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Symmetry-protected hidden order and magnetic neutron Bragg diffraction by URu2Si2
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 046003 (http://iopscience.iop.org/0953-8984/26/4/046003) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 132.239.1.231 This content was downloaded on 16/06/2017 at 21:14 Please note that terms and conditions apply.
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 046003 (9pp)
doi:10.1088/0953-8984/26/4/046003
Symmetry-protected hidden order and magnetic neutron Bragg diffraction by URu2Si2 D D Khalyavin1 , S W Lovesey1,2 , A N Dobrynin2 , E Ressouche3 , R Ballou4 and J Flouquet3 1 2 3 4
ISIS Facility, STFC Oxfordshire OX11 0QX, UK Diamond Light Source Ltd, Oxfordshire OX11 0DE, UK SPSMS, UMR-E CEA/UJF-Grenoble 1, INAC, Grenoble F-38054, France Institut N´eel, CNRS and Universit´e Joseph Fourier, BP 166, Grenoble F-38042, France
E-mail: [email protected] Received 1 November 2013, revised 25 November 2013 Accepted for publication 28 November 2013 Published 6 January 2014 Abstract
We investigate how the order parameter of a continuous phase transition can be protected from view by symmetry in a magnetic crystal. The symmetry in question forbids atomic displacements and formation of magnetic dipoles, rendering the order parameter invisible in standard x-ray and magnetic neutron Bragg diffraction. Analysis of the allowed magnetic space-groups reveals exact properties of the hidden order parameter. We demonstrate that Bragg spots forbidden by the chemical structure can unveil magnetic hidden order. The method is applied to URu2 Si2 , which has been thoroughly investigated in the past few decades using all manner of experimental techniques. Starting from the established chemical structure of URu2 Si2 , we have performed a critical analysis of available data for magnetic neutron Bragg diffraction. Keywords: hidden order, URu2 Si2 , neutron diffraction (Some figures may appear in colour only in the online journal)
1. Introduction
candidates for the hidden order that have been aired, at one time or another, include electronic multipoles with rank 2 (time-even, quadrupole), 3 (time-odd, octupole) and 5 (time-odd, triakontadipole) [6–8]. Symmetry-protected hidden order of a continuous phase transition does not feature in the vast literature on URu2 Si2 , to the best of our knowledge. We start from the premise, derived from many experimental observations, that, in the absence of an applied magnetic field, a hidden order parameter (i) does not allow any coupling to atomic displacements, (ii) it does not allow a non-zero magnetic dipole moment at U sites and (iii) it does not induce any molecular fields on Ru and Si sites. Requirement (i) follows from the observation that the phase transition does not induce a modulation of the chemical structure, (ii) follows from the absence of evidence for an ordering of magnetic dipoles, and (iii) follows from the absence of NMR signals on ligand nuclei.
