods-Svensson (WS) and simple subtraction (SS) models. At T > Tλ , 2ΓQ is the width parameter of the DHO function fitted to the total S(Q, ω) and does not represent a mode width. Adapted from (Andersen et al., 1994b).

D. Normal Liquid 4 He

In this subsection, we discuss the dynamical structure factor, S(Q, ω), of normal liquid 4 He at temperatures T ≥ Tλ . The bottom frame of Fig. 2.9 shows the scattering intensity, proportional to S(Q, ω), at wave vector Q = 1.1 ˚ A−1 in the normal liquid at pressure p = 20 bar (where Tλ = 1.93 K). S(Q, ω) is clearly very broad in ω and largely independent of temperature, quite unlike S(Q, ω) in the Bose condensed phase. Fig. 2.10, (left hand side) shows the spectral function A(ϕ, ω) at constant scattering angle ϕ that corresponds to Q ≃ 1.95 ˚ A−1 . Shown is A(ϕ, ω) at two temperature above Tλ = 2.17 K. A(ϕ, ω) is clearly a broad function of ω at T ≥ Tλ . A(ϕ, ω) is also an odd function of ω and must go to zero at ω = 0. Fig. 2.3 shows S(Q, ω) in the superfluid and normal phase at p = 20 bar for Q values between Q = 2.4 ˚ A−1 and Q = 3.6 ˚ A−1 . Again, S(Q, ω) is broad in ω at T ≥ Tλ . Particularly at Q ≥ 3.0 ˚ A−1 S(Q, ω) extends up to very high energy ω. These S(Q, ω) are typical of S(Q, ω) in normal liquid 4 He at wave vectors, 0.8 < Q < 3.6 ˚ A−1 . The broad S(Q, ω) of normal liquid 4 He is shown in


Page 16 of 48 16

2 centre and 1/2 heights at 2.5 K

10

2∆

4

SVP Stirling/Glyde

Normal Liquid He

SVP Gibbs et al 10 bar Gibbs et al 20 bar Gibbs et al

1 Normal T=2.5 K

31.2 bar MCM-41 48.6 bar MCM-41

5

0

1

2 Q (Å−1)

3

4 0 -1

ce

pte

dM

Fig. 2.16. For wave vectors Q up to Q ≃ 2.4 ˚ A−1 , the energy of the broad maximum in S(Q, ω) at T ≥ Tλ is similar to the P-R mode energy at T ≤ Tλ . However, the FWHM of S(Q, ω) in the normal liquid is 2−3 meV (1 Thz = 4.135 meV) or 2000-3000 times the width of the PR mode at T = 1.0 K in the Bose condensed phase (F˚ ak et al., 2012; Mezei, 1980; Mezei and Stirling, 1983). At Q > 2.5 ˚ A−1 , the energy of the maximum in S(Q, ω) lies ∼ well above the P-R mode energy. Since the S(Q, ω) of normal liquid at Q ≥ 1 ˚ A−1 is so broad, S(Q, ω) does not represent a single mode. Rather it is similar to the broad response observed in many classical fluids, as discussed below. Nonetheless, the usual way to represent S(Q, ω) in the normal phase, as in liquid metals, is to fit a DHO (Eq. 2.17) to the total S(Q, ω) with ωQ , ΓQ , and ZQ as free parameters. The DHO provides a good fit but the parameters are simply fit parameters and do not represent a mode. For example, the best fit ZQ is very large in the normal phase (ZQ > ∼ 100). The ZQ ceases to represent the intensity of S(Q, ω) or of a mode in S(Q, ω) since the integrated intensity in S(Q, ω) cannot exceed S(Q) and S(Q) ≤ 1. A broad P-R like energy dispersion curve such as shown in Fig. 2.16 is observed in many classical and other quantum liquids. The general P-R shape, and particularly a roton-like minimum, arises from correlations induced by the interatomic potential. For example, a broad zero sound mode is observed in normal liquid 3 He. Pines and collaborators (1981) have emphasized that the atomic correlations induced by interatomic potential are similar in liquid 3 He and 4 He. Thus the dy-

1

2

(meV)

(meV)

FIG. 2.17 S(Q, ω) at the roton wave vector (Q = 1.9 - 2.0 ˚ A−1 ) in normal liquid 4 He as function of pressure. Shown is the DHO function fitted to the observed S(Q, ω) at SVP (Gibbs et al., 1999; Stirling and Glyde, 1990), at 10 bar (Gibbs et al., 1999), 20 bar (Gibbs et al., 1999; Talbot et al., 1988), and to liquid 4 He in MCM-41 at 31.2 bar (Bossy et al., 2008b) and 48.6 bar (Bossy et al., 2012b). At high pressure S(Q, ω) peaks near ω = 0 as in nearly classical liquids such as neon (Buyers et al., 1975) and argon (Sk¨ old et al., 1972).

an

FIG. 2.16 The dispersion curve of the maximum and the halfheights (limit of hatched area) of S(Q, ω) in the normal liquid phase at T = 2.5 K and SVP. The black points with half widths are from Stirling and Glyde (1990) and Andersen and Stirling (1994) and the curve at Q > 2.3 ˚ A−1 is from Glyde et al. (1998). The open red squares are an interpolation developed in this review from the above data. The P-R energy dispersion curve at low T (blue and red triangles) is also shown.

0

us

0

cri pt

) (Arb. Units)

20 bar Talbot et al

S(Q,

ω (meV)

Low T

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

namic response in the normal phases of the two liquids is quite similar. The phonon dispersion curve of solid 4 He (or any bcc solid) along the [111] direction has the P-R form with a roton like minimum. This direction is special because the first reciprocal lattice vector in this direction is at a very large Q value so that the phonon dispersion curve can go through a roton minimum without the energy ωQ being forced to go to zero by a reciprocal lattice point. In other symmetry directions, ωQ is forced to go to zero at wave vectors in the roton region. Indeed, Nozières (2004) has suggested that the roton minimum in the liquid, induced by interatomic correlations, is the “ghost” of a Bragg point in the corresponding solid. The roton minimum is at the peak of S(Q). A P-R like dispersion curve is observed in liquid metals (Scopigno et al., 2005), particularly liquid potassium (Cabrillo et al., 2002). The mode in liquid metals is very broad for Q ≥ 1 ˚ A−1 as in Fig. 2.16 for normal liquid. As discussed by Kalman et al. (2010), the shape of the dispersion curve is common to many materials and arises from interatomic correlations common to these materials. In liquid metals and liquid Ne and Ar, S(Q, ω) also has a large peak at ω = 0 arising from diffusive motions in the liquids. To emphasize the similarity of normal liquid 4 He to liquid Ne and Ar, we show S(Q, ω) in normal liquid 4 He as a function of pressure in Fig. 2.17. Shown is the best fit DHO function to S(Q, ω) at the “roton” Q between SVP and a pressure of 46.8 bar. The latter pressure is achieved (Bossy et al., 2012a, 2008a) by confining the liquid in MCM-41. The confinement stabi-

Page 17 of 48


A. Landau Theory

In his landmark paper, Landau (1941) proposed that liquid 4 He supported collective excitations of two types. The first was a collective sound or density mode of energy ωQ = cQ, where c is the sound velocity and Q is the wave vector. The second was a collective rotation of the whole fluid of energy ωQ = ∆+~Q2 /2m where ∆ is an energy gap and Q begins at Q = 0. The first was denoted a “phonon”, the second a “roton”. With this, the term “roton” was introduced. Because liquid 4 He is a strongly interacting liquid, Landau argued that the liquid could not support independent single particle excitations. The argument is similar to that used today (Giamarchi, T., 2004; Haldane, 1981; Imambekov et al., 2012) to argue that 1D liquids support only collective modes at low energy. Based on these two collective modes, the absence of any other low energy modes and the postulate that the curl of the superfluid velocity, vs , is zero (∇ × vs = 0), Landau (1941) developed an enormously successful theory of superfluidity that is still used today (see section 4.H). To fit new data better, Landau (1947) modified his dispersion curve in 1947. He moved the “roton” mode to begin at a finite Q value, Q = QR , so that ωQ = ∆+~(Q − QR )2 /2m. He joined the phonon and roton mode excitations into a single dispersion curve that is shown in Fig. 1.1. He stated,

ce

pte

dM

an

In this section we review the theory of the collective response of liquid 4 He. We begin with the remarkably accurate P-R mode energy dispersion curve proposed by Landau (1941; 1947). We move to the microscopic formulation developed by Feynman (1954) by Feynman and Cohen (1956) and the Correlated Basis Function (CBF) theory (Campbell et al., 2015; Feenberg, 1969; Jackson and Feenberg, 1961, 1962; Krotscheck, 1985; Lee and Lee, 1975; Manousakis and Pandharipande, 1986). The CBF theories may be viewed as systematic extensions of the Feynman and Feynman and Cohen theories. We discuss recent variational Shadow Wave Function and diffusion Monte Carlo calculations of the P-R mode energy and the dynamical structure factor, S(Q, ω). We sketch sum rules for S(Q, ω) and the information they provide on the nature of the P-R mode. Finally, we discuss the field theory (FT) formulation of the dynamical susceptibility, χ(Q, ω), and S(Q, ω) which shows that there is a single particle component within S(Q, ω) and the P-R mode when there is Bose-Einstein condensation (BEC). Models of the temperature dependence of S(Q, ω) based on the FT formulation in which the intensity in the P-R mode is related to the condensate fraction, n0 , are reviewed. At wave vectors Q ≤ 2.5 ˚ A−1 the physical picture of the P-R mode is a density mode modified by interaction with other density modes. At Q → 0 (e.g. Q < ∼ 0.5 ˚ A−1 ), S(Q, ω) is confined to a single peak and this peak is well represented by a single, independent density mode. There is no multi-P-R mode component, SM (Q, ω), in S(Q, ω). At higher Q, interaction between density modes and creation of more than one density mode in the scattering event leads to a mult-P-R mode component in S(Q, ω) (see Eq. (2.1)). It also leads to a density mode modified by interaction. At Q ≥ 2.5 ˚ A−1 the presence of the quasiparticle component in S(Q, ω) proportional to the condensate fraction, n0 , which is in S(Q, ω) at all Q, begins to be observable. At much higher Q (Q ≥ 20 ˚ A) the condensate fraction can be determined from the magnitude of this quasiparticle component, as discussed in section 4. The CBF and Monte Carlo methods used to calculate the P-R mode energy at T = 0 K could, in principle, be equally well used to describe modes in a dense Bose liquid that is normal at T = 0 K. However, these same formula-

cri pt

3. INTERPRETATION: COLLECTIVE EXCITATIONS

tions predict BEC and superfluidity in liquid 4 He at low temperature. For example, both BEC and superfluidity is predicted using path integral Monte Carlo (PIMC) methods (Boninsegni et al., 2006a; Ceperley, 1995; Pollock and Ceperley, 1987). In the PIMC model, superfluidity and BEC arise from atomic exchange over long distances. In this sense superfluidity and BEC are independent properties that are a consequence of exchange over long distance. (Ceperley, 1995). BEC and superfluidity co-exist in the bulk liquid. They can be separated in porous media (Bossy et al., 2012a; Diallo et al., 2014; Glyde et al., 2000a; Plantevin et al., 2001; Taniguchi et al., 2011; Yamamoto et al., 2004, 2008)

us

lizes the liquid phase against solidification. As pressure is increased, the peak position of S(Q, ω) moves toward zero, from approximately 0.6 meV at SVP to near zero at 46.8 bar. Liquid 4 He becomes more classical at higher pressure. Under pressure, the S(Q, ω) of normal liquid 4 He peaks at ω ≃ 0 at Q ≃ 2.0 ˚ A−1 as observed (Buyers et al., 1975; Sköld et al., 1972) in liquid Ne and Ar. This response suggests that at Q ≃ 2.0 ˚ A−1 only diffusive 4 motions remain in liquid He under pressure as observed in many classical and nearly classical liquids.

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

“... With such a spectrum it is of course impossible to speak strictly of rotons and phonons as qualitatively different types of elementary excitations. It would be more correct to speak simply of the long wave (small Q) and short wave (Q in the neighborhood of QR ) excitations.” Landau (1947) clearly envisaged the collective excitation to be the same at all Q values. The term “roton” therefore no longer had a meaning of a rotation, but the term has survived. In this review, following Landau, we take “roton” to mean simply that part of the dispersion curve at wave vectors in the region Q ≈ QR (QR = 1.95 ˚ A−1 at SVP). We interpret the excitation as a density mode


Page 18 of 48 18

F

2

C FC

1

B. Bijl Feynman Theory

BF ωQ = (~Q2 /2m)/S(Q),

(3.1)

j

dM

(R = r1 , r2 , ..., rN ) as the excited state in which a single density mode of wave vector Q has been created in the liquid. Feynman argued, in analogy with small density oscillations, that the static structure factor, S(Q), at low Q should have the form lim S(Q) = ~Q/2mc.

Q→0

(3.3)

pte

Thus as Q → 0, the energy, ωQ given by (3.1) reduces to ωQ = cQ as expected for long wave density fluctuations. At large Q, S(Q) → 1 and (3.1) reduces to ωQ = ~Q2 /2m. In this way Feynman derived the energy dispersion relation of a “pure” density mode that had a sound mode energy at low Q, had a roton minimum at the Q value where S(Q) has a maximum and had a free particle energy at large Q. In the Landau and Feynman picture, the excitations of liquid 4 He are a gas of independent density modes. S(Q, ω) reduces to a delta function at all Q if the excited states are exactly density modes, SBF (Q, ω) = S(Q)δ(ω − ωQ ).

ce

Expt.

0

1

2

3

−1

Q (Å )

FIG. 3.1 Calculated Phonon-roton mode energy dispersion curves, ωQ . Shown is the Bijl-Feynman (F) energy (Feynman, 1954), the Feynman and Cohen (FC) energy (Feynman and Cohen, 1956), the Correlated Basis Function energy of Campbell, Krotschek and Lichtenegger (2015). Open red circles are observed values (Andersen et al., 1994a; Glyde et al., 1998).

an

with eigenfunction f (r) = eiQ·r ∑. The density operator in first quantization is ρ(r) = j δ(r − rj ). The Fourier ∑ transform ρ(Q) of ρ(r) is ρ(Q) = j e−iQ·rj . Thus we may interpret the Feynman wave function, ∑ ψF = eiQ·rj ϕ(R) = ρ† (Q)ϕ(R), (3.2)

0

us

Bijl (1940) and Feynman (1954) presented powerful arguments that the excited state wave function, ψ, of a dense, Bose liquid ∑ containing a single excitation should take the form ψ = j f (rj )ϕ. In ψ, ϕ is the ground state wave function. ψ is essentially the sum of single particle wave functions f (r) for each atom involved in the excitation. Feynman (1954) derived a variational equation for f (r). This equation had a minimum energy eigenvalue,

cri pt

ωQ (meV)

in a dense Bose liquid where the condensate fraction is small and the quasiparticle component in the mode is small. The excitation becomes a quasiparticle mode in a dilute Bose gas where the condensate fraction is large. In a Bose gas, the mode at low Q is denoted quasiparticle sound (Nozières and Pines, 1990).

(3.4)

Reversing the argument, with the assumption of the form (3.4), the Bijl-Feynman energy (3.1) can be derived as the ratio of the first moment (f -sum rule) to zeroth moment (S(Q)) of S(Q, ω)(see section 3.G). The zeroth moment is the definition of S(Q) in terms of S(Q, ω).

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Using field theory techniques (see section 3.J), Gavoret and Nozieres (1964) have shown that in the long wave limit Q → 0, S(Q, ω) in a Bose liquid containing BEC takes the delta function form (3.4) exactly. The result is independent of the level of atomic interaction or other physical assumptions. They also show that at Q → 0 the excitation energy is ωQ = cQ. With these results and the two sum rules noted above, we may show that at Q → 0, S(Q) is given exactly by (3.3). Thus at Q → 0 the Feynman theory is exact and the total excitation is exactly a density mode described by (3.2). Also using field theory methods, we may show that within the density mode there is a quasiparticle contribution of weight proportional to n0 . The quasiparticle excitations, defined as the poles of the single particle Green function, have the same energy, ωQ , as the density mode when there is BEC. As discussed in the experimental section 2, the observed S(Q, ω) reduces to a delta function expression (3.4) at Q → 0. The multi-P-R component, SM (Q, ω), is unobservable. Thus as Q → 0 we may interpret the mode of liquid 4 He as an independent density mode. The presence of a multi-P-R component SM (Q, ω) at higher Q is the signal that the P-R mode is not exactly an independent density mode at higher Q. Similarly, the energy (3.1), often denoted the Bijl-Feynman energy, while exact at Q → 0, lies well above the observed ωQ at Q values in the roton region (see Fig. 3.1). This again shows the mode is not exactly an independent density mode at higher Q.

Page 19 of 48


j

l= /j

where g(r) is the correlation function between pairs of atoms and ϕ(R) is again the ground state wave function. Feynman and Cohen (FC) proposed a g(r) = A(Q · r)/r3 that has a dipole-like form. To proceed, FC expanded the exponential containing g(r) and retained terms up to first order in g(r). The excited state was, ψB (R) = ρ†B (Q)ϕ(R),

(3.6)

where, ρ†B (Q) =

∑

eiQ·rj [1 + i

j

∑

3 ]. A(Q · rjl )/rjl

(3.7)

l= /j

cri pt

To improve the P-R energy, ωQ , Feynman and Cohen (1956) proposed an improved excited state wave function. This contained both single particle and pair correlations, ∑ ∑ ψFC = ( exp[iQ · rj + i g(rjl )])ϕ(R), (3.5)

single density mode with two other density modes, the modified single density energy, ωQ , up to second order lies close to but slightly above the FC energy in Fig. 3.1. If the perturbation theory is extended to fourth order (using the BF states (3.8)), an energy that is below the FC energy and very close to the observed ωQ in the roton region has been obtained by Lee and Lee (1975). Early CBF theory has been extensively reviewed by Woo (1976), Chester (1975) and Campbell (1978). Jackson (1973; 1974a) has evaluated S(Q, ω) using CBF states, an evaluation that illustrates clearly the observed structure of S(Q, ω) discussed in the experimental section 2.A.1. At lowest order, the state containing a single mode is the BF state given by Eq. (3.8). The corresponding single density mode Green function is G0 (q, ω) = [ω − ϵ0 (q) + iη]−1 where again ϵ0 (q) is the BF energy Eq. (3.1) and iη is a small imaginary part added to the energy. The mode interacts with pairs of other modes. The G(q, ω) including this interaction is, [ ]−1 G(q, ω) = ω + iη − ϵ0 (q) − Σ0 (q, ω) ,

us

C. Feynman and Cohen Theory

(3.9)

where,

Σ0 (q, ω) = −

1∑ |⟨lm|δH|q⟩|2 G02 (lm, ω), 2

dM

D. Correlated Basis Function Theory

pte

Feenberg and collaborators (Feenberg, 1969; Jackson and Feenberg, 1961, 1962) have developed a correlated basis function (CBF) theory of the ground state and excited states of liquid 4 He. The formulation emphasizes taking account of correlations between atoms, such as single particle and pair correlations in Eq. (3.5), in the wave functions and states of the liquid. These correlations are induced principally by the hard core of the inter-atomic potential. The zero order excited states used as a basis are generally the Bijl-Feynman states given by Eq. (3.2). These are the independent density mode states given by,

ce

|q⟩ = ρ† (q)|0⟩.,

(3.8)

BF with energy ϵ0 (q) = ωQ , the Bijl-Feynman (BF) energy, Eq. (3.1). Interaction between the density modes given by Eq. (3.8) is incorporated and a Brillouin-Wigner perturbation theory for the mode energy including this interaction is developed. Incorporating interaction of a

(3.10)

lm

an

ψB (R) is denoted the excited state including “backflow”. The ψB can be shown (Miller et al., 1962) to include creation of both single density modes (as in ψF ) and pairs of density modes (see p. 157 (Glyde, 1994a)). Thus in addition to improving the mode energy, a full implementation of ψFC in S(Q, ω) will lead to a multi-P-R components SM (Q, ω) in S(Q, ω)(Miller et al., 1962). The FeynmanCohen (FC) energy ωQ , obtained using Eq. (3.6) and Eq. (3.7) is shown in Fig. 3.1 and is much lower at the roton minimum than the Bijl-Feynman energy Eq. (3.1). The FC theory represents a significantly improved description at higher Q.

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

is the mode self energy and G02 (lm, ω) is the two mode Green function of the pair of other modes. Σ0 (q, ω) = ∆(q, ω) − iΓ(q, ω) has a real, ∆(q, ω), and an imaginary, Γ(q, ω), part. The DSF S(Q, ω) ∝ − π1 ImG(q, ω)∆(Q−q) for the interacting single density mode is, 1 Γ(Q, ω) π [ω − ϵ0 (Q) − ∆(Q, ω)]2 + [Γ(Q, ω)]2 = ZQ δ(ω − ωQ ) ω ≤ ωQ (3.11)

S1 (Q, ω) = ZQ

where ωQ = ϵ0 (Q) + ∆(Q, ωQ ). In the second equation of (3.11) we have used the property that 1∑ |⟨lm|δH|Q⟩|2 ImG02 (Q, ω) 2 lm 1∑ = |⟨lm|δH|q⟩|2 δ(ω − ϵ0 (l) − ϵ0 (m)) 2

Γ(Q, ω) =

lm

= 0.

ω ≤ ϵ0 (Q)

(3.12)

At T = 0 K, Γ(Q, ω) is zero at energies ω outside the two excitation band, ϵ0 (l) + ϵ0 (m). The two excitation band DOS is small for energies below that of two rotons, 2∆. It is zero for energies less than or equal to ωQ , (Γ(Q, ωQ ) = 0) and S(Q, ω) is a delta function at ω = ωQ which defines the P-R mode energy. At low Q in the phonon region where there is some upward dispersion in ωQ , Γ(Q, ωQ ) is not exactly zero (Mezei and Stirling, 1983). As emphasized recently (Beauvois et al., 2016), the two mode DOS is small but not zero at energies ωQ < ω < 2∆. At ω < 2∆, one or both of the two modes must be phonons.