Specific heat data for the uranium compound URu2 Si2 shows it undergoes two mean-field-like phase transitions, one at ≈1.0 K and a second at To ≈ 17.5 K [1–3]. Bulk superconductivity occurs at the low temperature transition, while the physical property of the transition at To is a puzzle to this day; for a review of developments in the period 1985–2013 see, for example, Mydosh and Oppeneer [4], and Chandra et al [5]. There is no consensus on the order parameter at To , because neither a modulation of the chemical structure or an ordering of magnetic dipoles is visible. In the absence of a visible, conventional order parameter it is customary to use the term hidden order for the enigmatic phase appearing at To . It is not decided whether 5f-electrons in URu2 Si2 should be treated as localized or itinerant, and 0953-8984/14/046003+09$33.00
1
c 2014 IOP Publishing Ltd Printed in the UK
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By its very nature, symmetry formulated hidden order is independent of a specific model of the correlated electrons, with concomitant conclusions valid for either localized or itinerant electrons, in S–L or j–j coupling schemes. Moreover, candidate magnetic space-groups for URu2 Si2 can be applied to an interpretation of data gathered on the material using any experimental technique. We choose the technique of magnetic neutron Bragg diffraction to illustrate symmetry protected, hidden order in URu2 Si2 . For an experiment of this type, the quantity of interest is a unit-cell structure factor for uranium ions, constructed from a knowledge of the symmetry of sites used by U ions and translation symmetry of U ions in a unit cell. Symmetry of the chemical structure, tetragonal I4/mmm-type, includes body-centring that imposes the extinction rule h + k + l even on Bragg spots, where h, k, l are integer Miller indices. Magnetic Bragg spots that satisfy h + k + l odd are also allowed and, not surprisingly, they identify the symmetry of the hidden order parameter. A recent investigation of URu2 Si2 at 2 and 25 K, exploiting a sophisticated neutron Bragg diffraction technique, inspired the present study [9]. A single crystal of URu2 Si2 was placed in a strong magnetic field (9.6 T) aligned with the tetrad axis of the crystal (c-axis). We re-visit the published diffraction data, and confront them with exact expressions for the neutron diffraction amplitude compatible with allowed magnetic space-groups. We prove, beyond reasonable doubt, that available magnetic neutron Bragg diffraction data do not answer the question of hidden order in URu2 Si2 , and propose an experiment with the potential to settle the question. Chemical structure, definition of atomic multipoles and electronic structure factors for magnetic neutron diffraction are subjects of section 2. Thereafter, in section 3, we formulate symmetry-protected hidden order, and derive five candidate magnetic space-groups for investigation by neutron diffraction. A general expression for the scattering amplitude of neutrons is given in section 4, with pertinent examples discussed in an appendix. Section 5 contains our findings from tests of our five candidates, derived in section 3, against data gathered by Ressouche et al [9] on URu2 Si2 in a magnetic field parallel to the c-axis. A discussion of results is given in section 6.
Figure 1. (Left panel) Tetragonal, I4/mmm-type chemical structure
of URu2 Si2 . (Right panel) Orthonormal coordinate system and the polar angles, equation (2.3), defining the scattering vector S = (h, k, l)—shown for the (1, 1, 1) plane—used in calculations.
Angular brackets h· · ·i in a multipole denote an expectation value, or time average, of the enclosed operator, a spherical tensor with projections Q0 that obey −K 0 ≤ Q0 ≤ K 0 . 0 Parity-even multipoles engaged in neutron diffraction, hTQK0 i, 0
0
0
K i and a possess a complex conjugate hTQK0 i∗ = (−1)Q hT−Q 0 0
time signature (−1)K [10]. Unit-cell structure factors for magnetic diffraction are derived from [10], X 0 0 9QK0 = exp(id · S)hTQK0 id , (2.1) d
where the sum is over U ions at positions d in the unit cell, and the Bragg wavevector S = (h, k, l) is depicted in figure 1. The scattering amplitude for magnetic neutron Bragg diffraction is a scalar product of an electronic dipole hQ⊥ i and the neutron spin. The operator Q⊥ is a sum of electron spin and linear momentum field operators, and we use a universal expression [10], XX ˆ K00 (KQK 0 Q0 |1p). (2.2) hQ⊥,p i = (4π )1/2 YQK (S)9 Q KQ K 0 Q0 0 ˆ In (2.2), 9QK0 is the electronic structure factor (2.1), YQK (S) 0 0 is a spherical harmonic, and (KQK Q |1p) a Clebsch–Gordan ˆ is coefficient [12]. The unit vector for spherical harmonics, S, derived from the Bragg wavevector S, namely,
2. Material properties and multipoles
Sˆ = (h, k, l)/|(h, k, l)|
The chemical structure of paramagnetic URu2 Si2 is represented by a centrosymmetric, tetragonal space group I4/mmm ˚ c = 9.538 A ˚ at (#139), with lattice constants a = 4.112 A, 25 K [9]. Body-centring imposes an extinction rule h + k + l even, where h, k, l are integer Miller indices. Uranium ions use sites 2a that have point group symmetry 4/mmm (D4h ). Figure 1 shows the chemical structure and our orthonormal coordinates. Electronic spin and orbital degrees of freedom in the ground-state of a material are often expressed in terms of multipoles with discrete symmetries. Multipoles observed by magnetic neutron diffraction are parity-even and time-odd. We elect to use spherical multipoles, and these have a rank K 0 that is an odd integer.