Page 20 of 48 20

|q⟩B = ρ (q)B |0⟩,

(3.13)

ρ†B

0.5

S(k,hω)

cri pt

hω (meV)

1.0

ρ = 0.022 (Å 3)

0.0 0.0 2.0 1.5 1.0 0.5

1.0

2.0

3.0 k (Å 1)

0.0 0.0

1.0

2.0

4.0

5.0

S(k,hω)

ρ = 0.026 (Å 3)

3.0 k (Å 1)

4.0

5.0

ce

pte

dM

an

where is given by (3.7). The states |q⟩B are used instead of the BF states (3.8). The zero order energy is the FC energy. At second order of this CBF theory, the P-R mode energy agrees well with observed value, comparable to that obtained using the BF states taken to fourth order (Lee and Lee, 1975). Manousakis and Pandharipande (1986) have also evaluated the dynamic susceptibility using the FC states as basis states. In this formulation, the dynamic susceptibility χ(Q, ω) and S(Q, ω) have the same structure as χ(Q, ω) obtained in the Field Theory Formulation (see below) and as obtained for anharmonic solids. (see section 11.3 of Glyde, 1994a). Campbell et al. (2015) have developed an extended equation of motion for the density dynamical susceptibility of liquid 4 He at T = 0 K within CBF theory. Earlier formulations have appeared (Campbell and Krotscheck, 2009; Clements et al., 1996; Krotscheck, 1985; Saarela, 1986). The inter-atomic correlations that are included in the ground state have been discussed (Campbell, 1973; Campbell and Pinski, 1979; Fabrocini et al., 2002). In the excited state, single atom, pair and triplet interatomic correlations are included (Campbell et al., 2015). Since the dynamic susceptibility χ(Q, ω) is calculated, S(Q, ω) as well as the P-R mode energy can be evaluated. The theory for the P-R energies is also reformulated in terms of a Brillouin-Wigner perturbation theory. The aim is to make contact with earlier formulations and to clarify approximations. Fig. 3.1 shows the P-R dispersion curve calculated by Campbell et al (2015) in liquid 4 He at SVP (p = 0)(density ρ = 0.0219 ˚ A−3 ). Fig. 3.2 shows the calculated P-R dispersion curve and the intensity in S(Q, ω) at energies above the P-R mode at SVP and a pressure p ≃ 24 bars (ρ = 0.026 ˚ A−3 ). The calculated P-R energy agrees extremely well with experiment at all k and

1.5

us

†

2.0

hω (meV)

In Eq. (3.11) S1B (Q, ω) is the broad component of the single density mode response that lies chiefly at energies ω > 2∆ where Γ(Q, ω) ̸= 0. S1B (Q, ω) was identified in the observed S(Q, ω) in Eq. (2.3) and in Fig. 2.11. At lower Q (Q . 2.5 ˚ A−1 ), S1B (Q, ω) is small compared to ZQ δ(ω − ωQ ). At Q & 2.5 ˚ A−1 , S1B (Q, ω) is large compared to ZQ δ(ω − ωQ ) and most of the single mode response lies in S1B (Q, ω) at energies ω > 2∆ (see Figs. 2.3 and 2.11). In addition to S1B (Q, ω) there can be direct excitation of two density modes by the neutron, denoted S2 (Q, ω). S2 (Q, ω) is also proportional to the two density mode density of states in (3.12) and contributes predominantly at ω > 2∆. Thus in SM (Q, ω) = S1B (Q, ω) + S2 (Q, ω) + · · · both S1B (Q, ω) and S2 (Q, ω) are proportional to ImG2 (Q, ω). The two cannot be readily separated in data. Manousakis and Pandharipande (1984) have developed a CBF theory similar to that developed by Feenberg and collaborators but one which uses the FC states (which include backflow) as the basis states,

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

FIG. 3.2 Phonon-roton mode energy, ωk , (solid red line) and intensity in the multi-P-R part, SM (Q, ω), of S(Q, ω) at energies ω > ωk calculated using Correlated Basis Function methods (Campbell et al., 2015). The blue open circles are the observed ωk (Donnelly et al., 1981). Upper frame is saturated vapour density (ρ = 0.022 ˚ A−3 ) and lower frame the −3 ˚ solidification density (ρ = 0.026 A ).

pressures. Particularly, at k ≥ 2.5 ˚ A−1 the calculated P-R energy plateaus at a value of approximately 2∆ as observed, a feature that was difficult to reproduce in the past. The P-R energy ωQ cannot exceed 2∆. At SVP, the observed roton energy is ∆= 0.742 meV, 2∆ = 1.484 meV. At SVP the calculated ωQ at the roton Q, ωQ = ∆, lies somewhat above the observed value. Correspondingly the calculated P-R energy at high Q, Q & 2.5 ˚ A−1 , where ωQ comes up to 2∆ lies somewhat above the observed value. However, the feature that the maximum value of ωQ is 2∆ is well reproduced. The calculations show clearly that the roton energy ∆ decreases with increasing pressure, as observed. Also, the maxon energy increases with increasing pressure as discussed in section 2.B. At p = 20 bars, the maxon energy ωQ approaches 2∆ and becomes limited by 2∆. At p > 20 bars ωQ at the maxon is limited by 2∆ and ωQ actually decreases with pressure at p > 20 bar. This pressure dependence and the role of ∆ in determining the shape of the dispersion curve is nicely illustrated in the calculations. At the maxon Q (Q = 1.1. ˚ A−1 ), the observed multiP-R intensity (e.g. S1B (Q, ω)) in SM (Q, ω) at ω > ωQ increases with pressure. The weight ZQ in the P-R mode

Page 21 of 48


E. Shadow Wave Functions

dM

Moroni et al.(1998a) have evaluated the energy, ωQ , and weight, ZQ , of the P-R mode using variational Monte Carlo methods. Impressive agreement with experiment is obtained using a shadow wave function (SWF) as the variational function. To begin, ground state (GS) properties were first calculated using the GS SWF, ∫ Ψ(R) = Φp (R) dS Θ(R, S) Φs (S), (3.14)

ce

pte

where R = r1 ...rN are the atomic variables and S = s1 ...sN are shadow (atomic) variables. The Φp (R) and Φs (S) are Jastrow ∑ wave functions between pairs of atoms. Θ(R, S) = e− i u(ri −si ) is a coupling function between the atom and the shadow atom variables where u(r − s) is a Gaussian. The Jastrow and coupling functions have variational parameters. The excited state containing a P-R mode of wave vector k is represented by the SWF, ∫ Ψk (R) = Φp (R) dS Θ(R, S) Φs (S)δk , (3.15) where

δk =

∑ j

∑

eik·[sj +

us

cri pt

This is a Feynman-Cohen like P-R mode state, Eq. (3.5) in the shadow variables. The pair correlations in the FC like state in Eq. (3.16) are approximately Gaussian and λ(s) is a short range function in the shadow variables. The P-R energy is defined as the difference in energy between the excited state containing a P-R mode and the GS. The calculated P-R mode energy agreed well with experiment, especially at the roton wave vector Q ≃ −1 2˚ A . Excellent agreement for the weight, ZQ , in PR mode, S1 (Q, ω) =ZQ δ(ω− ωQ ) is also obtained. The SWF ωQ lies above the observed P-R energy at (1) the maxon wave vector (Q ≃ 1.1 ˚ A−1 ) (especially at higher pressure) and (2) wave vectors beyond −1 the roton (Q & 2.5 ˚ A ). These discrepancies may arise from the following issue. In the variational SWF method, S(Q, ω) is effectively assumed to take the form S(Q, ω) = ZQ δ(ω− ωQ ) and all the intensity is ascribed to the P-R mode. As seen in section 2, S(Q, ω) = ZQ δ(ω− ωQ ) + SM (Q, ω) where SM (Q, ω) is the multiP-R component (which includes the broad component of S1 (Q, ω)) at energies ω > 2∆. At the maxon Q and −1 Q & 2.5 ˚ A , SM (Q, ω) is large. The SWF excited state includes SM (Q, ω) since multi-P-R excitations are included in a FC-like state, Eq. (3.16). The broad response contributes at higher energy, chiefly at ω > 2∆. By expressing all of this correct response as a single mode at ωQ , may result in ωQ values that are too high at Q values where the broad response is large. At the roton Q where the multi-P-R response is much smaller, good agreement of the SWF ωQ with experiment is obtained.

an

˚−1 decreases with pressure. The observed at Q = 1.1 A increase of SM (Q, ω) with pressure at the maxon Q is well reproduced in Fig. 3.2. The multi-P-R intensity is much larger at higher pressure (ρ = 0.025 ˚ A−3 ) (where ωQ approaches 2∆) than at SVP (ρ = 0.022 ˚ A−3 ), as is observed. In contrast, the intensity in SM (Q, ω) observed above the roton is smaller at higher pressure. This change with pressure is also well reproduced in Fig. 3.2. In the maxon region (0.6 < Q < 1.4 ˚ A−1 ) peaks in the intensity of SM (Q, ω) are shown in Fig. 3.2, as observed (Andersen et al., 1994b; Beauvois et al., 2016). The calculations predict some new intensity in S(Q, ω) at energies above the P-R energy at Q ≃ 0.8 -1.0 ˚ A−1 (at SVP but not at 20 bars) as is observed (Beauvois et al., 2016). The new intensity looks like an extension of the sound mode ωQ = cQ to higher Q values, denoted a ”ghost phonon”. The intensity is attributed to a finite Γ(Q, ω) in Eq. (3.12) at energies immediately above the P-R mode arising from contributions to the two mode DOS from two nearly parallel phonons. The calculations also predict some intensity at higher Q in S(Q, ω) at energies immediately above the P-R mode, as observed. Essentially, the calculated intensity at higher energies ω above the P-R mode is small but not zero for ω ≤ 2∆ and “switches on dramatically at ω = 2∆ at T = 0 K when two modes in the roton region can contribute to the two mode DOS.

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

l= /j (sl −sj )λ(|sl −sj |)

].

(3.16)

F. Diffusion Monte Carlo

Boronat and Casulleras (1997) present an interesting Diffusion Monte Carlo (DMC) evaluation of the PR mode energy. The P-R mode energy is defined as the difference in energy of the liquid when it contains a single mode and the ground state (GS). DMC can provide an exact value of the GS energy. Beginning with the GS wave function, Ψ0 (R), Boronat and Casulleras calculate the liquid energy when it contains a single PR mode. They use both the Bijl-Feynman (3.2) and the Feynman-Cohen (FC) (3.5) excited state wave function. They write the FC state, Eq. (3.5), as, C ΨF q (R) =

N ∑

eiq·¯rj Ψ0 (R),

(3.17)

j=1

∑ where ¯rj = rj + l=/j g(rjl )rjl . Backflow pair correlations C are included in the second term of r¯i . ΨF q (R) is used as a guiding function in the DMC calculation of the excited state energy. The oscillatory exponential in Eq. (3.17) introduces a sign problem in evaluating the excited state energy, as ubiquitous in Fermi systems. The sign problem is addressed using both the fixed node (FN) approximation


Page 22 of 48 22

−∞

pte

dM

are the most useful. At high wave vector where ∫ single atom response dominates, central moments dω (ω − ωR )n S(Q, ω) where ωR ≡ ϵQ ≡ ~Q2 /2m are often more useful. Three of the moments, Eq. (3.18), are, ∫ ∞ m0 = dω S(Q, ω) = S(Q) (3.19) −∞ ∫ ∞ m1 = dω ωS(Q, ω) = ~Q2 /2m ≡ ϵQ (3.20) −∞ ∫ ∞ n~ ~ lim m−1 = lim dω ω −1 S(Q, ω) = = Q→0 Q→0 −∞ 2χT 2mc2 (3.21)

The zeroth moment, m0 , is really the definition of the static structure factor S(Q) = S(Q, t = 0) = 1 N ⟨ρ(Q)ρ(−Q)⟩, Eq. (1.5). S(Q) is observed in X-ray scattering measurements, for example in which all inelastic scattering (ω ̸= 0) and elastic scattering (ω = 0) is collected together. The first moment, m1 , or f -sum rule, is a number of atoms or states sum rule and is independent of density, magnitude of interaction or phase of the system. The inverse first moment, m−1 , in the limit Q → 0, is denoted the compressibility sum rule. χT = [−V (∂p/∂V )T ]−1 is the isothermal compressibility where p and V are the pressure and the liquid volume, respectively. In a liquid, χT = 1/nmc2 where n = N/V is the number density, m the mass and c the sound velocity. Closed expressions for the second moment, m2 ,

ce

BF ωQ =

ϵQ ~Q2 m1 = = , m0 S(Q) 2mS(Q)

(3.22)

without any assumption on the frequency dependence of BF S(Q, ω). To equate ωQ given by Eq. (3.22) to the observed mode energy, ωQ , in S(Q, ω) we have to assume that S(Q, ω) = S(Q)δ(ω − ωQ ) (to introduce ωQ ). The moments m0 and m1 using this S(Q, ω) and Eq. (3.22) BF then leads to m1 /m0 = ωQ and ωQ = ωQ . We may obtain an improved value of ωQ , if we assume (1) that there is a multi-P-R mode component, SM (Q, ω), in S(Q, ω), as needed to describe experiment in Eq. (2.1), S(Q, ω) = ZQ δ(ω− ωQ ) + SM (Q, ω), and (2) that SM (Q, ω) contributes at energies ω ≥ ωQ . Then we can show from the moments and Eq. (3.22) BF BF that ωQ is an upper bound to ωQ , ωQ ≥ ωQ . At Q → 0, S(Q)= ~Q/2mc at T = 0. In this limit, Eq. (3.22) BF gives ωQ = cQ independently of the nature of S(Q, ω). Since ω = cQ is an extension of the sound mode and BF ωQ ≥ ωQ , this strongly suggests that ωQ = cQ and S(Q, ω) = S(Q)δ(ω − ωQ ) at Q → 0. While not proofs, the sum rule arguments provide limits and persuasive arguments on ωQ and the structure for S(Q, ω). Boronat et al. (1995) have presented several other important examples and rigorous bounds on ωQ and χ(Q) based on sum rules. As an introduction to the field theory formulation of S(Q, ω) that follows in the next section, we present the contribution to the f -sum rule arising from (1) the condensate state (k = 0) and (2) the momentum states (k ̸= 0). As shown in the following section 3.H, S(Q, ω) can be separated into a “singular” (S) part and a “regular” part (R),

an

Frequency moments or sum rules of S(Q, ω) provide important bounds on S(Q, ω), on the P-R mode and the static susceptibility, χ(Q). They are particularly useful in developing and checking models of S(Q, ω). At low wave vector where the liquid responds collectively, moments of the coherent DSF S(Q, ω), ∫ ∞ mn = dω ω n S(Q, ω), (3.18)

cri pt

G. Moments of S(Q, ω)

(Feenberg, 1969) and third moment, m3 , (Puff, 1965) have been obtained and evaluated (Boronat et al., 1995). The third moment is denoted the Puff (1965) sum rule. Stringari (1992) has discussed these moments for liquid 4 He extensively. Jackson (1974b) has developed a formulation of S(Q, ω) that explicitly fulfils sum rules. As noted by Boronat et al. (1995) and others, the BijlFeynman energy, ωBF , is rigorously given by the ratio of the first to the zeroth moment,

us

(Reynolds et al., 1982) and released node (RN) method (Ceperley and Alder, 1984). The RN method can provide an exact excited state energy but with increased uncertainty. Using the backflow state Eq. (3.17) and the RN method, Boronat and Casulleras obtain a ωQ in −1 good agreement with experiment up to Q ≃ 2.3 ˚ A . At −1 Q & 2.5 ˚ A where the multi-P-R components in the excited state and S(Q, ω) are large, the calculated energy lies above the observed P-R mode energy. This may be expected, as discussed in the section above, since the FC wave function contains multi-P-R mode components which contribute at high energy and which dominate the −1 excited state energy at higher Q & 2.5˚ A . The agreement with experiment is excellent at lower Q, Q . 2.3 ˚ A−1 .

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

S(Q, ω) = SS (Q, ω) + SR (Q, ω).

(3.23)

The singular part, SS (Q, ω), arises from k = 0 state and is proportional to the single quasiparticle Green function and the condensate fraction, n0 . The regular part, SR (Q, ω), arises from the k ̸= 0 states above the condensate. With this separation of S(Q, ω), the f -sum rule defined in Eq. (2.4) and in Eq. (3.19) is (Glyde, 1994b, 1995), m1 = fS + fR = ϵQ = ~Q2 /2m,

(3.24)

fS = n0 ϵQ − n0 hQ

(3.25)

where

Page 23 of 48


is the contribution from SS (Q, ω) involving the condensate (k = 0) and

We review the field theory formulation in the following section.

(3.26)

hQ = Σ11 (Q, ∞) − Σ12 (Q, ∞) − µ (3.27) ′ ⟨ ( 1 ∑ ˜−(q+Q) B ˜(q+Q) = − v(q) A˜−q A˜q + B 2V q )⟩ −1 + ρ′ (−q)Aq N0 2 (3.28)

The field theory method is a microscopic theory of the dynamics of Bose liquids in which the dynamical susceptibility and dynamical structure factor are expressed in terms of quasiparticle as well as collective excitations. In the general microscopic theory of Bose fluids (Bogoliubov, 1947; Dupuis and Sengupta, 2007; Gavoret and Nozières, 1964; Griffin, 1993; Hohenberg and Martin, 1964; Hugenholtz and Pines, 1959; Nepomnyashchy and Nepomnyashchy, 1975; Nepomnyashchy, 1992; Pistolesi et al., 2004), the Hamiltonian and other properties are written in terms of the annhilation and creation operators of the quasiparticles, (4 He atoms dressed through interaction with other quasiparticles). As applied to the DSF, S(Q, ω), the density operator ρ† (q) = ρ(−q) = ∑ † k ak+q ak is written as a sum of single quasiparticle

creation (a†k+q ) and annihilation (ak ) operators. In this form the density mode creation operator ρ† (q)) is a sum of quasiparticles annihilated from state k (ak ) and created in state k + q (a†k+q ). The density mode is a sum of coherent particle (k + q), hole (k) (PH) states each separated by the same wave vector q. A key advantage of the field theory method is that it reveals the role of the condensate in S(Q, ω). This is critical in understanding the temperature dependence of S(Q, ω) and why S(Q, ω) is so different in the Bose condensed phase and the normal liquid phase (T > Tλ ) where n0 = 0. An important part of the temperature dependence of S(Q, ω) (the intensity in the P-R mode) arises from the temperature dependence of the condensate fraction, n0 (T ) = N0 (T )/N . In contrast, the broadening of the P-R mode is largely explained through interaction of the P-R mode with other P-R modes in the liquid. (Bedell et al., 1984; F˚ ak et al., 2012; Khalatnikov, 1965; Landau and Khalatnikov, 1949) We can also draw on FT methods to obtain exact results for S(Q, ω) and to develop models that can be used to reproduce the observed S(Q, ω). For example, we may show that S(Q, ω) contains a term proportional to n0 (T ) (see below). When there is BEC we have macroscopic occupation of the k = 0 state, (N0 ∼ N ∼ 1020 , e. g. N0 /N ≃ 0.07 in liquid 4 He at SVP). In this event, the operator nature of ak or a†k for k = 0 can be neglected. That is, in the expressions N0 = ⟨a†0 a0 ⟩ and N0 + 1 = ⟨a0 a†0 ⟩, the unit term is negligible compared to N0 . Thus the order of the operators a0 and a†0 is unimportant and their commutation relations can be ignored. The a0 and a†0 can √ be replaced by numbers, a0 = a†0 = N0 , as usual in Bose liquid theory. (Bogoliubov, 1947; Gavoret and Nozières, 1964; Griffin, 1993; Hugenholtz and Pines, 1959) When there is BEC we may replace the operators a0

ce

pte

dM

an

where Σ11 (Q, ω) and Σ12 (Q, ω) are single quasiparticle + ˜ self energies and A˜k = ak + a+ −k , B = ak − a−k and ′ ρ (q) is the density operator involving states above the condensate (k ̸= 0) only (see Eq. (3.29). The ak are the usual quasiparticle annhilation operators familiar in Bose and Fermi liquid theory. The magnitude of hQ is not accurately known in the collective region at wavevectors Q < 2.5 ˚ A−1 but < hQ ∼ |µ| is expected where the chemical potential is µ ≃ −7 K (Glyde, 1994b, 1995). At higher Q, Q ≥ 2.5 ˚ A−1 , we expect ϵQ to be much greater than hQ . For example, at Q = 2.5 ˚ A−1 , using ~2 /2m = 6.06 K ˚ A2 , we have ϵQ ≃ 40 K so that ϵQ ≫ hQ and fS ≃ n0 ϵQ . At much higher Q where the impulse approximation holds, we may show that fS = n0 ϵQ and that fS arises from the condensate component, n0 δ(k), of the momentum distribution n(k). More precisely, at Q → 0 in the phonon region where ωQ = cQ and S(Q, ω) lies entirely in a single peak, we expect hQ to be small compared to ϵQ . The Hugenholtz-Pines theorem (Gavoret and Nozières, 1964; Griffin, 1993; Hugenholtz and Pines, 1959) states that Σ11 (0, 0) − Σ12 (0, 0) − µ = 0. If the frequency dependence of Σ11 (0, ω) and Σ12 (0, ω) is similar, we expect hQ to be near zero. In this case, a fraction n0 of the intensity in the P-R mode at Q → 0 arises from the condensate and a fraction (1 − n0 ) from excitations involving states above the condensate. The condensate and above condensate components both contribute within the P-R peak and, because of the strong interatomic interaction and the strong collective response, the two components are inseparable. As Q increases into the maxon and roton regions, the condensate component in S(Q, ω) is still not visible at T = 0 K. However, it does play a central role in the temperature dependence of S(Q, ω), as discussed below. At Q ≥ 2.5 ˚ A−1 , where hQ is definitely small with respect to ϵQ , collective response is weakening and single particle like response and macroscopic occupation of the condensate k = 0 state begins to be observable in S(Q, ω) at low T as well as at finite temperature (Sakhel and Glyde, 2004). At high Q, Q ≥ 20 ˚ A−1 , the condensate component in S(Q, ω) is used to measure the condensate fraction (Diallo et al., 2014; Glyde, 2013; Glyde et al., 2000a; Griffin et al., 1995).

H. Field Theory Methods

cri pt

is the contribution from SR (Q, ω) involving the states above the condensate. In Eq. (3.25), n0 = N0 /N and

us

fR = (1 − n0 )ϵQ + n0 hQ

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Page 24 of 48 24

√ N0 in ρ(q) as follows, ∑ † ak−q ak ρ(q) =

and a†0 by

For example, if we substitute Eq. (3.29) for ρ(Q) into χ(Q, t), the Fourier transform χ(Q, ω) of χ(Q, t) may be written in closed form as,

k

= a†−q a0 + a†0 aq +

′ ∑

χ(Q, ω) = χS (Q, ω) + χR (Q, ω) (3.34) = Λα (Q, ω)Gαβ (Q, ω)Λβ (Q, ω) + χR (Q, ω),

a†k−q ak

=

′ ∑ √ a†k−q ak N0 (aq + a†−q ) +

√ N0 Aq + ρ′ (q).

A†q

where

k

Λα (Q, ω) = (3.29)

a†q +a−q

[ ] ∫ ¯ ¯ ¯ ¯ ¯ ¯ n0 1 + i dkG2γ (−k)G1δ (k + Q)Pγδα (k, Q)

(3.35) ¯ = Q, ω; and Gαβ (Q, ω) are the four and k¯ = k, ω; Q Green functions of Eq. (3.33) obtained by substituting AQ = aQ + a†−Q into Eq. (3.33). i.e. Gαβ (Q, t) = β α α −iθ(t)⟨[aα Q (t), aQ (0)]⟩ where ak = ak (α = 1) and ak = † ak (α = 2). The initial field theory formulations based on perturbation theory noted above were plagued by divergences in the infrared limit (the limit Q → 0, ω → 0). A unified treatment of the IR limit has been achieved using renormalization group methods (Dupuis and Sengupta, 2007; Pistolesi et al., 2004). An excellent formulation and summary of the field theory expressions for χ(Q, ω) has been presented by Szwabinski and Weyrauth (2001). This includes implementation of the expressions to calculate S(Q, ω) and ωQ directly for comparison with other theory and experiment. Pashitskii and collaborators (2002b; 2010; 2002a) have similarly made explicit calculations of P-R mode energies based on FT formulations. In summary, a central result of field theory is that the single quasiparticle Green function Gαβ (Q, ω) appears directly in the dynamical susceptibility, χ(Q, ω), with weight proportional to the condensate fraction, n0 . The term containing Gαβ (Q, ω) is denoted the singular part, χS (Q, ω). However, χR (Q, ω) also has a sharp component at low Q since, for example, the normal liquid (where χS (Q, ω) is zero) supports a well defined sound mode at low Q. At higher Q (Q ≥ 0.7 ˚ A) the normal liquid supports only broad response. There is also a coupling between χ(Q, ω) and Gαβ (Q, ω) via the condensate. This coupling plays a key role in establishing a sharp mode in the Bose condensed phase (T < Tλ ) at higher Q where there is no mode (only broad response) in the normal phase. The coupling is articulated using the “dielectric” formulation of χ(Q, ω). This formulation and the coupling is sketched below.

an

In Eq. (3.29), = creates a single quasiparticle excitation of wave vector q (by creating a quasiparticle of wave veotor q (a†q ) or annihilating one of wave vector −q (a−q ). The ρ′ (q) is the usual density operator involving the k ̸= 0 states only. The Aq was first introduced by Bogoliubov to describe the quasiparticle modes in a Bose gas (where N0 ≃ N is large and ρ′ (q) is negligible). In this limit the density mode (ρ(q)) is a quasiparticle mode (Aq ). In liquid 4 He where N0 ≪ N and ρ′ (q) dominates, the mode (ρ(q)) is predominantly a density mode in the k ′ ̸= 0 states. There is, however, a small quasiparticle component in S(Q, ω) which plays a critical role in the temperature dependence of S(Q, ω) of liquid 4 He. Substituting Eq. (3.29) for ρ(q) in the DSF, we see that S(Q, t) contains a term proportional to n0 ,

√

us

=

cri pt

k

1 ⟨ρ(Q, t)ρ(−Q, 0)⟩ N √ N0 = n0 ⟨AQ (t)A†Q (0)⟩ + [⟨AQ (t)ρ′ (−Q, 0)⟩ N ] 1 +⟨ρ′ (Q, t)A†Q (0)⟩ + ⟨ρ′ (Q, t)ρ′ (−Q, 0)⟩ N = SS (Q, t) + SR (Q, t) (3.30)

dM

S(Q, t) =

pte

The first term in Eq. (3.30), the quasiparticle correlation function of momentum k = Q, is proportional to n0 . ′ There is also a cross √ term containing AQ and ρ (−Q, 0) proportional to N0 . The “singular” term SS (Q, t) is the sum of the first three terms in Eq. (3.30). The term is labelled singular because we expect poles in the Fourier transform, SS (Q, ω), of SS (Q, t). Finally, there is a “regular” term, SR (Q, t) =

1 ′ ⟨ρ (Q, t)ρ′ (−Q, 0)⟩, N

(3.31)

ce

involving ρ′ (Q) only. In the normal phase where n0 = 0, the total S(Q, t) reduces to SR (Q, t). While S(Q, t) in Eq. (3.30) is illustrative, it is much easier and more rewarding to evaluate corresponding dynamic susceptibility, χ(Q, t) = −iθ(t)⟨[ρ(Q, t)ρ(−Q, 0)]⟩,

(3.32)

and the single quasiparticle Green function G(Q, t) = −iθ(t)⟨[AQ (t), A†Q (0)]⟩.

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(3.33)

I. Dielectric Formulation of χ(Q, ω)

The dynamical susceptibility, χ(Q, ω), and the single quasiparticle Green function, Gαβ (Q, ω), that appears in χ(Q, ω) in Eq. (3.34), can be expressed in terms of a dielectric function ϵ(Q, ω). Using this dielectric formulation, it can be shown that χ(Q, ω) and Gαβ (Q, ω) share a common denominator when there is BEC. This means

Page 25 of 48


(3.36)

ϵ(Q, ω) = 1 − v(Q)χ(Q, ˜ ω).

(3.37)

where

dM

χ(Q, ˜ ω) is simply that part of χ(Q, ω) that cannot be divided into two parts separated by a single interaction line, v(Q). The regular χR (Q, ω), defined in Eq.

¯ = χ(Q)

∑

¯ Q) = χ0 (k,

¯ k

¯ k

pte

(3.42)

Eq. (3.42) has exactly the same form as Eq. (3.36) in which I(Q) plays the role of v(Q) and I(Q) is a well defined approximate interaction for helium. This comparison also suggest that if v(Q) is chosen to be close to I(Q), then χ(Q, ˜ ω) will be close to χ0 (Q, ω). Specifically, the functions χ, ˜ as χ0 , should be suitable functions to represent by simple model functions in a model representation of χ(Q, ω). An example of this is reviewed

ce

cri pt

where

˜χ Ñ ˜Λ ˜ χ ˜ D ˜R + Λ = , ϵ C

(3.38)

G=

˜ + Λv ˜ QΛ ˜ N ϵR N = D C

(3.39)

˜ C = ϵR D = ϵD

(3.40)

is a denominator common to χ(Q, ω) and Gαβ (Q, ω). χ(Q, ω) and Gαβ (Q, ω) share a common denominator only in the Bose condensed phase where n0 ̸= 0. When n0 = 0 the χ(Q, ω) and Gαβ (Q, ω) separate into independent functions. The DF was first developed for the charged Bose gas (Ma and Woo, 1967). In this case the Fourier transform of the Coulomb interaction, v(Q) = 4π/Q2 , is well defined. Helium atoms, however, interact via a potential that has a steeply repulsive hard core. The Fourier transform v(Q), is generally not well defined. We may readily identify a well defined v(Q) in helium by comparing the DF with the exact equation for χ(Q, ω),

∑ ∑ ¯ Q)I( ¯ k¯′ , Q)χ( ¯ Q) ¯ k, ¯ k¯′ , Q)] ¯ ¯ + χ0 (k, [χ0 (k,

¯ = Q,ω; k¯ = k,ω; χ(Q) ¯ = χ(Q, ω) and χ0 (Q) ¯ where Q 0 = χ (Q, ω) is the dynamical susceptibility for a non¯ k¯′ , Q) ¯ interacting Bose gas (the Lindhard function). I(k, is the full, renormalized four point interaction. If ¯ k¯′ and ω, then we ignore the dependence of I on k, ′ ¯ ¯ k , Q) ¯ → I(Q) and Eq. (3.41) reduces to: I(k, χ(Q, ω) = χ0 (Q, ω) + χ0 (Q, ω)I(Q)χ(Q, ω).

and

χ=

an

χ = χ ˜ + χv(Q) ˜ χ ˜ + χv(Q) ˜ χv(Q) ˜ χ ˜ + ··· = χ ˜ + χv(Q)χ ˜ = χ/ϵ, ˜

(3.34), can be similarly expressed in terms of an irreducible χ ˜R (Q, ω) and a dielectric function ϵR (Q, ω) = 1 − v(Q)χ ˜R (Q, ω). As with χ(Q, ω), the χ(Q, ˜ ω) can be separated in ˜ αG ˜ αβ Λ ˜β + χ ˜ ∝ √n0 and the form χ ˜ = Λ ˜R where Λ ˜ αβ = N ˜αβ /D. ˜ A central result is that χ(Q, ω) and G Gαβ (Q, ω) = Nαβ /D may be written in the form,

us

that χ(Q, ω) and Gαβ (Q, ω) have the same poles. As a result the single quasiparticle and density response have the same sharp mode energies. Specifically this means that there are no independent single particle modes at low energy when there is BEC. Rather there is only a single combined quasiparticle and density mode that has phonon dispersion linear in Q at low Q. This result provides a microscopic justification for Landau’s insightful empirical proposal that liquid 4 He supports only a collective mode and no independent single particle excitations. When n0 is small, the collective mode of χ(Q, ω) is predominantly a density mode. When n0 is large (e.g. in a Bose gas) the mode is predominantly a quasiparticle mode (Bogoliubov, 1947; Nozières and Pines, 1990). The dielectric formulation (DF) has its origins in the diagrammatic studies of Bose liquids by Gavoret and Nozieres (1964) and Hohenberg and Martin (1965). It has been developed extensively in algebraic form by Ma and Woo (1967). Briefly, χ(Q, ω) is expressed (Griffin, 1993; Griffin and Cheung, 1973; Ma and Woo, 1967; Szépfalusy and Kondor, 1974) in terms of an irreducible χ(Q, ˜ ω) and an interaction v(Q) in the form,

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(3.41)

¯′ k

in section 3.K below on the temperature dependence of χ(Q, ω).

J. Exact results from field theory

Using field theory methods, important exact results for the dynamical susceptibility, χ(Q, ω), and the single quasiparticle Green function, Gαβ (Q, ω), have been obtained (Gavoret and Nozières, 1964; Glyde, 1994a; Griffin, 1993). Firstly, at T = 0 K χ(Q, ω) and the Gαβ (Q, ω) are exactly (~ = 1),

lim χ(Q, ω) =

Q,ω→0

1 nQ2 , m ω 2 − c2 Q2 + iη

(3.43)


Page 26 of 48 26

q,ω→0

1 ω2

−

c2 q 2

+ iη

,

(3.44)

where m is the mass, n = N/V is the number density and n0 = N0 /N is the condensate fraction. Using this ′′ χ(Q, ω) and S(Q, ω) = −nπ[nB (ω) + 1]χ (Q, ω), we obtain,

lim S(Q, ω) =

Q,ω→0

~Q δ(ω − cQ), 2mc

(3.45)

pte

dM

an

at T = 0 K. This exact relation shows that the BijlFeynman result for S(Q, ω), Eq. (3.3) and (3.4), is exact in the limit Q → 0. Also, using the zeroth moment sum rule, the exact result, Eq. (3.45), gives the static structure factor S(Q) = ~Q/2mc at Q → 0. The S(Q) in Eq. (3.3), deduced by Feynman using small oscillation arguments, reproduces the exact result. The exact form, Eq. (3.45), also fulfils the f -sum rule completely. Thus at Q → 0, S(Q, ω) is confined entirely to a delta function at energy ω = ωQ = cQ. At Q → 0, S(Q, ω) has no multi-P-R component and no intensity outside the delta function in Eq. (3.45). Using the exact result for G11 and the Josephson relation (Baym, 1968; Holzmann and Baym, 2007; Josephson, 1966; M¨ uller, 2015) (see Eq. (4.37))between the superfluid fraction, ρS /ρ, and the condensate fraction n0 = N0 /N , we may show that the superfluid fraction ρS /ρ = 1 at T = 0 K independently of the magnitude of n0 . This satisfying exact result is discussed more fully in section 4.H.3. Secondly, the atomic momentum distribution, n(k), contains a term arising from the coupling between the quasiparticles and the P-R modes via the condensate. This coupling term is in addition to the contribution, n0 δ(k), from the condensate. The coupling term, nc (k), contributes at low k and follows from the quasiparticle Green function Eq. (3.44) and the general relation between n(k) and the quasiparticle Green function (see page 154 of (Mahan, 1990), ∫ dω nk = ⟨a†k ak ⟩ = A11 (k, ω)nB (ω) (3.46) 2π

cri pt

lim G11 (q, ω) = n0 mc2

G11 (k, ω) at higher k but these are not known and have not been included to date. Models of n(k) typically take the form of Eq. (4.24), i.e. n(k) = n0 [δ(k) + f (k)] + A1 n∗ (k), where n0 δ(k) arises from the condensate, n∗ (k) from the k ̸= 0 states, 2 2 and n0 f (k) = nc (k)e−k /(2kc ) from the coupling term. A 2 2 cut off, e−k /(2kc ) , where kc = 0.5 ˚ A−1 , has been added to restrict n0 f (k) to low k where G11 (k, ω) is known and contributes. The role of nc (k) in measurements (Diallo et al., 2012; Glyde, 1994a; Glyde et al., 2000a) of n(k) and n0 is discussed in section 4.E. The coupling between quasiparticles and P-R modes via n0 manifests itself in both single quasiparticle properties (n(k)) and the collective response (P-R modes). Thirdly, the field theory analysis reveals the role of n0 (T ) in determining the intensity in the P-R mode as a function of temperature. This is particularly clear at higher wave vector Q ≥ 0.8 ˚ A where S(Q, ω) is a broad function of ω at T ≥ Tλ in the normal phase (no P-R mode in S(Q, ω) at T ≥ Tλ ). At Q ≥ 0.8 ˚ A, the intensity in the P-R mode at T ≤ Tλ is proportional to n0 (T ) and the mode disappears at T = Tλ (Glyde, 1994a; Glyde and Stirling, 1990; Griffin, 1993; Stirling and Glyde, 1990). At low Q there is a well defined sound mode in the normal liquid and the role of n0 (T ) is more difficult to identify.(Andersen and Stirling, 1994; Woods, 1966) At Q ≥ 0.8 ˚ A, S(Q, ω) is a very broad function of ω in the normal phase and the role of n0 (T ) in establishing a sharp P-R mode at T ≤ Tλ can be readily identified. Model calculations of the dependence of S(Q, ω) on n0 (T ) are reviewed below.

us

and

ce

where A11 (k, ω) = −2ImG11 (k, ω) is the spectral function corresponding to G11 (k, ω). Explicitly, from Eq. (3.44), A11 (k, ω) = (n0 mc2 /2~ck)2π[δ(ω − ck) − δ(ω + ck)]. Substituting A11 (k, ω) in Eq. (3.46) gives a contribution to the n(k) that is normalized to unity, n(k)= nk /(2π)3 n, of nc (k) = n0 (

1 mc ) [2nB (ck) + 1] 2~(2π)3 n k

(3.47)

A11 (k, ω) and G11 (k, ω) above represent the coherent part of the Green function. There could be additional contributions from the incoherent part of G11 (k, ω) and

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

K. Temperature dependence of S(Q, ω)

One of the most fascinating features of the dynamic structure factor, S(Q, ω), is its temperature dependence. This temperature dependence may be divided into two parts. Firstly, as temperature is increased, the P-R mode broadens and its energy is shifted through interaction with thermal P-R modes. These interaction processes are discussed in section 2.C. Secondly, as temperature is increased, the weight of the P-R mode in S(Q, ω) decreases. At higher Q, Q ≥ 0.8 ˚ A−1 , it is readily observable that this weight goes to zero at Tλ because S(Q, ω) is very broad in the normal liquid phase at Q ≥ 0.8 ˚ A−1 . In the normal liquid phase at Q ≥ 0.8 −1 ˚ A , S(Q, ω) changes little with temperature between Tλ and 4.2 K. Within precision, the thermal broadening of a PR mode in the roton region is well described by the process in which the roton interacts with a thermally created roton and the two scatter into two other P-R modes, a four P-R mode process. Since the number of thermal rotons goes to zero at T → 0, the mode width ΓQ → 0 at T → 0. The theory of this process, originally proposed by Landau and Khalatnikov (1949), has subsequently been

Page 27 of 48


an

1. Role of the condensate

pte

dM

We review a model (Glyde, 1992, 1994a; Glyde and Stirling, 1990; Griffin, 1993) of S(Q, ω) based on the dielectric formulation. The goal is to reproduce the temperature dependence of S(Q, ω). It is to show how the loss of intensity in the P-R mode with increasing temperature follows from the temperature dependence of the condensate fraction, n0 (T ) = n0 (0)[1 − T /Tλ ]γ . In the model, all parameters are held independent of temperature except n0 (T ). Particularly, at higher Q values, Q & 1˚ A−1 , S(Q, ω) is very broad in the normal phase, as it is in other normal liquids (liquid 3 He, liquid Ne). The model displays how S(Q, ω) can go from a function that has a sharp P-R peak at low temperature, where n0 (T ) is finite, to a broad function at T ≥ Tλ where n0 (T ) = 0. The model also displays the role of BEC in creating a sharp P-R peak in S(Q, ω) at low temperatures and Q&1˚ A−1 . In this model, the temperature dependence of the width of the P-R mode arising from P-R interactions is not included. In a complete model, both the temperature dependence of mode intensity and width should be described. In the dielectric formulation (see section 3.I), the dynamical susceptibility χ(Q, ω) is expressed in terms of an “irreducible” χ(Q, ˜ ω) in the form, χ = χ+ ˜ χv(Q)χ ˜ = χ/ϵ. ˜ where ϵ is the dielectric function, ϵ = 1 − v(Q)χ. ˜ When there is a condensate, χ ˜ (as χ) separates into the form ˜G ˜Λ ˜ +χ χ ˜=Λ ˜R , where χ ˜R is the regular part which in˜ is a quasiparvolves only states above the condensate. G √ ticle Green function and Λ = n0 [1 + P ] depends on the condensate fraction, n0 , and an interaction term (P). As

ce

us

refined (Bedell et al., 1982, 1984; F˚ ak et al., 2012; Khalatnikov, 1965) as discussed in section 2.C.2. While the mode width is well described by this scattering process, the temperature dependence of the mode energy, ωQ (T ) is not. Specifically, Fak et al. (2012) have recently shown that P-R mode interactions involving phonons as well as rotons are important. The shift of the energy, ωQ (T ), with temperature at low temperature (T < ∼ 1 K) is dominated by terms involving thermal phonons. At higher temperatures (T > ∼ 1.2 K) the term considered by LK involving rotons appears to be the most important for both ωQ (T ) and ΓQ (T ) (see section 2.C.2). At T ≥ 1.5 K, it is difficult to determine ωQ (T ) and ΓQ (T ) uniquely from data. The values extracted from experiment depend on the model of S(Q, ω) used to analyse the data (see section 2.C.4). As a result the theory of the temperature dependence of ωQ (T ) and ΓQ (T ) at temperatures 1.5 ≤ T ≤ 2.17 K remains poorly tested by experiment today. Below we review a model of S(Q, ω) that describes the second feature, the temperature dependence of the weight of the P-R mode in S(Q, ω) at T ≤ Tλ , and displays the role of condensate, n0 (T ).

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

cri pt

27

FIG. 3.3 The dynamical structure factor, S(Q, ω), at Q = 1.95 ˚ A−1 (roton region) of liquid 4 He at SVP. The upper frame shows the calculated S(Q, ω) obtained using Eq. (3.53) in which only the condensate fraction n0 (T) depends on temperature. The S(Q, ω) is folded with an instrument energy resolution FWHM = 0.0153 THz. The lower frame shows the observed S(Q, ω) (intensity) obtained using the same energy resolution (Stirling and Glyde, 1990) (from (Glyde, 1992)).

discussed in section 3.I, if the interaction v(Q) is interpreted as a local approximation to the full four point interaction in the exact equation for χ(Q, ω)(see Eqs. (3.41) and (3.42)), and the energy dependence of I(Q) is ignored, then we see that the irreducible functions χ ˜ is similar to the non-interacting functions χ0 . The essential point is that the irreducible functions χ ˜ (like χ0 ) are appropriate functions to model. In the non-interacting ˜ → √n0 and χ limit Λ ˜ → χ0 = n0 G0 + χ0R In the model (Glyde, 1992), we represent χ ˜R by a susceptibility, χ ˜R =

˜R ˜R N N = 2 , 2 ˜R ω − (¯ ω0 − α) + i2ωΓ0 D

(3.48)

˜ by a Green where ω ¯ 0 , Γ0 and α are parameters and G function ˜ ˜ N ˜=N = G . 2 2 ˜ ω −ω ¯ SP + i2ωΓSP D

(3.49)

where ω ¯ SP and ΓSP are parameters. The corresponding


Page 28 of 48 28

˜G ˜Λ ˜ +χ ϵ = 1 − v(Q)χ ˜ = 1 − vQ [ Λ ˜R ] ∆ α = 1− − ˜ ˜R D D

(3.50)

=

χ ˜R 1 χ ˜R = = −1 ϵR 1 − v(Q)χ ˜R χ ˜R − v(Q) ˜R N ω 2 − ω02 − i2ωΓ0

(3.51)

The corresponding S(Q, ω) = −π −1 [nB (ω) + 1]χ′′ (Q, ω), a DHO function, is fitted to data in the normal phase to determine the parameters ω0 (Q) and Γ0 (Q) of the model. In the Bose condensed phase where no ̸= 0, χ=

˜G ˜Λ ˜ +χ χ ˜ Λ ˜R = . ϵ ϵ

(3.52)

˜ = √n0 and the and ϵ is given by Eq. (3.50). Using Λ ˜ R and model functions Eq. (3.48) and Eq. (3.49) for X ˜ respectively, the S(Q, ω) reduces to, taking ΓSP = 0, G,

2 2 ˜R N 2ωΓ0 (ω 2 − ω ¯ SP ) [n(ω) + 1] 2 2 2 + ∆))2 ] 2 2 ¯ SP + ∆) − ∆α + (2ωΓ0 (ω 2 − ω ¯ SP π [(ω − ω ¯ 0 )(ω − ω

(3.53)

an

S(Q, ω) =

χ ˜ =

us

˜ and α = v(Q)N ˜R . The parameter where ∆ = n0 v(Q)N ∆(T ) is proportional to n0 (T ). ˜ and the dielecThe model functions χ ˜R and G tric function define the model. The parameters are ω ¯ 0 , Γ0 , ω ¯ SP , ΓSP , ∆(T ), α. The parameters ω ¯ 0 and Γ0 in χR are determined by fitting the model to data in the normal phase. The parameters ω ¯ SP , ∆(0) and α are determined by fitting the model to data at T → 0 K. The parameter ΓSP is assumed to be zero here for simplicity. These parameters are held independent of temperature except ∆(T ). The temperature dependence follows solely from the temperature dependence of ∆(T ) ∝ n0 (T ). ˜ = 0, χ In the normal phase, n0 = 0, Λ ˜=χ ˜R and ϵ =

ϵR = 1 − v(Q)χ ˜R . Using the model Eq. (3.48) for χ ˜R , the dynamical susceptibility is

cri pt

˜ = √n0 , dielectric function ϵ = 1 − v(Q)χ ˜ is, setting Λ

ce

pte

dM

The additional parameters, ω ¯ SP , ∆(0) and α, in S(Q, ω) at T < Tλ where n0 (T ) ̸= 0 are determined from fits to data at low temperature. The P-R mode peak is sharp in the model because the peak width goes 2 to zero at ω 2 = ω ¯ SP + ∆. S(Q, ω) is not very sensitive to α (∆ is small). S(Q, ω) in Eq. (3.53) clearly reduces to the normal phase S(Q, ω) when ∆ = 0. The temperature dependence of S(Q, ω) in Eq. (3.53) arises solely from ∆(T ) ∝ n0 (T ). ∆(T ) was set at the Bose gas result, ∆(T ) = ∆(0)[1 − T /Tλ ]3/2 . Fig. 3.3 shows the model S(Q, ω) given Eq. (3.53) compared with data at the roton wave vector Q=1.925 ˚ A. As ∆(T ) goes to zero at T = Tλ , the intensity in the sharp peak at low temperature drops and S(Q, ω) goes over to a broad function as observed. The roton is an interesting example because at low temperature since 75 % of the intensity lies in the sharp P-R peak. The model illustrates clearly how this intensity, confined to a sharp peak at T < Tλ goes over to broad intensity at T = Tλ as the condensate fraction n0 (T ) goes to zero. The model fits the data best at Q = 1.1 ˚ A−1 , the maxon region. At the maxon approximately 50% of the intensity lies in the P-R mode at T → 0 K. The model has been compared with data at both low Q (Andersen et al., 1994b; Glyde, 1992) and at higher Q(Pearce et al., 2001) . Fig. 2.11 shows the fit of the model to data at Q = 3.0 ˚ A−1 by Pearce et al.. The model reproduces the loss of intensity in the P-R peak with increasing T, as n0 (T ) goes to zero at T = Tλ = 1.93 K. Multi-PR components are not included in the model. A simpler but more complete model (Sakhel and Glyde,

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

2004) valid at higher Q values also reproduces the data in Fig. 2.11 well. In principle, the DF model for S(Q, ω) should be combined with a theory for the temperature dependence of the P-R line width such as the Landau-Khalatnikov theory. In bulk liquid 4 He, it is clear that the P-R line width increases with increasing T up to T = Tλ = TBEC . An interesting counter example is the temperature dependence of the roton in liquid 4 He at p = 34 bar. (Bossy et al., 2012a, 2008a; Pearce et al., 2004) At this pressure the roton mode peak disappears at T = 1.5 K (Bossy et al., 2012a) leaving only broad response at T > 1.5 K. The temperature T = 1.5 K is identified with TBEC , the “critical temperature” for BEC at p = 34 bar. The roton mode disappears without substantial broadening at p = 34 bar since 1.5 K is below the temperature at which substantially broadening takes place (see Figs. 2.12 and 2.15) (2ΓQ ≃ 0.025 meV at 1.5 K). This example suggests that the P-R mode can disappear without significant broadening of the mode if TBEC is sufficiently low.