= (sin θo cos φo , sin θo sin φo , cos θo ),
(2.3)
with polar angles θo and φo shown in figure 1. A specific example of the amplitude (2.2) appears in section 4, and confrontations with data are reported in section 5. 0 A multipole is a sum of unit tensors W (a,b)K , where a = 0 or 1 labels spin degrees of freedom, and b = 0, 1, . . . , 2l (l = 3 for f-electrons) labels orbital degrees of freedom. Multipoles 0 hTQK0 i engaged in neutron diffraction are a sum of three unit tensors, W (0,K )K for the purely orbital contribution, and 0 W (1,b)K with b = K 0 ± 1 [10] for the mixed spin–orbital contribution, about which we have more to say here and in section 4. 0
2
0
J. Phys.: Condens. Matter 26 (2014) 046003
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Table 1. Parity-even irreducible representations conjugated with the 4/mmm10 site symmetry and the multipoles transformed by them, i.e., all allowed multipoles for neutron diffraction. The third column shows the corresponding space group I4/mmm10 representations associated with 0(k = 0) and M (k = 1, 1, 1) points. Multipole rank is odd with a maximum K 0 = 7. Real, 0 , and imaginary, 00 , parts of a 0 0 0 multipole are defined with the phase convention hTQK0 i = hTQK0 i0 + ihTQK0 i00 . The third row, 03+ , defines our naive model, cf appendix. For Eg , f1 and f2 are two orthogonal basis functions.
Site symmetry irrep
Multipoles
Space group irrep
A1g (4/mmm)
hT45 i00 , hT47 i00 hT23 i00 , hT25 i00 , hT27 i00 , hT67 i00 hT01 i, hT03 i, hT05 i, hT45 i0 , hT07 i, hT47 i0 hT23 i0 , hT25 i0 , hT27 i0 , hT67 i0 f1 → hT11 i0 , hT13 i0 , hT33 i0 , hT15 i0 , hT35 i0 , hT55 i0 , hT17 i0 , hT37 i0 , hT57 i0 , hT77 i0 f2 → hT11 i00 hT13 i00 , hT33 i00 , hT15 i00 , hT35 i00 , hT55 i00 , hT17 i00 , hT37 i00 , hT57 i0 , hT77 i0
01+ , M+ 1
B1g (40 /mmm0 ) A2g (4/mm0 m0 ) B2g (40 /mm0 m) Eg (η1 , 0; m0y m0z mz ) (η1 , η2 η1 = η2 ; m0z m0−xy mxy ) (η1 , η2 η1 6= η2 ; 20z /m0z )
At present there is no direct evidence that a localized 5f2 configuration for U ions is appropriate [4, 7]. Indeed, at low temperatures an itinerant f-electron character seems to prevail [5, 6, 8]. Even so, many multipolar theories using a crystal-field potential of hidden order have been proposed [4, 7, 11]. On the other hand, Cricchio et al [6] studied hidden order in URu2 Si2 using density-function theory for itinerant electrons, and analysed the solution in terms of unit tensors. For the order parameter they proposed a ferromagnetic motif of multipoles K 0 = 5 with W (1,6)5 the dominant player. 0 Contributions to diffraction by W (1,b)K with rank K 0 = 5 (triakontadipole) and K 0 = 7 (octaeicosahecatontapole) merit (1,b)K 0 arises from further comment. First, the unit tensor WP the contribution to Q⊥ proportional to exp(iR · S)s, a mixture of spin, s, and orbital, R, electron variables. The simulation of URu2 Si2 by Cricchio et al [6] uses an itinerant model of electrons with j–j coupling, and the j = 5/2 manifold of states is found to be the most important in determining ground-state electronic properties. Later work of a similar nature [8] confirms that energy bands crossing the Fermi level have mainly j = 5/2, with the hidden order a time-odd, triakontadipole in a nematic motif. States with j = 5/2 do not generate multipoles beyond K 0 = 5, whereas Russell–Saunders coupling of f-electrons permits K 0 = 7 (a theory of magnetic neutron scattering by ions described by j–j coupling is given [21]).