4. SINGLE PARTICLE EXCITATIONS A. Introduction

In this section we discuss the single particle excitations of liquid 4 He. These may be thought of as excitations of the liquid in which the energy and momentum is concentrated on a single atom. If this energy is high, the excited

Page 29 of 48


us

cri pt

width of S(Q, ω). The structure factor, S(Q), also oscillates with Q. Extensive measurements of S(Q, ω) showing these oscillations in the range 5 ≤ Q ≤ 10 ˚ A−1 have been made (Andersen et al., 1997) and reviewed (Glyde, 1994b; Sokol, 1995). The oscillations of the peak position and width of S(Q, ω) can be related to the oscillations in S(Q). They can also be explained in terms of oscillations in the He-He atom scattering amplitude σ(Q) with Q. Within precision no oscillations in S(Q) are observed at Q ≥ 12 ˚ A−1 . This is usually taken as the Q value at which the coherent S(Q, ω) can be accurately approximated by the incoherent DSF, Si (Q, ω), defined in Eq. (1.6). As noted in the Introduction, the first measurements of the Bose-Einstein condensate fraction, n0 , were made using reactor source neutrons and at wave vector transfers in the range Q = 8−15 ˚ A−1 (Cowley and Woods, 1968; Mook, 1974, 1983; Mook et al., 1972; Sears et al., 1982; Svensson et al., 1980; Woods and Sears, 1977). The initial values ranged from 2 to nearly 20 % with most values between 7 and 15 %. Using the expressions for S(Q, ω) developed by Sears (Sears, 1969, 1970; Sears et al., 1982) which included Final State effects, values of n0 = 10 − 13 % were subsequently obtained (Mook, 1983; Sears et al., 1982). For reviews, see (Glyde and Svensson, 1987) and p. 367 (Glyde, 1994a). A second generation of measurements were made at the spallation neutron source IPNS at a higher wave vector Q = 23 ˚ A−1 (Snow and Sokol, 1990, 1995; Snow et al., 1992; Sosnick et al., 1991, 1990a, 1989, 1990b). At the higher Q value, FS effects were substantially smaller and a condensate fraction of n0 = 10 % at SVP was obtained. Measurements as a function of pressure were also made (Snow and Sokol, 1995). These measurements have been extensively reviewed (Ceperley, 1995; Glyde, 1994a; Griffin, 1993; Silver, 1989; Sokol, 1995). Drawing on the much more intense spallation neutron sources at the ISIS Facility, Rutherford Appleton Laboratory, UK, and at the Spallation Neutron Source (SNS) at Oak Ridge National Lab, USA, a third generation of measurements became possible Two instruments, MARI at the ISIS and ARCS at the SNS, are available in the Q range 20 ˚ A−1 < Q < 30 ˚ A−1 with sufficiently high energy resolution to determine the relatively narrow n(k) of condensed helium. Making measurements over a range of Q values also enabled substantial improvement. The n(k) is an intrinsic property of liquid 4 He and therefore independent of Q. In contrast, FS effects depend on Q. By making measurements as a function Q the FS interactions can be identified and separated from the intrinsic properties n(k) and n0 . This enabled an experimental determination of FS contributions. Also, the shape of n(k) could be determined. Some of these results including solid helium have been reviewed (Glyde, 2013). The VESUVIO instrument at ISIS offers scattering at much higher momentum and energy transfer but a broader energy resolution. The atomic kinetic energy, ⟨K⟩, of atoms, particularly of hydrogen (H), in a wide

ce

pte

dM

an

atom can respond as if nearly free and independent of its neighbours. At lower energy, the response of the atom is modified by interaction with the remainder of the liquid. In neutron scattering measurements, single particle response can be observed when the transfer of momentum, ~Q, and energy, ~ω from the neutron to the liquid is high. In this regime, the energy of the incoming neutron is also high (e.g. 1 eV) and its wavelength is short compared to the inter-atomic spacing in the liquid. The neutron scatters predominantly from the nucleas of a single He atom and the energy is transferred predominantly to a single atom, denoted the struck atom. When ~ω is large compared to the inter-atomic potential, the atom recoils largely independently of its neighbours. The limit in which the interaction with neighbours is neglected completely is denoted the Impulse Approximation (IA). The interactions of the struck atom with its neighbours following the scattering are denoted Final State (FS) effects. The goal of measurements at high ~Q and ~ω is to determine single particle properties such as the atomic momentum distribution, n(k). In liquid 4 He, it is especially to determine the Bose-Einstein condensate fraction, n0 = N0 /N , the fraction in the zero momentum (k = 0) state. Bose-Einstein condensation (BEC) is of great intrinsic interest as the origin of coherence in the liquid and of superfluidity. In addition, the Fourier transform of n(k) is the one-body density matrix (OBDM). The ˆ † (r)Ψ(0)⟩ ˆ OBDM is ρ1 (r) ≡ ⟨Ψ and n(r) ≡ n1 ρ1 (r) = ∫ ik·r ik·r dkn(k)e = ⟨e ⟩ where n = N/V . The OBDM can be calculated directly using many-body techniques such as diffusion Monte Carlo (DMC) and path integral Monte Carlo (PIMC). This provides a direct link between theory and experiment. Specifically, the observed intermediate scattering function (ISF) in the IA, JIA (s) = SIA (Q, t)eiωR t , is the OBDM for lengths s = vR t (vR = ~Q/m) along the scattering vector Q, ˆ i.e. n(s) = JIA (s) = ⟨e−ikQ s ⟩, as we show below r = sQ, in section 4.B. In classical systems, n(k) is a Gaussian function, the Maxwell-Boltzmann distribution. In classical systems momentum and position are statistically independent variables. In quantum systems, however, momentum and position do not commute and the shape of n(k) depends on the interatomic potential. Determining the shape of n(k) and the atomic kinetic energy of atoms in quantum systems is of fundamental interest. It is also of interest to determine the degree to which the Impulse Approximation is satisfied and the nature of Final State effects. To make contact with previous sections, we recall that the collective P-R mode terminates at Q = 3.6 ˚ A−1 . 4 Scattering from He is entirely coherent and the coherent DSF is always observed. However, at Q ≃ 4−5 ˚ A−1 , 4 liquid He begins to respond like a collection of inter˚−1 , the incoherent acting single particles. At Q > ∼ 4−5 A i = j term in S(Q, t) in Eq. (1.3) dominates. In the range 5 ≤ Q ≤ 10 ˚ A−1 , the coherent contributions arising from the i ̸= j terms in S(Q, t) are small. They are observed as small oscillations in the peak position and

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Page 30 of 48 30

cri pt

We may also write JIA (y) as,

B. Impulse approximation

pte

dM

We consider high energy incoming neutrons that scatter from 4 He atomic nuclei. We focus on high momentum ~Q and energy ~ω transfer from the neutron to the struck atom, high compared to the potential energy of interaction between the atoms in the liquid. In the Impulse Approximation (IA) we neglect the interatomic interaction entirely. Essentially, we assume that the impulse received by the struck atom from the neutron is much larger than the impulses received from neighbouring atoms. For a uniform or nearly uniform system we may use momentum states k to describe the liquid. Other states such as natural orbitals have been recently discussed (Glyde, 2013). We consider an atom initially in state k (of energy ϵ0 (k) = (~k)2 /2m). After the transfer of momentum ~Q, the struck atom is in final state k + Q of energy ϵ0 (k + Q) = ~2 (k + Q)2 /2m. In the IA the dynamical structure factor (DSF) is, ~ ∑ SIA (Q, ω) = Nk δ(~ω − [ϵ0 (k + Q) − ϵ0 (k)] N k ∫ ~ [Q2 + 2k · Q]) = dkn(k)δ(ω − 2m ∫ = dkn(k)δ(ω − ωR − vR kQ ). (4.3)

ce

ˆ is the initial momentum along Q, Since kQ = k · Q SIA (Q, ω) is the initial momentum distribution along Q integrated over the other two directions. In the IA, the only property of the liquid appearing in S(Q, ω) is kQ = y and the expectation value ⟨δ(y − kQ )⟩ can be taken over the initial momentum distribution, n(k). S(Q, ω) in the IA clearly depends only on the “y scaling” variable y = (ω − ωR )/vR and not separately on Q and ω. In this limit it is usual (Sears, 1984; West, 1975) to introduce the function JIA (y) defined as ∫ JIA (y) ≡ vR SIA (Q, ω) = dkn(k)δ(y − kQ )

an

The ⟨kα2 ⟩ is also the second moment of the incoherent DSF, Si (Q, ω), ∫ ∫ 1 2 2 dω(ω − ωR ) Si (Q, ω) = dyy 2 J(Q, y) ⟨kα ⟩ = vR 2 (4.2) In the second equality we have introduced the “y scaling variable” y ≡ (ω − ωR )/vR , where ωR = ~Q2 /2m and vR = ~Q/m, which will be discussed extensively below. We have also written the incoherent DSF as J(Q, y) ≡ vR Si (Q, ω). FS effects do not contribute to the second moment so that J(Q, y) can be replaced by the IA, JIA (y), for calculation of this second moment (see section 4.D and Eq. (4.32)). In the following sections we discuss expressions for the incoherent DSF usually expressed in the form J(Q, y). We begin with the Impulse Approximation.

process” in which energy is transferred from the neutron to the liquid. We have also used δ(ax) = a1 δ(x) and defined a recoil frequency ωR = ~Q2 /2m (the actual recoil frequency of non-interacting atoms in the initial state k = 0), a recoil velocity, vR = ~Q/m, and ˆ for the initial atomic momentum k along the kQ = k · Q scattering vector Q. Introducing the “y scaling” variable y ≡ (ω − ωR )/vR , the delta function in (4.3) is δ(ω − ωR − vR kQ ) = v1R δ(y − kQ ) and the DSF is, ∫ 1 1 SIA (Q, ω) = dkn(k)δ(y − kQ ) = ⟨δ(y − kQ )⟩ vR vR (4.4)

us

variety of systems has been determined (Andreani et al., 2005). The n(k) of H is nearly ten times broader than that of 4 He. Also, assuming an isotropic n(k), the ⟨K⟩ is simply related to the second moment of n(k) along a direction α, ⟨kα2 ⟩, by ( 2)∫ 3~ 2 2 ⟨K⟩ = (3~ /2m)⟨kα ⟩ = dkn(k)kα2 . (4.1) 2m

where Nk is the number of atoms in initial state k and n(k) is the initial momentum distribution, normalized to unity. In the IA, we have included only the “Stokes

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

JIA (y) = ⟨δ(y − kQ )⟩ ∫ = dkx dky n(kx , ky , y) ≡ nL (y).

(4.5)

In Eq. (4.5) the z axis has been chosen parallel to Q. The nL (y) is denoted the Longitudinal momentum distribution. For a spherically symmetric n(k), nL (y) reduces to n(y), the 1D momentum distribution along one direction, e.g. y = kQ = kz . The length variable conjugate to y is s = vR t. The intermediate scattering function (ISF) conjugate to JIA (y) is, ∫ JIA (s) = dye−iys JIA (y) ∫ = ⟨ dye−iys δ(y − kQ )⟩ = ⟨e−ikQ s ⟩.

(4.6)

Indeed, the IA to the ISF, JIA (s) = ⟨e ⟩, can also be a starting point and can be simply derived (Glyde, 2013). ikQ s

C. One body density matrix

JIA (s) is the one body density matrix (OBDM) of the ˆ along the scattering vecliquid for displacements r = sQ tor Q. Explicitly, the OBDM is defined as ˆ † (r2 )Ψ(r ˆ 1 )⟩ ρ1 (r1 , r2 ) ≡ ⟨Ψ

(4.7)

Page 31 of 48


Comparing n(r) in Eq. (4.9) with JIA (s) in Eq. (4.6) above, we see that the DSF in the IA, JIA (s) = ⟨e

−ikQ s

⟩ = n(s)

(4.10)

0.5

0.0

-4

-2

0

2

4

-1

y (Å )

FIG. 4.1 Final-State broadening functions, R(Q, y), at Q = 23 ˚ A−1 of liquid 4 He at saturated vapour pressure (SVP) : experiment (solid line) and calculated, (Silver, 1989) (open circles, Sil), (Carraro and Koonin, 1990) (dotted line, CK) and (Mazzanti et al., 1996) (dashed line, Maz) (from (Glyde et al., 2000b)).

D. Final state effects

dM

an

ˆ along Q. is the OBDM for displacements r = sQ From (4.5) and (4.10) we see that a measurement of JIA (y) provides the longitudinal momentum distribution nL (y) directly and determination of JIA (s), the OBDM. Particularly, if there is BEC and n(k) has a condensate term, n0 δ(k), then from Eq. (4.5), JIA (y) will have a delta function component, JIA (y) = n0 δ(y), at y = 0. From (4.6) and (4.10), the corresponding OBDM will have a constant term of magnitude n0 , JIA (s) = n0 . Models of the OBDM are simpler to construct than models of nL (y). Models of the intermediate scattering function J(Q, s) including final state (FS) interaction effects are especially easier to construct than the corresponding J(Q, y), as discussed below.

1.0

cri pt

where Nk = ⟨nk ⟩ = ⟨a†k ak ⟩ is the number of atoms in state k, as in Eq. (4.3). Defining the dimensionless OBDM n(r) = ρ1 (r)/n (n = N/L), we have ∫ 1 n(r) ≡ ρ1 (r) = dkn(k)eik·r = ⟨eik·r ⟩ (4.9) n

Exp't Sil CK Maz

us

k

1.5

R(Q,y) (Å)

where Ψ† (r2 ) is the fluid operator that creates a particle at point r2 . For a uniform system ρ1 (r1 , r2 ) = ρ1 (r1 −r2 ) depends only on the separation between r1 and r2 . Intro∑ ˆ ducing the operators a†k and ak , Ψ(r) = V 1/2 k ak eik·r , we have N 1 ∑ Nk eik·(r1 −r2 ) (4.8) ρ1 (r1 − r2 ) = V N

ce

pte

The Impulse Approximation (IA) is a high momentum transfer, ~Q, short scattering time approximation. However, the Q value at which the IA in liquid 4 He becomes valid is not accurately known. In the Q range 20 ≤ Q ≤ 30 ˚ A−1 where extensive measurements have been made, it is known that final state (FS) interactions, arising from the interaction of the recoiling struck atom with its neighbours, make significant contributions to S(Q, ω). These FS contributions have to be included in the DSF to obtain accurate values of n(k) and the condensate fraction. Extrapolations of FS functions determined at Q = 30 ˚ A−1 to higher Q (e.g. Q ≃ 150 −1 ˚ A ) suggest that FS effects are significantly smaller at Q ≥ 150 ˚ A−1 but not negligible. However, this has not been adequately tested. To incorporate FS effects, we note that Q = 12 ˚ A is generally accepted as the Q value at which the coherent ISF S(Q, t) can be accurately approximated by the incoherent ISF Si (Q, t) defined in Eq. (1.6). For example, the incoherent static structure factor is Si (Q) =

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

˚−1 , the oscillations in Si (Q, t = 0) = 1. At Q & 12 A S(Q) arising from coherent contributions become unobservable and S(Q) has saturated to S(Q) = 1. We begin with Si (Q, t). As in the IA, we use the length variable s = vR t and define a function J(Q, s) that reduces to the IA in the limit s → 0, J(Q, s) ≡ Si (Q, t)eiωR t .

(4.11)

J(Q, s) can be expressed in a form resembling the IA (4.10) (Rahman et al., 1962) (see p. 326 (Glyde, 1994a)), J(Q, s) = ⟨Ts e−i

∫s 0

ds′ kQ (s′ )

⟩.

(4.12)

Here kQ (s) = eiHt kQ e−iHt (t = s/vR ) is in the Heisenberg representation. Ts is the “time” ordering operator that requires earlier times (shorter s) to be on the right in expansions of J(Q, s). Clearly, J(Q, s) depends separately on Q and s. The nature of the IA can be clarified by comparing J(Q, s) and JIA (s) in (4.12) and (4.10). Firstly, the IA is valid at short scattering time (short s). If we expand kQ (s) around s = 0, k(s) = ′ kQ (0) + kQ s + · · · , then if terms beyond kQ (0) = kQ are negligible, then J(Q, s) in Eq. (4.12) reduces to the IA in (4.10). Final state effects, differences between J(Q, s) and JIA (s), arise from interactions of the recoiling struck atom with its neighbours which change kQ (s) from its initial value kQ (0). If the struck atom travels only a short distance s = vR t, within the scattering time, then k(s) ≃ kQ (0) and the IA is accurate. Physically, if s is short, the struck atom’s potential energy will


Page 32 of 48 32

cri pt

an

2 4 where α ¯ n are cumulants of kQ , α ¯ 2 = ⟨kQ ⟩, α ¯ 4 = ⟨kQ ⟩− 2 2 ∗ 3⟨kQ ⟩ , · · · . If n (s) is at least approximately Gaussian, then the higher terms in α ¯ 4 and α ¯ 6 will be small. Similarly, J(Q, s) in Eq. (4.12) can be expanded in cumulants (Glyde, 1994a,b), [ ] ∑ (−i)n n J(Q, s) = exp µ ¯n s (4.14) n! n

The formulation of FS effects in terms of a FS broadening function R(Q, y) in Eq. (4.16) was first proposed by Gersch and Rodrigez (1973). When formulated in position space as above, J(Q, s) = JIA (s)R(s) emerges as a simple product with the FS broadening function given by a series expansion. The convolution formulation is particularly useful if the momentum distribution contains a sharp feature such as a condensate component, n(k) = n0 δ(k). In this case, from (4.6) and (4.16), JIA (y) has a term n0 δ(y) and J(Q, y) has a term n0 R(Q, y), respectively. The IA is reached if R(Q, y) → δ(y) and R(Q, s) → 1. A number of important calculations of the FS broadening function R(Q, y) defined in Eq. (4.16) have been made. Fig. 4.1 shows three of these calculated R(Q, y) at Q = 23 ˚ A−1 compared with the series expression Eq. (4.17) obtained by fitting the series expression to data (denoted Expt.). All the R(Q, y) have a similar shape. The calculated R(Q, y) are all narrower than the R(Q, y) given by the series expression Eq. (4.16). The observed condensate fractions obtained using the calculated R(Q, y) in the data analysis are all smaller than or the same as n0 obtained using the series expression Eq. (4.17) (Glyde et al., 2000a). Specifically the n0 obtained using the Carraro and Koonn R(Q, y) was the same within error as that obtained using Eq. (4.17). These FS functions have been reviewed extensively elsewhere (Silver, 1988) and Chap. 19 (Glyde, 1994a).

us

change little from its s = 0 value, for smoothly varying potentials and the potential can be ignored. Similarly, ′ if kQ = (dkQ /dt)/vR and higher derivatives are small, then k(s) ≃ kQ (0)). Clearly, if there is a perfectly hard ′ core potential between atoms (kQ → ∞) then the IA will never be accurately reached. For smoothly varying potentials, the IA is approached as s → 0. This picture of FS effects suggests the J(Q, s) may be well represented at high Q by a series in powers of s with a finite number of terms needed. For example, not including BEC, JIA (s) can be expanded in cumulants,   ∑ (−i)n ∗ (s) = ⟨e−ikQ s ⟩ = exp  α ¯ n sn  n∗ (s) = JIA n! n even [ ] 1 1 1 = exp − α ¯ 2 s2 + α ¯ 4 s4 − α ¯ 6 s6 + · · · , (4.13) 2 4! 6!

dM

where µ ¯2 = α ¯2, µ ¯3 = β¯3 = a¯3 /(λQ) µ ¯4 = α ¯ 4 + β¯4 with 2 2 ¯ β4 = a¯4 /(λQ) (λ = ~ /m), and higher terms. The IA appears in J(Q, s). The IA is given by the terms in α ¯n in Eq. (4.14). The terms in J(Q, y) with coefficients β¯n represent the FS contributions. The FS terms decrease in magnitude with increasing Q. For example, a¯3 = ⟨∇2 v(r)⟩/6 and a¯4 = ⟨(∇Q v)2 ⟩/3 are constants. Since all terms in both (4.13) and (4.14) appear in an exponential, J(Q, s) can be written as a product, ∗ J(Q, s) = JIA (s)R(Q, s),

(4.15)

pte

where R(Q, s) contains the FS terms and is denoted the FS broadening function. In Eq. (4.14) we could retain all the terms in expansion of JIA (s) plus the condensate term, i.e. expand only R(Q, y). In this case we can ∗ replace JIA (s) by the full JIA (s) in Eq. (4.15). When expressed in the conjugate y variable, Eq. (4.15) is a convolution integral, ∫ J(Q, y) = dy ′ JIA (y ′ )R(Q, y − y ′ ). (4.16)

ce

In the series formulation Eq. (4.14), the FS function, R(Q, s), is obtained as a series, [ ] i ¯ 3 i ¯ 5 1 ¯ 6 R(Q, s) = exp β3 s − β5 s + β6 s + · · · 3! 5! 6! (4.17) In Eq. (4.17) the term in s4 has been omitted since we have found β¯4 to be negligible in liquid 4 He (Diallo et al., 2012; Glyde et al., 2011a, 2000a).

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1. Additive Approach

Final State corrections to the IA can also be expressed as terms added to the IA. This formulation of FS effects was extensively developed by Sears (Sears, 1969, 1984) and has been widely used to analyse data from many early measurements (Mook, 1983; Sears et al., 1982; Svensson et al., 1980; Woods and Sears, 1977). This additive approach (AA) to FS effects can be obtained from (4.15) by expanding the exponential in R(Q, s) of eq. (4.17) to give, J(Q, s) = JIA (s)[1 +

i ¯ 3 1 β3 s + β¯4 s4 + ...](4.18) 3! 4!

In this expression the full JIA (s) including the condensate is usually retained. FS terms in s3 and s4 appear as terms added to the IA. The AA can also be obtained from Eq. (4.14) by retaining the Gaussian term (s2 ) in the exponential and expanding the higher order terms to give, 1

2

J(Q, s) = e 2 µ¯2 s [1 +

i 1 µ ¯3 s3 + µ ¯4 s4 + ...](4.19) 3! 4!

In (4.19) a Gaussian IA has been assumed in the exponential (¯ µ2 = α ¯ 2 ). As a result, the µ ¯n can contain terms from JIA (s)(e.g. µ ¯4 = α ¯ 4 + β¯4 ). The AA is most useful when the n(k) (or JIA (y)) is broad compared to R(Q, y). That is, where n(s) is

Page 33 of 48

AUTHOR SUBMITTED MANUSCRIPT - ROPP-100857.R1 33 (ω−ω ′ )2

S(Q,ω) (meV−1)

0.05

S2

0 10

cri pt

~ SIA

S1

30 Energy Transfer (meV)

50

Q = 16.00 Å−1 T = 2.50 K Ei = 266 meV

0.02

S1

0 80

us

~ SIA

S2

120 160 Energy Transfer (meV)

dM

FIG. 4.2 Observed S(Q, ω) including the instrument resolution in liquid 4 He at SVP and low temperature (open circles)(reproduced from Andersen et al. (1997)). Shown is the Impulse Approximation in a Gaussian approximation, S˜IA (Q, ω), (dashed line) and the leading, S1 (Q, ω), and second order, S2 (Q, ω), FS contributions, proportional to µ3 and µ4 respectively, given by Eq. (4.21) fitted to the data with µ2 , µ3 and µ4 treated as free fitting parameters. At Q = 8 ˚ A−1 coherent effects are important, at Q = 16 ˚ A−1 they are negligible.

pte

relatively short ranged compared to R(Q, s). It is also useful in displaying the nature of the leading FS terms. The additive formulation can also be readily extended to lower Q, 4 < Q < 12 ˚ A−1 where coherent contributions are significant (Andersen et al., 1997; Glyde, 1994a). The coherent ISF S(Q, t) can be expressed in a form similar to Eq. (4.12) for J(Q, s) and expanded as in (4.14) to give, e

2

1 2 µ2 s

′ where ωR = ωR + µ1 = ωR /S(Q) and µn (Q) are cumulants that can be related to the central moments of S(Q, ω) in the usual way. The DSF corresponding to S(Q, t) in Eq. (4.20) obtained by Fourier transform is,

S(Q, ω) = S˜IA (Q, ω) + S1 (Q, ω) + S2 (Q, ω)

E. Model OBDM and momentum distribution

The cumulant expansion of the IA, n∗ (s), in Eq. (4.13) is an expanson in powers of s valid for a normal liquid or for the “normal” liquid part (k ̸= 0 states) of liquid 4 He. The condensate component of n(s) is long range in s and cannot be expanded in powers of s. As is usual with BEC, we have to add in the condensate component into n∗ (s) to complete n(s). Including BEC the total n(s) is, n(s) = JIA (s) = n0 [1 + f (s)] + A1 n∗ (s)

1 i [1 + µ3 t3 + µ4 t4 ] (4.20) 3! 4!

ce

S(Q, t) = S(Q)e

′ −iωR t

′ 2 where ωd2 = (ω − ωR ) /µ2 . In the Q range 4 < Q < 12 ˚ A−1 , coherent contributions to S(Q, ω) are significant and the µn (Q) are functions of Q and oscillate with Q. The expressions for the µn (Q) are complicated. However, (4.21) represents a convenient structure to fit to experiment with the µn (Q) treated as free fitting parameters (e.g. there are three µn (Q) in Eq. (4.20)). The oscillations in the µn (Q) cease and S(Q) → 1 at Q ≃ 12 ˚ A−1 where the incoherent regime is reached. In the incoherent regime S(Q, ω) in (4.21) simplifies to ′ n Si (Q, ω) with ωR = ωR , µn = vR µ ¯2 (vR = ~Q/m), and 2 ¯3 = β¯3 = a¯3 /(λQ) µ ¯4 = α ¯ 4 + β¯4 with µ ¯2 = α ¯ 2 = ⟨kQ ⟩, µ 2 2 ¯ β4 = α ¯ 4 /(λQ) (λ = ~ /m) as in J(Q, s) of Eq. (4.14). Fig. 4.2 shows S(Q, ω) observed in liquid 4 He at Q =8˚ A−1 and 16 ˚ A−1 with a fit of Eq. (4.21) to the data (Andersen et al., 1997). The S˜IA (Q, ω) term is the dom′ inant term, is Gaussian and is centered at ωR . S1 (Q, ω) ′ is the leading FS term, is odd in (ω − ωR ) and serves ′ chiefly to shift the peak position of S(Q, ω) below ωR . −1 −1 ˚ ˚ S1 (Q, ω) is clearly smaller at Q = 16 A than at 8 A . S2 (Q, ω) is the second order term and is symmetric in ′ (ω − ωR ). S2 (Q, ω) decreases less with increasing Q because the coefficient µ ¯4 = α ¯ 4 + β¯4 contains α ¯ 4 which is part of n(k) (the leading deviation of n(k) from a Gaussian) and α ¯ 4 is independent of Q. At Q = 16 ˚ A−1 coherent effects are negligible.

an

S(Q,ω) (meV−1)

R S˜IA (Q, ω) = S(Q)[2πµ2 ]−1/2 e− 2µ2 [ ] µ3 ωd2 ˜ ′ S1 (Q, ω) = − 2 (ω − ωR ) 1 − SIA (Q, ω) 2µ2 3 [ ] µ4 ω4 S2 (Q, ω) = 1 − 2ωd2 + d S˜IA (Q, ω) 3 8µ2 3 (4.22)

Q = 8.00 Å−1 T = 2.50 K Ei = 88 meV

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(4.21)

(4.23)

where n0 = N0 /N is the condensate fraction. The corresponding 3D momentum distribution is n(k) = n0 [δ(k) + f (k)] + A1 n∗ (k)

(4.24)

The first term is the condensate contribution. The second 2 2 term, n0 f (k) = nc (k)e−k /2kc , arises from the coupling between the collective and single particle excitations via the condensate. At low k the coupling is given by nc (k) =


Page 34 of 48

an

1 mc n0 ( 2~(2π) 3 n ) k [2nB (ck) + 1] (see Eq. (3.47)). A cut-off

function, e−k /2kc has been added to limit this term to small k values (Glyde, 1994a). As above, n∗ (s) is the “normal” component arising from the k ̸= 0 states. The f (s) is obtained as the Fourier transform of f (k). The condensate terms can be added into J(Q, s) = n(s)R(Q, s) following the argument given below Eq. (4.15). Also we expect that “atoms” excited out of the k = 0 state to a final state Q will suffer the same FS interactions as those excited out of states k ̸= 0 to final states k + Q (Q ≫ k). The constant A1 is chosen so that ∫ ∫ n(k) is normalized to unity, i.e. dynL (y) = dkn(k) = n(s = 0) = n0 [1 + f (0)] + A1 = 1. The model for n(k) in Eq. (4.24) was developed and tested in early measurements (Mook, 1983; Sears et al., 1982; Sokol and Snow, 1991; Sosnick et al., 1989; Svensson, 1984; Woods and Sears, 1977). The model has been reviewed by (Silver, 1989) and on p. 367 by (Glyde, 1994a). Sometimes the term f (s) is omitted. The condensate fraction is found to be ∼ 15% larger if f (s) is omitted (Glyde et al., 2000a). Fig. 4.3 shows the OBDM n(s) of liquid 4 He at SVP obtained from fits to the observed J(Q, y) at Q = 27.5 ˚ A−1 . At small s, n(s) is ap∗ proximately Gaussian [ 1 2 given ] by the leading term in n (s), ∗ 2 n (s) = exp − 2 ⟨kQ ⟩s . Important deviations from a Gaussian arising from the higher order terms in n∗ (s) in Eq. (4.13) are, however, observed (Glyde et al., 2011a, 2000a). The total n(s), which includes the condensate term n(s) = n0 [1 + f (s)], is shown as a red line Fig. 4.3. The FS function R(Q, s) has a simple form and goes to zero at s ≃ 5˚ A. Essentially, the R(Q, s) serves to cut off J(Q, s) = n(s)R(Q, s) at s ≃ 5˚ A. This means the condensate component in n(s) is observed over a limited 2

ce

pte

dM

2

us

FIG. 4.3 The one-body density matrix (OBDM), n(s) = n0 [1 + f (s)] + A1 n∗ (s), (solid red line), of liquid 4 He at SVP obtained from fits to data (see Eq. (4.23)). In n(s), n0 is the condensate fraction and n∗ (s) arises from the k > 0 states (dashed blue line). Final State function R(Q, s) at Q = 27.5 ˚ A−1 (solid green line) is also shown. The difference between n(s) and n∗ (s) shows n0 [1 + f (s)].

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

cri pt

34

FIG. 4.4 Upper frame: Observed dynamical structure factor, J(Q, y), including the instrument resolution in liquid 4 He at SVP at low temperature in the Bose condensed phase (T < Tλ )(blue points) and in the normal liquid phase (red points) (T > Tλ ) (Tλ = 2.17 K) (from Glyde et al. (2000b)). Lower frame: As the upper frame for liquid 4 He at 24 bar (Tλ = 1.86 K)(from Diallo et al. (2012)). J(q, y) in the Bose condensed phase has an increased peak height at y = 0 arising from the term n0 R(Q, y) in J(Q, y), providing direct evidence of the condensate n0 . The dashed line shows the instrument resolution.

range of s only, a range limited by FS effects. Clearly, the condensate fraction could be observed with higher precision if measurements could be made at higher Q where FS effects are smaller and R(Q, s) extends out to larger s. From the fits to data shown in Fig. 4.3 a condensate fraction n0 = 7.25% was obtained at T = 0.05 K.

Page 35 of 48


F. Measurements of the condensate, the momentum distribution and Final State effects 12 Exp't, Current

4

PIMC, Boninsegni et al (2006) DMC, Moroni et al

10

DMC-U Moroni & Boninsegni (2004)

8

cri pt

4

(Bulk)

6

n

0

(%)

Liquid He

4

2

0

0

1

2

3

us

Temperature (K)

pte

dM

an

Fig. 4.4 shows the observed J(Q, y) in liquid He in the Bose condensed phase (temperatures T < Tλ ) and in the normal phase (T > Tλ ) (Tλ = 2.17 K at SVP and Tλ = 1.86 K at 24 bar). At T < Tλ , the condensate contribution, n0 δ(k), in the momentum distribution, n(k), in Eq. (4.24) adds a term n0 δ(y) to the IA, JIA (y) in Eq. (4.5). When JIA (y) is substituted into Eq. (4.16), the condensate contributes a term n0 R(Q, y) to J(Q, y) where R(Q, y) is the FS function. The term n0 R(Q, y) contributes intensity to J(Q, y) chiefly at y = 0 in the Bose condensed phase that is not seen in the normal phase. R(Q, y) peaks at y = 0, as shown in Fig. 4.1. The condensate term n0 R(Q, y) also contributes at values of y away from y = 0. In Fig. 4.4, the condensate fraction is clearly significantly larger at SVP than at 24 bars. To obtain n0 , the model J(Q, s) = n(s)R(Q, s), with n(s) given by Eq (4.23) and R(Q, s) by Eq. (4.17), are Fourier transformed, convoluted with the instrument resolution and fitted to the observed J(Q, y). Measurement of J(Q, y) as a function of Q is needed so that the FS function R(Q, y), which is a function of Q, can be identified and separated from the IA which is independent of Q. Fig. 4.5 shows the temperature dependence of the condensate fraction n0 (T ) obtained from fits of the model J(Q, s) to data such as shown in Fig. 4.4. At SVP, the observed n0 (T ) drops from n0 = 7.25 ± 0.75% at 0 K to zero at Tλ = 2.17 K. The n0 (T ) calculated using Monte Carlo (MC) methods (Boninsegni et al., 2006a; Ceperley, 1995; Glyde et al., 2011b; Moroni and Boninsegni, 2004; Moroni et al., 1997; Rota and Boronat, 2012) agree well with the observed n0 (T ). The diffusion MC (T = 0 K) values of Moroni et. al.1997 are especially noteworthy since they obtained very good agreement (n0 = 7 %) and appeared before the observed values shown in Fig. 4.5. The function [ ] T γ n0 (T ) = n0 (0) 1 − ( ) (4.25) Tλ

ce

can be fitted to the observed n0 (T ) with n0 (0) and γ as free parameters. Eq. (4.25) is the Bose gas result for n0 (0) = 1 and γ = 3/2. The best fit to n0 (T ) at SVP gives n0 (0) = 7.25 ± 0.75% and γ = 5.5 ± 1.0. The best fit at 24 bar gives n0 (0) = 2.3 ± 0.2% and γ = 13 ± 2 (Diallo et al., 2012). The temperature dependence of n0 (T ) observed in liquid 4 He, especially at 24 bar, is quite different from that in a Bose gas, In strongly interacting liquid 4 He, the atoms are highly localized in space by the hard core of the potential between neighbours. This localization in space means the 4 He cannot also be highly localized in k space. The localization in space, at 24 bar especially, greatly reduces n0 below the Bose gas result n0 = 1 (complete localization in k space to k = 0). In addition, because of the

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

FIG. 4.5 Upper frame: Observed condensate fraction, n0 (T ), in liquid 4 He at SVP vs. temperature (blue points). Calculated values are (Rota and Boronat, 2012), (Moroni et al., 1997), (Moroni and Boninsegni, 2004), (Boninsegni et al., 2006a) (from Glyde et al. (2011a)). Lower frame: As upper frame at pressure p = 24 bar; Observed n0 (T ) (black dots) and calculated n0 (T ) (triangles) (from Diallo et al. (2012) .

strong interaction and low value of n0 (0), increasing the temperature initially has little impact in further reducing n0 (T ) until a temperature T near Tλ are reached. Thus, at low temperature n0 (T ) changes little with T until a T near Tλ is reached, especially at 24 bar. This temperature dependence is clearly well reproduced by path integral Monte Carlo (PIMC) calculations (Diallo et al., 2012; Rota and Boronat, 2012). Fig. 4.6 shows the condensate fraction, n0 (0), at low temperature as a function of pressure. The fraction shown at 24 bar in Fig. 4.6 (Glyde et al., 2011a) is n0 = 3.0 ± 0.75% which is somewhat larger than the more accurate value n0 = 2.3 ± 0.2% at 24 bar quoted


Page 36 of 48 36

10 PIGS, Rota Boronat DMC, Moroni (2004)

24 bar 3

n*(k) (Å )

Observed (2000) Observed (present)

6

n

0

(%)

SVP

0.08

PIMC, Boninsegni (2006)

0.04

4 0.00

2

0

cri pt

8

2

4

-1

20 p (bar)

0.08

24 bar

3

FIG. 4.6 Condensate fraction, n0 , at low temperature in liquid 4 He versus pressure. The solid circles are observed values from Glyde et al. (2011a), the lines are calculated values, PIGS by Rota and Boronat (2012) and DMC by Moroni and Boninsegni (2004). At SVP (p = 0), n0 calculated by Boninsegni et al. (2006b) (triangle) and observed previously (Glyde et al., 2000b) are also shown (from Glyde et al. (2011a)).

FIG. 4.7 The 3D momentum distribution, n∗ (k), for states k > 0, Eq. (4.13), at SVP and 24 bar (from (Glyde et al., 2011a)).

us

10

k (Å )

n*(k) (Å )

0

dM

pte

ce

Observed n*(k) Gaussian Component

0.04

an

above (Diallo et al., 2012). The n0 decreases from n0 = 7.25 ± 0.75% at SVP (p = 0) to n0 = 2.8 ± 0.2% at 24 bar. The pressure dependence of n0 is clearly well reproduced by PIMC calculations. The n0 in liquid 4 He at negative pressures (Moroni and Boninsegni, 2004) and high positive pressures (Vranjes et al., 2005) has been calculated using MC methods. The spinodal density, ρS , is the density at which the liquid is mechanically unstable, ρS = 0.0160 ˚ A−3 and p = −9.2 bar. At ρS , a condensate fraction n0 = 29% is predicted. The n0 in liquid 4 He under pressure up to 300 bar has been calculated (Vranjes et al., 2005). While n0 decreases with increasing p, it does not go to zero at high pressure. Rather n0 has a small, finite value at high pressure (e.g. n0 = 0.5% at 300 bar). It would be especially interesting to measure n0 (T ) at negative pressures where n0 is much larger than the SVP value. Fig. 4.7 shows the atomic momentum distribution n∗ (k) of atoms in the k ̸= 0 states observed in liquid 4 He at SVP and at p = 24 bar. The n∗ (k) is obtained from the model n∗ (s) of Eq. (4.13) that is fitted to the observed J(Q, y) with α ¯2, α ¯ 4 and α ¯ 6 as free fitting parameters. The n∗ (k) is significantly broader at 24 bar than at SVP. This again arises from the tighter localization of the atoms in space at 24 bar. Fig. 4.8 shows the full n∗ (k) at 24 bar and its Gaussian component. The Gaussian component is obtained by omitting the α ¯ 4 s4 and α ¯ 6 s6 terms in n∗ (s) and retaining only the leading, Gaussian term, n∗ (s) = exp[−¯ α2 s2 /2]. It is clear that deviations from Gaussian behaviour are significant. The deviation arises principally from the α ¯ 4 s4 term. These deviations, which are strictly a quantum effect, are well reproduced in PIMC calculations. Fig. 4.9 shows the Final State function, R(Q, s) and

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0.00

0

2

4 -1

k (Å )

FIG. 4.8 The 3D momentum distribution n∗ (k) for k > 0, Eq. (4.13), at 24 bar. The Gaussian component is obtained by neglecting the 4 th and 6 th order cumulants in n∗ (s) (α4 = α6 = 0) (from Glyde et al. (2011a)).

R(Q, y), of liquid 4 He at pressures 24 bar and SVP (p = 0). FS effects are somewhat more significant at 24 bar since R(Q, s) is shorter ranged at 24 bar and hence R(Q, s) cuts off J(Q, s) at smaller values of s. Essentially, the magnitude of interaction increases with increasing pressure. In a gas where there is no interaction, R(Q, s) = 1 and R(Q, y) = δ(y).