02+ , M+ 2 03+ , M+ 3 04+ , M+ 4 05+ , M+ 5
3.1. Local (or site) order parameters
Elements of symmetry that exist when URu2 Si2 is placed in a magnetic field are an essential set of references and, for the moment, we set aside the question of hidden order. (Magnetic space-groups are specified in terms of the Miller and Love and Belov–Neronova–Smirnova notations [13–15].) At this elementary level of investigation, magnetic properties of U ions are prescribed by symmetry found in the point group of a U site in the grey group (I4/mmm10 ) and the five parity-even irreps 0i+ with i = 1–5. Results are found in table 1; entries are all the parity-even and time-odd multipoles that are allowed to contribute in magnetic neutron diffraction. To explain the making of entries consider a particular case 02+ . The corresponding site symmetry is 40 /mmm0 of sites (2a) in the space group I40 /mmm0 derived from 02+ conjugated with 4/mmm10 . Entries in table 1 are parity-even multipoles that 0 satisfy elements of symmetry in 40 /mmm0 , namely, ImhTQK0 i ≡ 0
hTQK0 i00 with Q0 = ±2 and ±6 (examples of the steps involved in imposing elements of symmetry on a spherical multipole can be found in [16]). Multipoles listed in table 1 represent local (or site) order parameters either primary, if they drive the transition, or secondary, through a coupling to the primary one via the applied magnetic field. Global distortions induced from the local order parameters are determined by the space group representations subducing the site irreps and incorporating the translational cosets. A magnetic field parallel to the c-axis of I4/mmm has symmetry 03+ . In consequence, multipoles listed in table 1 for 03+ conjugated with 4/mmm10 are those that may be observed, irrespective of hidden order, and they define a naive model. Experimental observation of multipoles not present in the naive model will identify the hidden order, e.g., evidence of multipoles in the second line of entries, 40 /mmm0 , identify a hidden order parameter with 02+ symmetry.
3. Symmetry-protected hidden order
We assume that the phase transition with a hidden order parameter is continuous. Subsequent discussions are independent of a specific model of electronic correlations. The one and only reference to our knowledge about the amplitude for neutron diffraction, (2.2), is a restriction on the rank of multipoles, K 0 ; the rank is odd, with a maximum K 0 = 7 appropriate for actinide ions, and multipoles are time-odd (magnetic). A unit-cell structure factor for diffraction by uranium ions is constructed from a knowledge of the symmetry of sites used by U ions and translation symmetry of U ions in a unit cell.
3.2. Hidden order
To discuss possible symmetries of the system in the hidden order phase, the information about the translational symmetry 3
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Table 2. Irreducible representations of the I4/mmm10 space group with zero subduction frequencies, equation (3.1), for the mechanical and
magnetic reducible representations on the U, Ru and Si sites. The corresponding magnetic space-groups without and in a magnetic field parallel to the c-axis are given as well, along with the site symmetries for occupied atomic sites. Parity-odd candidates have identical neutron diffraction amplitudes that coincide with the naive model, and U ions use sites deprived of a centre of inversion symmetry. On the other hand, U sites are centres of inversion symmetry in parity-even candidates that feature in our discussion and section 5. Irrep. 01+ 01+ ⊕ 03+ 02+ 02+ ⊕ 03+ 04+ 04+ ⊕ 03+ 03− 03− ⊕ 03+ 04− 04− ⊕ 03+ M+ 1 + M+ 1 ⊕ 03 + M2 + M+ 2 ⊕ 03 − M2 + M− 2 ⊕ 03 M− 3 + M− 3 ⊕ 03 − M4 + M− 4 ⊕ 03
Magnetic space group
U (site symmetry)
Ru (site symmetry)
Si (site symmetry)
I4/mmm
4/mmm
4mm
I4/m
4/m..