1. PIMC calculations of Si (Q, ω)

Nakayama and Makri (2005) have made a most interesting calculation of Si (Q, ω) using PIMC methods. This is done by expressing Si (Q, ω) in terms of the atomic velocity correlation function, C(t), and calculating C(t) using PIMC. Specifically, by making a cumulant expansion directly of Si (Q, t) defined in (1.6) and retaining only the leading, Gaussian (Q2 ) term, Si (Q, t) can be

Page 37 of 48


Final State 0.8

Function

cri pt

R(Q,s) (

Å)

-1 Q=28 Å

0.4 P=24 bars P= SVP 0.0 0

2

4

6

s (Å)

1.2 P=24

Final State

P=SVP

Function -1 Q=28 Å

us

R(Q,y) (Å)

0.8

0.4

-0.4 -6

-4

-2

0

2

4

an

0.0

6

-1

y (Å

)

dM

FIG. 4.9 Final State function R(Q, s) given by Eq. (4.17) and its Fourier transform R(Q, y) at SVP and at 24 bar. The R(Q, s) goes to zero at smaller values of s and R(Q, y) is broader at higher pressure.

expressed as (Rahman et al., 1962), [

Si (Q, t) = e−iωR t exp −

2

Q 3

∫

t

dt′ (t − t′ )C(t′ )

]

(4.26)

0

pte

where C(t) is the velocity correlation function, C(t) =

1 ∑ ⟨ˆ vj (0).ˆ vj (t)⟩. N j

(4.27)

ce

Similar expressions for Si (Q, ω) in terms of the velocity have been used extensively by Sears. PIMC methods are usually applied to equilibrium or stationary properties such as superflow. When applied to dynamic properties, difficulties with oscillator behavior in the dynamics arise. As an approximation in calculating C(t) to avoid this problem, Nakayama and Makri retained only the classical path (classical particle trajectory) of the atoms as a function of time. However, quantum Bose symetrization, i.e. symmetrization with respect to particle interchange in their classical paths was

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

FIG. 4.10 The incoherent dynamical structure factor in liquid 4 He at SVP in the Bose condensed phase (T < Tλ ) (blue lines and circles) and in the normal liquid phase (T > Tλ )(red lines and circles) (Tλ = 2.17 K). The PIMC simulations (Nakayama and Makri, 2005) are the lines, data the circles. The vertical dashed line is the recoil energy, ωR . (a) Q = 8.0 ˚ A−1 : simulations are at T = 1.43 K, (ρ = 0.2186 ˚ A−3 ) (blue) and T = 2.5 K (ρ = 0.2179 ˚ A−3 ) (red); the data (Andersen et al., 1997) at T = 1.42 K (blue) and T = 2.5 K (red). (b) Q = 23.0 ˚ A−1 : simulations are at T = 0.35 K (ρ = 0.02210 ˚ A−3 ) (blue) and T = 3.33 K, (ρ = 0.02210) ˚ A−3 (red); data (Sosnick et al., 1989) at T = 0.35 K (blue) and T = 3.5 K (red). Figures from Nakayama and Maki (2005).

incorporated, especially long exchange paths. This approximation is said to become an exact quantum treatment in the short time (short trajectory) limit. Since, high Q and ω represents a short time limit, this method should be accurate for calculating Si (Q, ω) at high Q. Fig. 4.10 shows the calculated Si (Q, ω) at temperatures above Tλ (typically 3.5 K) and below Tλ (typi-


Page 38 of 48 38 3

(g/cm ) 0.2859

0.2502

0.2224

0.2001

0.1819

0.1668

45

0.1539

0.1429

Solid

GFMC Solid Superfluid Normal liquid

35

30

25

20

15

10 14

cri pt

Atomic Kinetic Energy (K)

PIMC Solid

40

16

18

20

22

24

26

28

3

V (cm /mole)

us

FIG. 4.11 Kinetic energy per atom in solid and liquid 4 He. The solid symbols are experimental values in the solid from Hilleke et al. (1984)(solid triangles), Celli et al. (1998) (solid circle), Blasdell et al. (1993)(solid squares) and Diallo et al. (2004)(solid star). The open circles and squares are calculations in the solid from Whitlock and Panoff (1987) and from Ceperley (1993; 1995). The open triangles and diamonds are measurements in the liquid (Glyde et al., 2011a).

an

cally 1 K) compared with experiment at two Q values. The essential feature is that Si (Q, ω) at T < Tλ shows an enhanced peak at ω ≃ ωR (i.e at y = ω − ωR ≃ 0) not seen at T > Tλ , a peak usually associated with BEC. In this formulation the enhanced peak at temperatures T < Tλ arises from including particle exchange, exchange that is necessary for superflow. If symmetrization of particles is not incorporated, the enhanced peak at T < Tλ disappears. It is facinating that particle exchange, usually associated with superflow, leads to contributions to Si (Q, ω) usually associated with BEC and indeed used to measure the condensate fraction, n0 . Specifically, when Bose particle exchange is incorporated, Si (Q, ω) shows a contribution associated with the term n0 R(Q, ω − ωR ) in Si (Q, ω) arising from BEC. Si (Q, ω) in Fig 4.10 shows other interesting features. The IA in the Gaussian limit is, from Eq. (4.21), ′ 2 Si (Q, ω)= [2πµ2 ]−1/2 e−(ω−ωR ) /2µ2 where µ2 = α2 = 2 2 2 vR α ¯ 2 = vR ⟨kQ ⟩. It is clear that Si (Q, ω) in (4.26) includes dynamics beyond the IA, i.e. Final State contributions. The FS effects shift the peak in Si (Q, ω) to lower ω below ωR . The shift arises chiefly from the asymetric FS term S1 (Q, ω) in Eq. (4.21), proportional to 3 µ3 = β3 = vR a¯3 /(λQ). Also the shape of Si (Q, ω) in the wings is different at T < Tλ and T > Tλ . This reflects the contribution of the FS function in the term n0 R(Q, ω − ωR ) of Si (Q, ω) proportional to n0 at T < Tλ .

Recent measurements of the kinetic energy of liquid He in the superfluid (BC) and normal liquid as a function of molar volume are shown in Fig. 4.11. ⟨K⟩ is lower in the BC phase because A1 = 1 − n0 (T )[1 + f (s = 0)] ≃ 1 − n0 (T ) is less than one and because α ¯ 2 (BC) < α ¯ 2 (N ). The estimate of the condensate fraction, n0 , from the kinetic energies (Sears, 1983) in the Bose condensed (BC) and normal (N) phases (given by Eqs. (4.28) and (4.29), respectively) can be obtained by substituting A1 = 1 − n0 (T )[1 + f (s = 0)] ≃ 1 − n0 (T ) in Eq. (4.28) and 2 eliminating 3~ 2m using (4.29) to obtain n0 as, 4

dM

2. Kinetic energy and the condensate fraction

The atomic kinetic energy, ⟨K⟩, in liquid 4 He can be determined from the second moment of the atomic momentum distribution, n(k), as given by Eq. (4.1). In addition, the condensate fraction can be estimated from the a measurement of the ⟨K⟩ in both the Bose condensed (T < Tλ ) phase and in the normal liquid (T > Tλ ) phase (Sears, 1983). Specifically, using the model momentum distribution Eq. (4.24) determined by fits to experiment, the ⟨K⟩ given by Eq. (4.1) in the Bose condensed (BC) phase is ⟨K⟩BC =

3~2 2m

)

∫

(

3~2 2m

)

pte

(

A1

dkn

∗

(k)kα2

=

A1 α ¯ 2 (BC)

ce

(4.28) To obtain Eq. (4.28), we have assumed that the second moment of the n0 [δ(k) + f (k)]∫ term in n(k) is negligible. In Eq. (4.28), α ¯ 2 (BC) = dkn∗ (k)kα2 is the second ∗ moment of n (k) in the BC phase. In the normal phase (T > Tλ ), where n0 = 0 and A1 = 1, Eqs. (4.1) and (4.24) give, (

⟨K⟩N =

) 3~2 α ¯ 2 (N ). 2m

(4.29)

where α ¯ 2 (N) is the second moment of n∗ (k) in the normal liquid phase.

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

n0 = 1 −

α ¯ 2 (N ) ⟨K⟩BC . α ¯ 2 (BC) ⟨K⟩N

(4.30)

This expression is usually applied with the assumption that the second moment over the k ̸= 0 states is the same in the normal (N) and Bose condensed (BC) phases, i.e. α ¯ 2 (N ) = α ¯ 2 (BC). With this assumption, Eq. (4.30) reduces to (Sears, 1983), n0 = 1 −

⟨K⟩BC . ⟨K⟩N

(4.31)

Expression (4.31) usually gives (Andreani et al., 2005; Fielding et al., 1996; Mayers, J., 1997) an n0 at SVP that is typically 30% higher than n0 = 7.25 ± 0.75 % obtained by fitting models to the observed J(Q, y). This is because the physical assumption made in using Eq (4.31) is that all the drop in ⟨K⟩ in going from the normal to Bose condensed phase arises from the onset of BEC. It is assumed

Page 39 of 48


TABLE I Atomic kinetic energy, ⟨K⟩, of H in H2 O and D in D2 O at 300 K calculated by Moreh and Neminski (2010) and of 4 He in normal liquid 4 He at T = 2.3 K observed by Glyde et.al.(2011a).The corresponding RMS momentum 2 1/2 along an axis, σ = ⟨kα ⟩ , in the momentum distribution, assumed spherically symmetric, (⟨K⟩ = (3~2 /2m)σ 2 ) is also listed. For 4 He, ~2 /m = 12.13 K ˚ A2 . ⟨K⟩ (meV) ⟨Kx ⟩ = 20.1 ⟨Ky ⟩ = 36.1 ⟨Kz ⟩ = 55.1

dM

ce

pte

The last result holds because FS effects do not contribute to the first or second moment of J(Q, y). FS effects contribute first to the third moment. (The third moment of J(Q, y) is β¯3 ). With these results, ⟨K⟩ can be obtained directly from the second moment of an observed J(Q, y), either by direct integration of data or fitting a function to data, without involving FS effects. Most of the measurements have been made on the VESUVIO instrument at the ISIS Facility, RAL. The incident neutron energy is high, 1-800 eV, the momentum transfer is variable but high and the Impulse Approximation (IA) is usually assumed. In any case the sec2 ond moment of n(k), (⟨kQ ⟩ = α ¯ 2 = σ 2 ) along an axis parallel to Q, can be extracted from the second moment of J(Q, y)(see Eq. (4.32)). Data analysis and the instrument resolution of VESUVIO has been discussed (Blostein et al., 2005). To illustrate and compare with helium, we consider H in H2 O and D in D2 O. The ⟨K⟩ in these systems has been accurately calculated (Ceriotti and Manolopoulos, 2012; Lin et al., 2010; Moreh and Nemirovsky, 2010; Morrone and Car, 2008) as well as measured. The calculations of Lin et al. (2010) are especially interesting since the OBDM has been evaluated in the form of Eqs. (4.9) and (4.10) (as observed in JIA (s)) rather than from the usual definition Eq. (4.7)). Table I lists the calculated

σ

˚ A−1 σx = 4.38 σy = 5.88 σz = 7.26

TABLE II Atomic kinetic energy, ⟨Kα ⟩ and RMS momentum σα along an axis α of D in D2 O obtained from ∑ measurements of σα (Romanelli et al., 2013), ⟨K⟩ = α ⟨Kα ⟩, 2 ⟨Kα ⟩ = (~2 /2m)σα .

an

The measurements of n(k), BEC and the kinetic energy, ⟨K⟩, in liquid 4 He reviewed above may be viewed as part of measurements of the ⟨K⟩ and n(k) of atoms in a wide range of materials. The ⟨K⟩ of hydrogen (protons)(H) and deuterons (D) in water, in many liquids and solids and biomaterials, particularly, has been investigated (Andreani et al., 2005). As noted above in Eq. (4.1), the atomic kinetic energy, ⟨K⟩, in an isotropic fluid is simply related ( to the )second 2 moment α ¯ 2 = ⟨kQ ⟩ of n(k) as ⟨K⟩ = 3~2 /2m α ¯ 2 as given by (4.1) and (4.2). The second moment can be calculated from the DSF J(Q, y) as, ∫ ∫ 2 2 α ¯ 2 = ⟨kQ ⟩ = dyJ(Q, y)y = dyJIA (y)y 2 . (4.32)

H in H2 O D in D2 O Liquid 4 He

σ ˚−1 A 4.86 5.93 0.95

(K) 1720 1278 16.3

cri pt

G. Kinetic energies in condensed Matter

⟨K⟩ (meV) 148.2 110.1

us

that n∗ (k) does not change with temperature and is the same in the N and BC phases. This assumption is weakest when n0 is small, i.e. when the drop in ⟨K⟩ arising from BEC is small. Recent measurements (Glyde et al., 2011a) have shown that α ¯ 2 (N ) > α ¯ 2 (BC). Including this difference would reduce n0 obtained from Eq. (4.31) bringing it in better agreement with the n0 = 7.25 ± 0.75 %. The error introduced by assuming α ¯ 2 (N ) = α ¯ 2 (BC) is most significant at high pressure where n0 is small. For example, use of Eq. (4.31) and the ⟨K⟩ values quoted in Fig. 4.11 (Glyde et al., 2011a) would lead to a factor of two error in n0 at 24 bar. Nonetheless, Eq. (4.31) provides a simple method to estimate n0 .

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

⟨K⟩ of H in H2 O and D in D2 O at 300 K and the cor2 responding mean square momentum σ 2 = α ¯ 2 = ⟨kQ ⟩ of n(k). The σ in Table I are calculated from the ⟨K⟩ assuming a spherically symmetric n(k), i.e. for H in H2 O (⟨K⟩ = (3~/2m)σ 2 (m = 1 amu)). The ⟨K⟩ and σ of H in H2 O is clearly much larger than that of liquid 4 He. The calculated σ in Table I agree with observed values when a spherically symmetric n(k) is also assumed, e.g. for D in D2 O, σ = 5.83 ˚ A is observed giving a ⟨K⟩ = 106 K (Giuliani et al., 2011). The n(k) of H in H2 O and D in D2 O is not, however, spherically symmetric (Andreani et al., 2016; Romanelli et al., 2013). Table II lists the observed σα and the corresponding ⟨K⟩ along separate axes of D in D2 O (Romanelli et al., 2013). The σα are all still much greater than the corresponding σ = 0.95 ˚ A in normal liquid 4 He. The total observed ⟨K⟩ at 300 K (111.3 ± 3 meV) (Romanelli et al., 2013) obtained using a non-spherically symmetric n(k) agrees well with the total calculated ⟨K⟩ in Table I. Quantum contributions to the ⟨K⟩ of H and D at room temperature are very significant. For example, assuming the classical expression, ⟨K⟩ = (3/2)kB Te , for the ⟨K⟩ of H in H2 O at 300 K would require an effective temperature of Te = 1114 K (i.e. Te >> T ). Both the calculated (Moreh and Nemirovsky, 2010) and observed (Andreani et al., 2005; Giuliani et al., 2011) ⟨K⟩ are roughly independent of temperature up to 300 K suggesting large quantum contributions. The ⟨K⟩ then increases only by 20-25 % between 300 and 673 K. Calcula-


Page 40 of 48 40

1. Landau and Superfluidity

dM

an

In this subsection we sketch three theories of superfluidity. Among the vast literature on superfluidity, these three formulations may be viewed as separate, self standing theories of superfluidity. There are many connections between them. At the same time each provides an independent physical basis for superfluidity and a means of calculating the superfluid fraction ρS /ρ without drawing on other theories. Our goal in the sketch is to connect superfluidity with the measurements of BEC and P-R modes and with the concepts reviewed above. We consider 3D, bulk liquid 4 He excluding films (2D) where the Kosterlitz-Thouless theory clearly applies and 1D liquid (4 He in nanopores) where superfluidity is strictly a finite size effect often following Luttinger Liquid theory predictions (Del Maestro and Affleck, 2010; Del Maestro et al., 2011; Giamarchi, T., 2004; Haldane, 1981; Kulchytskyy et al., 2013; Vranjeˇs Markić and Glyde, 2015). A recent review on BEC and superfluidity emphasizes the connection with P-R modes (Vilchynskyy et al., 2013).

cri pt

H. Superfluidity, modes and BEC

the walls, eventually acquires the velocity of the walls of the pipe, vn = u. The superfluid component can remain at rest, vs = 0. The fluid mass flow in the laboratory frame is J = ρn vn = ρn u. The normal liquid density is defined as ρn = dJ/du and ρs = ρ − ρn is derived from ρn . J is the liquid momentum per unit volume. In the reference frame of the pipe, the superfluid component is moving ′to the left with velocity vs = −u and in that frame J = ρs vs = −ρs u. In his first key insight, Landau proposed that superfluid 4 He supported a well-defined, collective P-R mode and only this mode. There were no single particle excitations. Intitially he proposed (1941) separate phonon and roton branches but later (1947) joined the two branches into a single, unified collective P-R mode, as discussed in section 3.A. Single particle excitations were excluded because the fluid was strongly interacting and because single particle scattering would enable transfer of mass between the independent superfluid and normal liquid components. Modern measurements of the P-R mode (section 2) and theory of P-R modes (section 3) are core topics of this review. Landau calculated the fluid momentum J arising from thermally exciting the P-R modes. With an expression for J in terms of exciting P-R modes and J = ρn u above, Landau determined ρn (T ) in terms of thermal excitation of P-R modes and ρs (T ) as ρs (T ) = ρ−ρn (T ). At T = 0 K the fluid is in its ground state (no excitations), J = 0, ρn = 0, and the fluid is entirely superfluid (ρs = ρ). It is remarkable that the Landau theory using modern values of the P-R mode energy remains a reliable empirical method of calculating the superfluid fraction ρS /ρ and thermodynamic properties (e.g. specific heat) today. Since the liquid supports only a P-R mode that has a phonon energy, ωQ = cQ, at low Q, it becomes possible to translate the superfluid at a finite velocity without creating excitations (i.e. no dissipation). Specifically the fluid can move through a barrier and impurities can move within the fluid up to a critical velocity without exciting P-R modes. The requirement that superflow at a finite velocity be possible without creating excitations is denoted the Landau criterion for superfluidity. With exclusively P-R modes, superfluid 4 He meets this criterion. In a second key insight, Landau proposed that superfluid velocity field has no rotation, ∇ x vs = 0. With this property plus one other central equation (Leggett, 2006), Landau was able to extend the classical hydrodynamic theory of fluids to explain the fascinating properties of superfluid 4 He. Indeed, the theory was so successful that any role for BEC in explaining superfluidity in a strongly interacting fluid such as liquid 4 He seemed quite unnecessary and BEC was largely forgotten for many years. From today’s vantage point, the remarkable insights made by Landau can be shown to follow from BEC. As seen in the data reviewed here, BEC has been shown experimentally to exist in liquid 4 He beyond doubt. The condensate fraction can be small. As a consequence of BEC, liquid 4 He supports only a single combined den-

us

tions (Moreh and Nemirovsky, 2010) suggest that ⟨K⟩ is dominated by the zero point kinetic energy (ZPKE), defined as 1/2 the ZPE valid for a harmonic oscillator. Thus quantum contributions to ⟨K⟩ appear to dominate at 300 K. The σ and n(k) of H in DNA (Reiter et al., 2010), in proteins (lysozyme)(Senesi et al., 2007) and in hydration water (Reiter et al., 2012) has also been observed. The n(k) in hydration water is found to be quite different from that in bulk water. This is probably because the H in hydration water on a surface is in a spectrum of sites with a spectrum of n(k), a spectrum that cannot be well represented by a single scattering center or single n(k). Measurement of the ⟨K⟩ provides direct information on the quantum nature of H and D in condensed matter.

pte

The Landau (1941; 1947) theory of superfluidity begins with the two fluid model proposed by Tisza (1938; 1940; 1947). To the two fluid model Landau added remarkable physical insights on the nature of liquid 4 He and developed a complete theory of superfluidity. In Tisza’s model, the liquid is assumed to be composed of independent superfluid and normal liquid components. The mass density, ρ, and mass current density, J, are a direct sum of the two components,

ce

ρ = ρs + ρn J = ρs vs + ρn vn

(4.33) (4.34)

The ρs and ρn are defined empirically in terms of the observed superfluid phenomena. Consider superfluid 4 He in a pipe that is moving to the right with a velocity u. The normal liquid component, which has friction with

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 41 of 48


2. Path integrals, symmetrization and superfluidity

pte

dM

cri pt

an

The superfluid fraction, ρS /ρ, can be calculated microscopically using Monte Carlo methods, particularly path integral Monte Carlo (PIMC) (Boninsegni et al., 2006a; Ceperley, 1995; Pollock and Ceperley, 1987). In this formulation, superfluidity and a finite ρS /ρ arise from quantum exchange paths of the particles introduced when the density matrix is symmetrized with respect to interchange of particles. When the exchange paths become long enough that some paths traverse the length L of the sample, then a finite superfluid fraction is obtained. A finite ρS /ρ arises from Bose symmetrization that leads to particle exchanges over long distances. In the path integral (PI) formulation above, ρS /ρ does not arise from BEC nor a coherent condensate field. In this sense the PI formulation is an independent formulation of superfluidity. Specifically, from straightforward considerations of fluid flow, as in the Landau theory, it can be shown that the increase in free energy of the fluid, ∆Fvs , when the superfluid component has a velocity vs is (Pollock and Ceperley, 1987),

At the same time, the OBDM defined in Eq. (4.7) and the BEC condensate fraction can be calculated using exactly the same PI methods. The OBDM develops BEC only after the density matrix is symmetrized and long exchange paths develop. In the PI formulation BEC, as with superflow, arises from symmetrization and long quantum particle exchanges. If the OBDM is diagonalized to reveal occupation of states, we see that long particle exchanges and macroscopic occupation of a single state (BEC) occur together. Also, as reviewed in section 4.F.1, Nakayama and Makri 2005 have shown that including particle exchange in the incoherent DSF leads to an enhanced peak in Si (Q, ω) at T < Tλ that is usually interpreted as BEC in n(k) and is used to measure the condensate fraction. That is, including particle exchanges leads to contributions to Si (Q, ω) that look exactly like BEC. Si (Q, ω) and the OBDM are consistent, the condensate contributions appear only when Bose symmetrization and long particle exchanges are included. We could denote this an exchange formulation of BEC. PIMC provides the most accurate first principles, microscopic values of ρS /ρ (Boninsegni et al., 2006a; Ceperley, 1995; Pollock and Ceperley, 1987). In addition, PIMC methods provide the most accurate values of the OBDM, the momentum distribution, n(k), and the condensate fraction (Boninsegni et al., 2006a; Ceperley, 1995; Ceperley and Pollock, 1987; Moroni and Boninsegni, 1997, 2004; Rota and Boronat, 2012) for direct comparison with the data reviewed here. These properties arise directly from quantum exchange of particles over long distances.

us

sity/single particle mode as discussed in section 3.H. There are no single particle excitations at low energy. The P-R mode is also uniquely sharply defined at low T because there are no single particle modes. In the Bose condensed phase the P-R mode can decay to other P-R modes only. This unique sharpness and the temperature dependence of the P-R mode width has been verified using neutrons (see section 2.C.2) Secondly, if there √ is BEC, there will be a Bose condensate field Ψ(r) = N0 (r)eiϕ(r) where N0 is the number in the condensate state and ϕ(r) is the phase of the condensate state. The superfluid velocity (condensate velocity) is given by vs = (~/m)∇ϕ(r) so that the curl of vs must be zero. The existence of BEC shows that the Landau insights are correct. Indeed BEC can be the physical starting point of the two fluid model and Landau theory (e.g. p. 35 (Leggett, 2006)). However, as originally formulated, Landau theory is an independent theory and indeed the first theory of superfluidity.

∆Fvs ρS 1 = mv 2 . N ρ 2 s

(4.35)

The ∆Fvs can be calculated using PIs and the ρS /ρ is given by (Pollock and Ceperley, 1987),

ce

mkB T L2−d ρS =( 2 ) ⟨W 2 ⟩, ρ ~ dρ

(4.36)

where W is the number of times an exchange path winds around the model system of dimensions d and sides of length L, W = 0, ± 1, ± 2, .. The ⟨W 2 ⟩ can be evaluated numerically using PIMC methods and a finite ρS /ρ arises from long exchange paths.

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

3. BEC and superfluidity

In this formulation, superfluidity is a consequence of BEC. A starting point is that liquid 4 He at temperatures T < Tλ has a Bose condensate, as shown conclusively in the measurements reviewed above. At T < Tλ , there is a macroscopic fraction, n0 = N0 /N , in one single particle state. We denote this state by χ0 (r) = |χ0 (r)|eiϕ(r) . We introduce the condensate field (order parameter) √ Ψ(r) = N0 χ0 (r). This condensate field has velocity vs = (~/m)∇ϕ(r), as discussed above. This formulation uses the two fluid model Eq. (4.33). A normal fluid velocity and normal fluid density are defined in terms of the observed mass flow as in the Landau theory (see p. 76 (Leggett, 2006)). The superfluid density is obtained as ρS (T ) = ρ − ρn as before. From Galilean invariance the vs is the superfluid velocity in the two fluid model. With ∇ x vs = 0 and Landau’s second key equation (see p. 84 (Leggett, 2006)) we have all the equations needed to derive the remarkable Landau theory of superfluidity. The increase in the Free Energy (when there is a superfluid component) giving Eq. (4.35) could also be calculated as before. In this formulation, it appears that an independent calculation of ρS /ρ is


Page 42 of 48 42

(4.37)

dM

an

The Green function G11 (q, t) is defined below Eq (3.35) and the limit of G11 (q, ω) needed in the Josephson relation is given by Eq. (3.44) at T = 0. Substituting Eq. (3.44) into the Josephson relation we find that ρS /ρ = 1 at T = 0 K exactly for all Bose fluids independently of the magnitude of the interaction and the value of n0 . Specifically, Eqs. (4.37) and (3.44) show that the superfluid fraction always goes to unity at T = 0 K (as proposed by Landau (1941) even though n0 can be very small at T = 0 K (e.g. 2 − 3 % in liquid 4 He near solidification). Clearly, also ρS /ρ = 0 when n0 = 0. The superfluid density can in principle be derived microscopically using linear response theory, as used to obtain the Josephson relation, This has been achieved in Bose gases at temperatures near Tλ and in some special cases. But it has proved difficult to obtain general expressions for ρS (T )/ρ in this manner.

cri pt

1 ρS = −n0 m lim 2 , q→0 q G11 (q, ω = 0) ρ

Bosons. As in other dense liquids, normal liquid 4 He supports a sound mode at low Q, of energy ωQ = cQ (Cowley:71,Stirling:90,Gibbs:99). The sound mode is observed in S(Q, ω) as a narrow, well-defined peak at low Q. At the same time, S(Q, ω) in normal liquid 4 He broadens rapidly as Q increases. At Q ≥ 1 ˚ A−1 , S(Q, ω) is broad with width comparable to its energy and cannot be interpreted as a single mode (see Figs. 2.16 and 2.17). At Q≃2˚ A, S(Q, ω) is so broad that S(Q, ω) extends to ω < 0 (see Fig. 2.17). However, the peak position of this very broad response has a P-R like energy dispersion, as shown in Fig. 2.16. This dispersion follows from the correlations in the atomic motion introduced by the repulsive core of the interatomic potential. This energy dispersion is common to a wide variety of liquids reflecting the common nature of the interatomic correlations in the liquids, as emphasized in section 2.D and Refs. (Kalman et al., 2010; Nozières, 2004; Pines, 1981). The position in Q of the minimum in ωQ at Q ≃ 2 ˚ A−1 corresponds to the maximum in S(Q) and roughly to the position of the first reciprocal lattice vector of the corresponding solid. Equally, in all these normal quantum or nearly classical liquids, S(Q, ω) is very broad at Q ≥ 1 ˚ A−1 , as in Fig. 2.16. The unique feature of Bose-condensed, superfluid 4 He is that the width, 2ΓQ , of the P-R mode goes to zero at T → 0 even at higher Q values out to the end point Q = 3.6 ˚ A−1 . At T ≤ 1K, 2ΓQ is extremely small (e.g. 2ΓQ ≤ 10−3 meV at Q ≃ 2 ˚ A−1 ). Below 1K, the width cannot be resolved with current neutron spectrometers (see section 2.C.2 ). When there is BEC the single particle and density excitations have the same energy (see sections 3.H and 3.I). In the presence of BEC there are no independent single particle excitations to which the P-R mode can decay. The P-R mode can decay only via interaction with other P-R modes. At most Q values this decay requires a thermal P-R mode and hence 2ΓQ → 0 as T → 0. At very low Q, where there is upward dispersion in ωQ (see section 2.A.4), a temperature independent three phonon decay process becomes possible which introduces a small temperature independent width at low Q (Mezei and Stirling, 1983). There is a consensus (see sections 3.H, 3.I and 3.K.1) that BEC is responsible for the uniquely narrow P-R mode in superfluid 4 He at T ≤ Tλ . In contrast in solid helium, longitudinal phonons can decay to other phonons via a temperature independent three phonon process. In normal liquid 3 He, the zero sound mode can decay to particle-hole states. In these two systems the mode widths become independent of temperature and remain of the order of 1 meV at T → 0. In Bose-condensed liquid 4 He at T ≤ Tλ , P-R modes decay and suffer energy shifts via processes involving other P-R modes, as introduced originally by Landau and Khalatnikov and extended recently by Fak et al. (2012) (see section 2.C.2). These processes do not explicity involve BEC. In contrast, the temperature depen-

us

needed using, for example, linear response theory as discussed below. In addition, since there is BEC, we know from the work reviewed above in section 3.I that there will be only a single combined density/single particle P-R mode. The mode has sound dispersion ωQ = cQ at low Q (as shown in section 3.J. Thus as a consequence of BEC, the liquid fulfils the Landau criterion for superfluidity. The liquid can be translated at a finite velocity without exciting the characteristic modes of the fluid. We may also independently connect the superfluid fraction ρS /ρ and the condensate fraction, n0 = N0 /N , using the Josephson relation and some exact results from section 3.J. The Josephson relation (Baym, 1968; Holzmann and Baym, 2007; Josephson, 1966; M¨ uller, 2015) is,

5. CONCLUDING REMARKS AND FUTURE OPPORTUNITIES

ce

pte

In this section we make some concluding remarks on the P-R mode and the role of BEC in establishing a welldefined P-R mode in the superfluid phase. We summarize measurements of BEC and atomic momentum distributions and comment on the role of the P-R mode and BEC in establishing superfluidity. We also identify some opportunities for advancement of the field.

A. Concluding remarks 1. The P-R mode

We remark firstly on the origin and interpretation of the P-R mode beginning with normal liquid 4 He. Liquid 4 He is a dense quantum liquid of strongly interacting

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 43 of 48


dence of the weight of the P-R mode in S(Q, ω) appears to be controlled largely by the condensate fraction as suggested by Field theory methods (see section 3.H) and models discussed in section 3.K.1.

shown to follow from the coherence created in the fluid by BEC. This offers a consistent approach to superfluidity which is reviewed in section 4.H.3. Particularly, the consistent formulation by Leggett (2006) in which both superfluidity and superconductivity follow directly from BEC is very appealing.

B. Some future opportunities

us

The P-R mode energy, ωQ , in superfluid 4 He at SVP shows anomalous or upward dispersion at low Q (see section 2.A.4). Based on thermodynamic properties, this upward dispersion is predicted (Sridhar, 1987) to cease at p ≃ 15 - 20 bar. It would be interesting to measure ωQ directly at 0.1 ≤ Q ≤ 0.8 ˚ A−1 at higher pressure, as has been done at SVP (Stirling, 1983), to determine whether anomalous dispersion persists or not at higher pressure. It would be equally interesting to determine at what Q ZQ first falls below S(Q). Donnelly et al. (1981) produced numerical, interpolated values of ωQ in superfluid 4 He at SVP based on the measurements of Cowley and Woods (1971). These numerical ωQ have been widely used. It would be useful to produce revised numerical, interpolated values of ωQ at SVP and 20 bar drawing on much new data obtained since 1981. In section 2.C.3 and in Figs 2.13 to 2.15, we saw that the temperature dependence of ωQ , ΓQ , and ZQ at T ≥ 1.7 K are not uniquely determined. Specifically, the ωQ , ΓQ , and ZQ depend on the model used to analyse the data (see section 2.C.3). Quite different values of ΓQ and ZQ are obtained using the WS and SS models. Improved models are needed which enable more model independent values of ωQ , ΓQ , and ZQ to be obtained as a test of existing theory. In section 3.K.1, a model of S(Q, ω) that illustrated how the weight of the P-R mode in S(Q, ω) depended on the condensate fraction was reviewed. This is a complicated model. A simpler model with fewer parameters would be a step forward. In section 2.D and Fig. 2.17 we saw that the peak position of S(Q, ω) at Q ≃ 2 ˚ A−1 in normal liquid 4 He moved from ω ≃ 0.55 meV at SVP toward ω ≃ 0 at 48.6 bar. In nearly classical liquids such liquid Ne and Ar, S(Q, ω) peaks near ω = 0. As pressure increases normal liquid 4 He appears to move toward becoming a semi-classical liquid. It would be interesting to demonstrate/reproduce this movement of S(Q, ω) from that of a normal quantum to nearly classical liquid using PIMC. A PIMC calculation of S(Q, ω) using imaginary time and Laplace transform should provide sufficient precision because the temperature is relatively high and a broad S(Q, ω) is anticipated. In the measurement of BEC and n∗ (k), the Final State (FS) function remains the least accurately determined component. Progress could proceed by more precise measurement of the FS function at existing Q values or by

ce

pte

dM

an

In section 4 we reviewed measurements of BEC, the atomic momentum distribution, n∗ (k), and final state (FS) effects. BEC definitely exists in superfluid 4 He at T ≤ Tλ . At SVP (p ≃ 0) and T → 0, the condensate fraction is n0 = 7.25 ± 0.75 % with generous error bars. This value is confirmed in more than one experiment and in PIMC and DMC calculations (see section 4.F and Fig. 4.6). The n0 drops to n0 = 2.8 ± 0.20 % at p = 24 bars near the solidification line. In the solid phase, n0 = 0.0 ± 0.30 % (Diallo et al., 2012). Essentially, the OBDM ˆ † (s)Ψ(0)⟩, ˆ n(s) = (1/n)ρ1 (s) ≡ ⟨Ψ defined in Eq. (4.7) and written in the model form n(s) = n0 [1 + f (s)] + A1 n∗ (s) (see Eq. (4.23)), is determined by fits to data. The observed atomic momentum distribution, n∗ (k), shows clear deviations from a Gaussian with the fourth and sixth cumulants of n∗ (k) well determined from data. The FS function R(Q, y) determined from data is generally broader in y than the calculated R(Q, y) (see Fig. 4.1), although use of the calculated R(Q, y) of Cararro and Koonin (Carraro and Koonin, 1990) leads to the same values of n0 . The FS function appears to be the least well determined component at present but changes in it are not expected to change the observed values of n0 significantly. Neutron scattering remains to date the only technique available to observe n(k) and BEC. At the end of section 4, theories of superfluidity were reviewed to illustrate the connection of superfluidity to the P-R mode and BEC. There is not a single view on the origin of superfluidity. For example, in the original Landau theory superflow follows from the existence of the P-R mode and the absence of single particle modes (see section 4.H.1). At a more microscopic level, as noted above, the existence of only a P-R mode and no independent single particle modes can be shown to follow from BEC. However, as originally formulated, the Landau theory is a complete theory of superfluidity that does not draw on BEC. In contrast in the microscopic path integral (PI) formulation of liquid helium, superfluidity and BEC are two properties of the liquid that follow from long-range quantum exchanges of particles in the fluid (section 4.H.2). With the onset of long range exchanges, there is onset of long range order in the OBDM (BEC) and onset of a finite superfluid fraction. PIMC provides the most fundamental calculation of both the condensate fraction and the superfluid fraction. In the PI formulation, superflow does not follow from BEC; rather the two are co-existing properties. Thirdly, given that BEC is a clearly observed property of liquid 4 He, as seen in section 4.F, superflow can be

cri pt

2. BEC, the momentum distribution and final state effects

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Page 44 of 48 44

6. ACKNOWLEDGMENTS

pte

dM

It is a pleasure to acknowledge valuable discussions and productive collaborations with D. Abernathy, F. Albergamo, K. H. Andersen, P. Averbuch, R. T. Azuah, G. Baym, J. Boronat, J. Bossy, D. M. Ceperley, D. R. Daughton, S. O. Diallo, J. DuBois, B. F˚ ak, M. Holzmann, O. Kirichek, N. Mulders, P. Nozières, J. Ollivier, J. V. Pearce, O. Plantevin, R. Rota, A. R. Sakhel, H. Schober, A. A. Shams, W. G. Stirling, J. Taniguchi, J. W. Taylor, L. Vranjeˇs Markić, Z. Zuhrianda. I am indebted to B. F˚ ak, E. Krotscheck and for sending figures and to Z. Zuhrianda for assistance in calculations and in figure and text preparation for the manuscript. The hospitality of the Institut Laue Langevin, Grenoble is gratefully acknowledged where much of this review was written. This work was supported by the US DOE, Office of Basic Energy Sciences under contract No. ER46680.

References

Albergamo, F., J. Bossy, P. Averbuch, H. Schober, and H. R. Glyde, 2004, Phys. Rev. Lett. 92, 235301. Allen, J. F., and H. Jones, 1938, Nature 141, 243. Allen, J. F., and A. Misener, 1938, Nature 141, 75. Andersen, K. H., J. Bossy, J. C. Cook, O. G. Randl, and J.-L. Ragazzoni, 1996, Phys. Rev. Lett. 77, 4043. Andersen, K. H., and W. G. Stirling, 1994, J. Phys. Condens. Mat. 6, 5805. Andersen, K. H., W. G. Stirling, and H. R. Glyde, 1997, Phys. Rev. B 56, 8978.

ce

us

cri pt

Andersen, K. H., W. G. Stirling, H. R. Glyde, R. T. Azuah, S. M. Bennington, A. D. Taylor, Z. A. Bowden, and I. Bailey, 1994a, Physica B 197, 198. Andersen, K. H., W. G. Stirling, R. Scherm, A. Stunault, B. F˚ ak, H. Godfrin, and A. J. Dianoux, 1994b, J. Phys. Condens. Mat. 6, 821. Andreani, C., D. Colognesi, J. Mayers, G. F. Reiter, and R. Senesi, 2005, Adv. Phys. 54, 377. Andreani, C., G. Romanelli, and R. Senesi, 2016, J. Phys. Chem. Lett. 7(12), 2216. Azuah, R. T., S. O. Diallo, M. A. Adams, O. Kirichek, and H. R. Glyde, 2013, Phys. Rev. B 88, 024510. Balibar, S., 2007, J. Low Temp. Phys. 146(5-6), 441. Baym, G., 1968, in Mathematical Methods In Solid State And Superfluid Theory, edited by R. C. Clarke and G. H. Derrick (Oliver and Boyd, Edinburgh), chapter 3, pp. 121–156. Baym, G., D. H. Beck, and C. J. Pethick, 2013, Phys. Rev. B 88, 014512. Baym, G., D. H. Beck, and C. J. Pethick, 2015a, J. Low Temp. Phys. 178, 200. Baym, G., D. H. Beck, and C. J. Pethick, 2015b, Phys. Rev. B 92, 024504. Beauvois, K., C. E. Campbell, J. Dawidowski, B. F˚ ak, H. Godfrin, E. Krotscheck, H.-J. Lauter, T. Lichtenegger, J. Ollivier, and A. Sultan, 2016, Phys. Rev. B 94, 024504. Bedell, K., D. Pines, and I. Fomin, 1982, J. Low Temp. Phys. 48, 417. Bedell, K., D. Pines, and A. Zawadowski, 1984, Phys. Rev. B 29, 102. Bertaina, G., M. Motta, M. Rossi, E. Vitali, and D. E. Galli, 2016, Phys. Rev. Lett. 116, 135302. Bijl, A., 1940, Physica 7(9), 869. Blasdell, R. C., D. M. Ceperley, and R. O. Simmons, 1993, Z. Naturforsch. A 48, 433. Blostein, J. J., J. Dawidowski, and J. R. Granada, 2005, Phys. Rev. B 71, 054105. Bogoliubov, N. N., 1947, J. Phys. U.S.S.R. 11, 23. Boninsegni, M., N. V. Prokof’ev, and B. V. Svistunov, 2006a, Phys. Rev. Lett. 96, 070601. Boninsegni, M., N. V. Prokof’ev, and B. V. Svistunov, 2006b, Phys. Rev. E 74, 036701. Boronat, J., and J. Casulleras, 1997, Europhys. Lett. 38, 291. Boronat, J., J. J. Casulleras, F. Dalfovo, S. Moroni, and S. Stringari, 1995, Phys. Rev. B 52(2), 1236. Bose, S. N., 1924, Z. Phys. 26, 178. Bossy, J., K. H. Andersen, J. C. Cook, and O. R. Randl, 2000, Physica B 284, 27. Bossy, J., J. Ollivier, H. Schober, and H. R. Glyde, 2012a, Europhys. Lett. 98, 56008. Bossy, J., J. Ollivier, H. Schober, and H. R. Glyde, 2012b, Phys. Rev. B 86, 224503. Bossy, J., J. V. Pearce, H. Schober, and H. R. Glyde, 2008a, Phys. Rev. Lett. 101, 025301. Bossy, J., J. V. Pearce, H. Schober, and H. R. Glyde, 2008b, Phys. Rev. B 78, 224507. Bossy, J., H. Schober, and H. R. Glyde, 2015, Phys. Rev. B 91, 094201. Bryan, M. S., T. R. Prisk, R. T. Azuah, W. G. Stirling, and P. E. Sokol, 2016, Europhys. Lett. 115(6), 66001. Buyers, W. J. L., V. F. Sears, P. A. Lonngi, and D. A. Lonngi, 1975, Phys. Rev. A 11, 697. Cabrillo, C., F. J. Bermejo, M. Alvarez, P. Verkerk, A. MairaVidal, S. M. Bennington, and D. Mart´ın, 2002, Phys. Rev. Lett. 89, 075508.

an

measurements at higher Q where FS effects are smaller. Measurement a higher Q with the required energy resolution requires new neutron scattering instruments. Higher statistical precision is possible on existing instruments. However, the values of the condensate fraction and shape of the n∗ (k) reviewed here are not likely to change much with improved precision. A reliable calculated FS function is also a path to improvement. In section 4.F.1, we reviewed a PIMC calculation (Nakayama and Makri, 2005) of S(Q, ω) at high Q and ω. PIMC determination of S(Q, ω) at high Q and ω is possible since S(Q, t) at short t only is needed. It would be interesting to pursue these calculations as a method of extracting the condensate fraction and FS effects from PIMC. That is, the PIMC S(Q, ω) clearly contains condensate and FS contributions. In addition, a calculation of the OBDM provides J(Q, s) in the Impulse Approximation. A comparison of the full J(Q, s)(S(Q, t)) with the IA might offer an alternative method of obtaining the FS function. Essentially, the ability to calculate functions at short t or short s by PIMC may offer an opportunity to obtain improved understanding of the appearance of BEC and FS contributions in S(Q, ω). A major (blue sky) advance would be to observe the phase of the condensate.

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 45 of 48


us

cri pt

F˚ ak, B., and J. Bossy, 1998b, J. Low Temp. Phys. 113(3 - 4), 531. F˚ ak, B., T. Keller, M. E. Zhitomirsky, and A. L. Chernyshev, 2012, Phys. Rev. Lett. 109, 155305. F˚ ak, B., L. P. Regnault, and J. Bossy, 1992, J. Low Temp. Phys. 89, 345. Farhi, E., B. Fak, C. M. E. Zeyen, and J. Kulda, 2001, Physica B 297, 32. Feenberg, E., 1969, Theory of Quantum Fluids (Academic Press, New York). Fernandez-Alonso, F., and D. L. Price, 2013, Neutron Scattering, volume 44 (Academic Press). Feynman, R. P., 1954, Phys. Rev. 94, 262. Feynman, R. P., and M. Cohen, 1956, Phys. Rev. 102, 1189. Fielding, A. L., D. N. Timms, A. C. Evans, and J. Mayers, 1996, J. Phys. Condens. Mat. 8(38), 7205. Gavoret, J., and P. Nozières, 1964, Ann. Phys. 28, 349. Gersch, H. A., and L. J. Rodriquez, 1973, Phys. Rev. A 8, 905. Giamarchi, T., 2004, Quantum Physics in One Dimension (Oxford Univerrsity Press, Oxford). Gibbs, M. R., K. H. Andersen, W. G. Stirling, and H. Schober, 1999, J. Phys. Condens. Mat. 11, 603. Ginzburg, V. L., 2009, On superconductivity and superfluidity (Springer-Verlag, Berlin Heidelberg). Giuliani, A., F. Bruni, M. A. Ricci, and M. A. Adams, 2011, Phys. Rev. Lett. 106, 255502. Glyde, H. R., 1984, Condensed Matter Research Using Neutrons (Plenum Press, New York), chapter 5, NATO ASI Series B, p. 95. Glyde, H. R., 1992, Phys. Rev. B 45, 7321. Glyde, H. R., 1994a, Excitations in Liquid and Solid Helium (Oxford University Press, Oxford). Glyde, H. R., 1994b, Phys. Rev. B 50, 6726. Glyde, H. R., 1995, Phys. Rev. Lett. 75, 4238. Glyde, H. R., 2013, J. Low Temp. Phys. 172, 364. Glyde, H. R., R. T. Azuah, and W. G. Stirling, 2000b, Phys. Rev. B 62, 14337. Glyde, H. R., S. O. Diallo, R. T. Azuah, O. Kirichek, and J. W. Taylor, 2011a, Phys. Rev. B 83, 100507. Glyde, H. R., S. O. Diallo, R. T. Azuah, O. Kirichek, and J. W. Taylor, 2011b, Phys. Rev. B 84, 184506. Glyde, H. R., M. R. Gibbs, W. G. Stirling, and M. A. Adams, 1998, Europhys. Lett. 43, 422. Glyde, H. R., O. Plantevin, B. Fak, G. Coddens, P. S. Danielson, and H. Schober, 2000a, Phys. Rev. Lett. 84, 2646. Glyde, H. R., and W. G. Stirling, 1990, in Phonons 89, edited by S. Hunklinger, W. Ludwig, and G. Weiss (World Scientific, Singapore), volume 2 of Proceedings of the Third International Conference on Phonon Physics and the Sixth International Conference on Phonon Scattering in Condensed Matter. Glyde, H. R., and E. C. Svensson, 1987, in Methods of Experimental Physics: Neutron Scattering, edited by D. L. Price and K. Sk¨ old (Academic Press), volume 23 of Methods of Experimental Physics, chapter 13, pp. 303 – 403. Godfrin, H., M. Meschke, H. J. Lauter, A. Sultan, H. M. B¨ ohm, E. Krotscheck, and M. Panholzer, 2012, Nature 483, 576. Golub, R., D. Richardson, and K. Lamoreaux, 1991, UltaCold Neutrons (Taylor and Francis Group, Oxford). Gordillo, M. C., and J. Boronat, 2016, Phys. Rev. Lett. 116, 145301. Griffin, A., 1993, Excitations in a Bose-Condensed Liquid

ce

pte

dM

an

Campbell, C. E., 1973, Phys. Lett. A 44, 471. Campbell, C. E., 1978, in Progress in Liquid Physics, edited by C. A. Croxton (Wiley, New York), chapter 6. Campbell, C. E., and E. Krotscheck, 2009, Phys. Rev. B 80(17), 174501. Campbell, C. E., E. Krotscheck, and T. Lichtenegger, 2015, Phys. Rev. B 91, 184510. Campbell, C. E., and F. J. Pinski, 1979, Nucl. Phys. A 328, 210. Carraro, C., and S. E. Koonin, 1990, Phys. Rev. Lett. 65, 2792. Celli, M., M. Zoppi, and J. Mayers, 1998, Phys. Rev. B 58(1), 242. Ceperley, D. M., 1995, Rev. Mod. Phys. 67, 279. Ceperley, D. M., and B. J. Alder, 1984, J. Chem. Phys. 81, 5833. Ceperley, D. M., and E. L. Pollock, 1987, Can. J. Phys. 65, 1416. Ceriotti, M., and D. E. Manolopoulos, 2012, Phys. Rev. Lett. 109, 100604. Chester, G. V., 1975, in The helium liquids: proceedings of the fifteenth Scottish Universities Summer School in Physics, 1974, edited by J. Armitage and I. Farquhar (Academic Press, London), NATO Advanced Study Institute Series, p. 1. Clements, B. E., H. Godfrin, E. Krotscheck, H. J. Lauter, P. Leiderer, V. Passiouk, and C. J. Tymczak, 1996, Phys. Rev. B 53, 12242. Cowley, R. A., and A. D. B. Woods, 1968, Phys. Rev. Lett. 21, 787. Cowley, R. A., and A. D. B. Woods, 1971, Can. J. Phys. 49, 177. Daunt, J. G., and K. Mendelssohn, 1938a, Nature 141, 911. Daunt, J. G., and K. Mendelssohn, 1938b, Nature 142, 475. Del Maestro, A., and I. Affleck, 2010, Phys. Rev. B 82, 060515. Del Maestro, A., M. Boninsegni, and I. Affleck, 2011, Phys. Rev. Lett. 106, 105303. van Delft, D., 2008, Physics Today 61(3), 36. van Delft, D., and P. Kes, 2010, Physics Today 63(9), 38. Diallo, S. O., R. T. Azuah, D. L. Abernathy, R. Rota, J. Boronat, and H. R. Glyde, 2012, Phys. Rev. B 85, 140505. Diallo, S. O., R. T. Azuah, D. L. Abernathy, J. Taniguchi, M. Suzuki, J. Bossy, N. Mulders, and H. R. Glyde, 2014, Phys. Rev. Lett. 113(21), 215302. Diallo, S. O., J. V. Pearce, R. T. Azuah, F. Albergamo, and H. R. Glyde, 2006, Phys. Rev. B 74, 144503. Diallo, S. O., J. V. Pearce, R. T. Azuah, and H. R. Glyde, 2004, Phys. Rev. Lett. 93, 075301. Donnelly, R. J., J. A. Donnelly, and R. N. Hills, 1981, J. Low Temp. Phys. 44, 471. Dubbers, D., and M. G. Schmidt, 2011, Rev. Mod. Phys. 83, 1111. Dupuis, N., and K. Sengupta, 2007, Europhys. Lett. 80(5), 50007. Einstein, A., 1924, Sitzungsber. Kgl. Preuss. Akad. Wiss., Phys. Math. Kl. 261. Fabrocini, A., S. Fantoni, and E. Krotscheck, 2002, Introduction to Modern Methods of Quantum Many-Body Theory and their Applications, Advances in Quantum Many-Body Theory, volume 7 (World Scientific, Singapore). F˚ ak, B., and K. H. Andersen, 1991, Phys. Lett. A 160, 468. F˚ ak, B., and J. Bossy, 1998a, J. Low Temp. Phys. 112, 1.

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Page 46 of 48 46

us

cri pt

densed Matter, volume 1 (Clarendon Press, Oxford). Ma, S. K., and C. W. Woo, 1967, Phys. Rev. 159, 165. Mahan, G. D., 1990, Many Particle Physics (Plenum, New York), 2nd edition. Manousakis, E., and V. R. Pandharipande, 1984, Phys. Rev. B30, 5062. Manousakis, E., and V. R. Pandharipande, 1986, Phys. Rev. B 33, 150. Maris, H. J., 1973, Phys. Rev. A 8, 2629. Maris, H. J., 1977, Rev. Mod. Phys. 49, 341. Maris, H. J., and W. E. Massey, 1970, Phys. Rev. Lett. 25, 220. Marshall, W., and S. W. Lovesey, 1971, Theory of Thermal Neutron Scattering (Oxford University Press, Oxford). Mayers, J., 1997, J. Low Temp. Phys. 109, 135. Mazzanti, F., J. Boronat, and A. Polls, 1996, Phys. Rev. B 53, 5661. Mezei, F., 1980, Phys. Rev. Lett. 44, 1601. Mezei, F., and W. G. Stirling, 1983, in 75th Jubilee Conference on Helium-4, edited by J. G. M. Armitage (World Scientific, Singapore), p. 111. Miller, A., D. Pines, and P. Nozières, 1962, Phys. Rev. 127, 1452. Mineev, V. P., 1980, Pis’ma Zh. Eksp. Teor. Fiz. 32, 509. Mook, H. A., 1974, Phys. Rev. Lett. 32, 1167. Mook, H. A., 1983, Phys. Rev. Lett. 51, 1454. Mook, H. A., R. Scherm, and M. K. Wilkinson, 1972, Phys. Rev. A 6, 2268. Moreh, R., and D. Nemirovsky, 2010, J. Chem. Phys. 133(8), 084506. Moroni, S., and M. Boninsegni, 1997, Europhys. Lett. 40, 287. Moroni, S., and M. Boninsegni, 2004, J. Low Temp. Phys. 136, 129. Moroni, S., D. Galli, S. Fantoni, and L. Reatto, 1998a, Phys. Rev. B 58(2), 909. Moroni, S., G. Senatore, and S. Fantoni, 1997, Phys. Rev. B 55, 1040. Morrone, J. A., and R. Car, 2008, Phys. Rev. Lett. 101, 017801. M¨ uller, C. A., 2015, Phys. Rev. A 91, 023602. Nakayama, A., and N. Makri, 2005, Proc. Nat. Acad. Sci. U.S.A. 102(12), 4230. Nepomnyashchy, A. A., and Y. A. Nepomnyashchy, 1975, Sov. Phys. JETP Lett. 21, 1. Nepomnyashchy, Y. A., 1992, Phys. Rev. B 46, 6611. Szwabi´ nski, J., and M. Weyrauch, 2001, Phys. Rev. B 64, 184512. Nozières, P., 2004, J. Low Temp. Phys. 137(1), 45. Nozières, P., and D. Pines, 1990, Theory of Quantum Liquids: Superfluid Bose Liquids, volume 2 (Westview Press (Perseus Books Group), Redwood City, CA). Onnes, H. K., 1911, Proc. R. Netherlands Acad. Arts Sci. (KNAW) 13, 1274. Palevsky, H., K. Otnes, K. E. Larsson, R. Pauli, and R. Stedman, 1957, Phys. Rev. 108, 1346. ´ A., S. V. Mashkevich, and S. I. Vil’chinskii, Pashitski, E. 2002b, Phys. Rev. Lett. 89, 075301. ´ A., S. I. Vil’chinskii, and A. V. Chumachenko, Pashitski, E. 2010, Low Temperature Physics 36(7), 576. ´ A., S. I. Vil’chinskii, and S. V. Mashkevich, Pashitski, E. 2002a, Low Temperature Physics 28(2), 79. Pearce, J. V., R. T. Azuah, B. F˚ ak, A. R. Sakhel, H. R. Glyde,

ce

pte

dM

an

(Cambridge University Press, Cambridge). Griffin, A., and T. H. Cheung, 1973, Phys. Rev. A 7, 2086. Griffin, A., D. Snoke, and S. Stringari (eds.), 1995, BoseEinstein Condensation (Cambridge University Press, Cambridge, England). Grigoriev, P. D., O. Zimmer, A. D. Grigoriev, and T. Ziman, 2016, Phys. Rev. C 94, 025504. Haldane, F. D. M., 1981, Phys. Rev. Lett. 47, 1840. Henshaw, D. G., 1958, Phys. Rev. Lett. 1, 127. Hilleke, R. O., P. Chaddah, R. O. Simmons, D. L. Price, and S. K. Sinha, 1984, Phys. Rev. Lett. 52(10), 847. Hohenberg, P. C., and P. C. Martin, 1964, Phys. Rev. Lett. 12, 69. Hohenberg, P. C., and P. C. Martin, 1965, Ann. Phys. 34, 291. Holzmann, M., and G. Baym, 2007, Phys. Rev. B 76, 092502. Hugenholtz, N., and D. Pines, 1959, Phys. Rev. 116, 489. Imambekov, A., T. L. Schmidt, and L. I. Glazman, 2012, Rev. Mod. Phys. 84, 1253. Jackson, H. W., 1973, Phys. Rev. A 8, 1529. Jackson, H. W., 1974a, Phys. Rev. A 10, 278. Jackson, H. W., 1974b, Phys. Rev. A 9, 964. Jackson, H. W., and E. Feenberg, 1961, Ann. Phys. 15, 266. Jackson, H. W., and E. Feenberg, 1962, Rev. Mod. Phys. 34, 686. Josephson, B. D., 1966, Phys. Lett. 21, 608. Kalman, G. J., P. Hartmann, K. I. Golden, A. Filinov, and Z. Donko, 2010, Europhys. Lett. 90(5). Kamerlingh Onnes, H., 1908, Proc. R. Netherlands Acad. Arts Sci. (KNAW) 10, 445. Kapitza, P., 1938, Nature 141, 74. Keesom, W. H., 1942, Helium (Elsevier, Amsterdam). Keesom, W. H., and K. Clusius, 1932, in Proc. R. Netherlands Acad. Arts Sci. (KNAW), volume 35, p. 307. Keesom, W. H., and A. P. Keesom, 1932, in Proc. R. Acad. Amsterdam, volume 31, p. 90. Khalatnikov, I. M., 1965, An Introduction to the Theory of Superfluidity (Benjamin, New York). Kikoin, A. K., and B. G. Lasarew, 1938, Nature 141, 912. Krotscheck, E., 1985, Phys. Rev. B 31, 4258. Krotscheck, E., and T. Lichtenegger, 2015, J. Low Temp. Phys. 178(1), 61. Kulchytskyy, B., G. Gervais, and A. Del Maestro, 2013, Phys. Rev. B 88, 064512. Landau, L. D., 1941, J. Phys. U.S.S.R. 5, 71. Landau, L. D., 1947, J. Phys. U.S.S.R. 11, 91. Landau, L. D., and I. M. Khalatnikov, 1949, Zh. Eksp. Teor. Fiz. 19, 637. Lee, D. K., and F. J. Lee, 1975, Phys. Rev. B 11, 4318. Leggett, A. J., 2006, Quantum Liquids: Bose Condensation and Cooper Pairing in Condensed Matter Systems (Oxford University Press, Oxford,). Lin, L., J. A. Morrone, R. Car, and M. Parrinello, 2010, Phys. Rev. Lett. 105, 110602. Lin-Liu, Y. R., and C. W. Woo, 1973, J. Low Temp. Phys. 14, 317. London, F., 1938, Nature 141, 643. London, F., 1950, Superfluids Vol. 1: Macroscopic Theory of Superfluid Superconductivity, volume 1 of Structure of Matter (Wiley, New York). London, F., 1954, Superfluids Vol. 2: Macroscopic Theory of Superfluid Helium, volume 2 of Structure of Matter (Wiley, New York). Lovesey, S. W., 1984, Theory of Neutron Scattering from con-

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 47 of 48


us

cri pt

Sk¨ old, K., J. M. Rowe, G. Ostrowski, and P. D. Randolph, 1972, Phys. Rev. A 6, 1107. Smith, A. J., R. A. Cowley, A. D. B. Woods, W. G. Stirling, and P. Martel, 1997, J. Phys. C: Solid State 10, 543. Snow, W. M., and P. E. Sokol, 1990, J. Low Temp. Phys. 80, 197. Snow, W. M., and P. E. Sokol, 1995, J. Low Temp. Phys. 101(5), 881. Snow, W. M., Y. Wang, and P. E. Sokol, 1992, Europhys. Lett. 19, 403. Sokol, P. E., 1995, in Bose Einstein Condensation, edited by A. Griffin, D. Snoke, and S. Stringari (Cambridge University Press, Cambridge), p. 51. Sokol, P. E., and W. M. Snow, 1991, in Excitations in Two-Dimensional and Three-Dimensional Quantum Fluids, edited by A. Wyatt and H. J. Lauter (Plenum Press, New York), volume 257 of NATO Advanced Study Institute, Series B: Physics, p. 47. Sosnick, T. R., W. M. Snow, R. N. Silver, and P. E. Sokol, 1991, Phys. Rev. B 43, 216. Sosnick, T. R., W. M. Snow, and P. E. Sokol, 1990a, Phys. Rev. B 41, 11185. Sosnick, T. R., W. M. Snow, P. E. Sokol, and R. N. Silver, 1989, Europhys. Lett. 9, 707. Sosnick, T. R., W. M. Snow, P. E. Sokol, and R. N. Silver, 1990b, Europhys. Lett. 9, 707. Squires, J. L., 1978, Introduction to the Theory of Thermal Neutron Scattering (Cambridge University Press, Cambridge). Sridhar, R., 1987, Physics Reports 146(5), 259 , ISSN 03701573. Stirling, W. G., 1983, in 75th Jubilee Conference on Helium-4, edited by J. G. M. Armitage (World Scientific, Singapore), p. 109. Stirling, W. G., 1991, in Excitations in Two-Dimensional and Three-Dimensional Quantum Fluids, edited by A. Wyatt and H. J. Lauter (Plenum Press, New York), volume 257 of NATO Advanced Study Institute, Series B: Physics, p. 25. Stirling, W. G., and H. R. Glyde, 1990, Phys. Rev. B 41, 4224. Stringari, S., 1992, Phys. Rev. B 46(5), 2974. Svensson, E. C., 1984, in Proceedings of the 1984 Workshop on High-Energy Excitations in Condensed Matter, Los Alamos National Laboratory (LA-10227-C), volume 2, p. 456. Svensson, E. C., V. F. Sears, A. D. B. Woods, and P. Martel, 1980, Phys. Rev. B 21, 3638. Szépfalusy, P., and I. Kondor, 1974, Ann. Phys. 82, 1. Talbot, E. F., H. R. Glyde, W. G. Stirling, and E. C. Svensson, 1988, Phys. Rev. B 38, 11229. Taniguchi, J., R. Fujii, and M. Suzuki, 2011, Phys. Rev. B 84(13), 134511. Tilley, D. R., and J. Tilley, 1990, Superfluidity and Superconductivity (IOP Publishing Ltd), 3rd edition edition. Tisza, L., 1938, Nature 141, 913. Tisza, L., 1940, J. Phys. Radium 1, 350. Tisza, L., 1947, Phys. Rev. 72, 838. Vilchynskyy, S. I., A. I. Yakimenko, K. O. Isaieva, and A. V. Chumachenko, 2013, Low Temperature Physics 39(9), 724. Vranjes, L., J. Boronat, J. Casulleras, and C. Cazorla, 2005, Phys. Rev. Lett. 95, 145302. Vranjeˇs Markić, L., and H. R. Glyde, 2015, Phys. Rev. B 92, 064510. West, G. B., 1975, Phys. Rep. 18, 263. Whitlock, P. A., and R. M. Panoff, 1987, Can. J. Phys. 65,

ce

pte

dM

an

and W. G. Stirling, 2001, J. Phys. Condens. Mat. 13, 4421. Pearce, J. V., S. O. Diallo, H. R. Glyde, R. T. Azuah, T. Arnold, and J. Z. Larese, 2004, J. Phys. Condens. Mat. 16, 4391. Penrose, O., and L. Onsager, 1956, Phys. Rev. 104, 576. Pines, D., 1981, Physics Today 34(11), 106. Pines, D., and C.-W. Woo, 1970, Phys. Rev. Lett. 24, 1044. Pistolesi, F., 1998, Phys. Rev. Lett. 81, 397. Pistolesi, F., C. Castellani, C. Di Castro, and G. C. Strinati, 2004, Phys. Rev. B 69, 024513. Pitaevskii, L. P., 1959, Zh. Eksp. Teor. Fiz. 36(4), 830. Plantevin, O., B. F˚ ak, H. R. Glyde, N. Mulders, J. Bossy, G. Coddens, and H. Schober, 2001, Phys. Rev. B 63, 224508. Pollock, E. L., and D. M. Ceperley, 1987, Phys. Rev. B 36, 8343. Price, D. L., 1978, in The Physics of Liquid and Solid Helium: Part II, edited by K. H. Bennemann and J. B. Ketterson (Wiley, New York), volume II, p. 675. Price, D. L., and F. Fernandez-Alonso (eds.), 2015, Neutron Scattering - Magnetic and Quantum Phenomena, volume 48 (Elsevier), 1st edition. Price, D. L., and K. Sk¨ old, 1987, Methods of Experimental Physics, Neutron Scattering (Academic Press, New York). Prisk, T. R., N. C. Das, S. O. Diallo, G. Ehlers, A. A. Podlesnyak, N. Wada, S. Inagaki, and P. E. Sokol, 2013, Phys. Rev. B 88, 014521. Puff, R. D., 1965, Phys. Rev. 137, A406. Rahman, A., K. S. Singwi, and A. Sj¨ olander, 1962, Phys. Rev. 126, 986. Reiter, G. F., A. I. Kolesnikov, S. J. Paddison, P. M. Platzman, A. P. Moravsky, M. A. Adams, and J. Mayers, 2012, Phys. Rev. B 85, 045403. Reiter, G. F., R. Senesi, and J. Mayers, 2010, Phys. Rev. Lett. 105, 148101. Reynolds, P. J., D. M. Ceperley, B. J. Alder, and W. A. Lester, 1982, J. Chem. Phys. 77, 5593. Romanelli, G., M. Ceriotti, D. E. Manolopoulos, C. Pantalei, R. Senesi, and C. Andreani, 2013, J. Phys. Chem. Lett. 4(19), 3251. Rota, R., and J. Boronat, 2012, J. Low Temp. Phys. 166(1-2), 21. Rugar, D., and J. S. Foster, 1984, Phys. Rev. B 30, 2595. Ruvalds, J., and A. Zawadowski, 1970, Phys. Rev. Lett. 25, 333. Saarela, M., 1986, Phys. Rev. B 33, 4596. Sakhel, A. R., and H. R. Glyde, 2004, Phys. Rev. B 70, 144511. Schmidt-Wellenburg, P., K. H. Andersen, and O. Zimmer, 2009, Nucl. Instrum. Meth. A 611, 259. Schutz, K., and K. M. Zurek, 2016, Phys. Rev. Lett. 117, 121302. Scopigno, T., G. Ruocco, and F. Sette, 2005, Rev. Mod. Phys. 77, 881. Sears, V. F., 1969, Phys. Rev. 185, 200. Sears, V. F., 1970, Phys. Rev. A 1, 1699. Sears, V. F., 1983, Phys. Rev. B 28, 5109. Sears, V. F., 1984, Phys. Rev. B 30, 44. Sears, V. F., E. C. Svensson, P. Martel, and A. D. B. Woods, 1982, Phys. Rev. Lett. 49, 279. Senesi, R., A. Pietropaolo, A. Bocedi, S. E. Pagnotta, and F. Bruni, 2007, Phys. Rev. Lett. 98, 138102. Silver, R. N., 1988, Phys. Rev. B 37, 3794. Silver, R. N., 1989, Phys. Rev. B 39, 4022.

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Page 48 of 48 48

cri pt

41, 974. Yamamoto, K., H. Nakashima, Y. Shibayama, and K. Shirahama, 2004, Phys. Rev. Lett. 93, 075302. Yamamoto, K., Y. Shibayama, and K. Shirahama, 2008, Phys. Rev. Lett. 100, 195301. Yarnell, J. L., G. P. Arnold, P. J. Bendt, and E. C. Kerr, 1958, Phys. Rev. Lett. 1, 9. Zawadowski, A., J. Ruvalds, and J. Solana, 1972, Phys. Rev. A 5, 399. Zsigmond, G., F. Mezei, and M. T. F. Telling, 2007, Physica B 388, 43.

ce

pte

dM

an

us

1409. Woo, C. W., 1976, in The Physics of Liquid and Solid Helium: Part I, edited by K. H. Bennemann and J. B. Ketterson (Wiley Interscience, New York), volume 1, p. 349. Woods, A. D. B., 1966, in Quantum Fluids, edited by D. F. Brewer (North Holland, Amsterdam). Woods, A. D. B., and R. A. Cowley, 1973, Rep. Prog. Phys. 36, 1135. Woods, A. D. B., P. A. Hilton, R. Scherm, and W. G. Stirling, 1977, J. Phys. C: Solid State Phys. 10(3), L45. Woods, A. D. B., and V. F. Sears, 1977, Phys. Rev. Lett. 39, 415. Woods, A. D. B., and E. C. Svensson, 1978, Phys. Rev. Lett.

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Elementary excitations and crossover phenomenon in liquids.

Probing electron-phonon excitations in molecular junctions by quantum interference.

Focus: Phase-resolved nonlinear terahertz spectroscopy--From charge dynamics in solids to molecular excitations in liquids.

Decay of fermionic quasiparticles in one-dimensional quantum liquids.

Designing Quantum Spin-Orbital Liquids in Artificial Mott Insulators.

Editorial--analysis in gases and liquids using quantum cascade lasers.

Quantum Chemical Modeling of Hydrogen Bonding in Ionic Liquids.

Low-energy electrodynamics of novel spin excitations in the quantum spin ice Yb₂Ti₂O₇.

Electronic excitations in a dielectric continuum solvent with quantum Monte Carlo: acrolein in water.

SrTiO3 interface.

Adiabatic quenches and characterization of amplitude excitations in a continuous quantum phase transition.

Localization and dynamics of long-lived excitations in colloidal semiconductor nanocrystals with dual quantum confinement.

All-electromagnetic control of broadband quantum excitations using gradient photon echoes.

classical calculations: time-domain approach.

Monolithic metallic nanocavities for strong light-matter interaction to quantum-well intersubband excitations.

Anomalous sequence of quantum Hall liquids revealing a tunable Lifshitz transition in bilayer graphene.

Interaction between phosphomolybdic anion and imidazolium cation in polyoxometalates-based ionic liquids: a quantum mechanics study.

Multiple excitations in photosynthetic systems.

Insights from quantum chemistry into piperazine-based ionic liquids and their behavior with regard to CO₂.

A quantum chemistry study for ionic liquids applied to gas capture and separation.

Nanoscale control of phonon excitations in graphene.

Collective excitations in 2D hard-disc fluid.

Mid-infrared spectroscopy for gases and liquids based on quantum cascade technologies.

Probing the interactions between ionic liquids and water: experimental and quantum chemical approach.