¯ 4m2 ¯ 4..
4..
I4 /mmm
4 /mmm
4 m2
40 mm0
Fm0 m0 m
m0 m0 m
m0 m0 2
I4 /mm m
4 /mm m
20 20 2 4¯ 0 m0 2
Im0 m0 m
m0 m0 m
I4/m0 mm
4/m0 mm
0
0
0
0
0
0
0 0
0
I4 /m mm ¯ 0 m0 I 42 0
0
0
0
PI 4/mmm
4/mmm
P4/mn0 c0
4/m..
PI 42 /mmc
4 /mmm
P42 /mn0 m0
m.m0 m0
PI 42 /nnm P42 /nm c
40 /m0 m0 m ¯ 0 20 4m
PI 4/nmm
4/m0 mm
0 0
0 0
0
420 20
PI 42 /nmc
4 /m mm ¯ 0 m0 42
P42 /nn0 m0
4 mm 4.. 40 mm0
¯ 4.. ¯40 m20
0
0
m0 m0 2
2.m0 m0 4mm
2.20 20 ¯ 4m2
P4/nn c
0
40 m0 m
2.20 20 ¯ 4m2
4 /m mm ¯ 0 m0 42 0
0
m0 m0 2 4¯ 0 m20
420 20
I42 2 0
¯0
0
4.. 40 mm0
¯ 4.. 4¯ 0 m0 2
2.m0 m0 40 m0 m
2m m . ¯ 4m2 0
0
change across the transition is necessary. At the present time, reliable information is not available and only indirect measurements [17, 18] indicate the M-point of symmetry (k = 1, 1, 1) as the most probable candidates for the propagation vector of the hidden order parameter (k = 1, 1, 1 ≡ 0, 0, 1, because reciprocal and real lattices are I-centred and translation by 1, 1, 0 is allowed). Let us point out the specific property of this point, namely, the eight out of ten irreducible representations of the I4/mmm10 space group associated with this propagation vector are one-dimensional (M± i , i = 1–4). A timeodd order parameter transforming accordingly to one of these representations does not allow coupling to any symmetry breaking physical quantity like atomic displacement, for instance, measurable in high-precision diffraction experiments. This can stimulate a hidden nature of the associated phase if the order parameter also does not generate a magnetic dipole on the U position and internal fields on the Ru and Si sites. This peculiarity of the M-point is also relevant to the 0-point (k = 0) whose eight irreps (0i± , i = 1–4) are one-dimensional as well and, therefore, we will restrict our consideration to these two propagation vectors. Any other special/general point or line/plane of symmetry involves only multidimensional irreps which imply coupling to physical quantities detectable in conventional diffraction and spectroscopic measurements.
2m0 m0 .
0
4mm
¯ 4.. ¯40 m20
40 mm0
2.20 20
2.m0 m0
4..
In consequence, an order parameter which (i) does not allow any coupling to atomic displacements, (ii) does not allow non-zero magnetic dipole moment at U sites and (iii) does not induce internal fields on the Ru and Si sites is ‘symmetry protected’ from a direct measurements using standard techniques. These symmetry conditions can be rigorously worked out from subduction frequencies, mν , of the irreps for the appropriate reducible representations on the U, Ru and Si sites, namely, X mν = (1/n(g)) χ (g)χ ∗ν (g), (3.1) g
where n(g) is the number of elements in the space group, χ (g) and χ ν (g) are characters of the element, g, in the reducible and irreducible representations, respectively. The irreps with zero frequencies are listed in table 2, along with the corresponding magnetic space-groups. All of them are one-dimensional and represent preferable candidates for the symmetry of the hidden order parameter. We take advantage of the experimental evidence of the continuous nature of the phase transition at To implying a single irreducible representation responsible for the symmetry breaking. The signature of the hidden order parameter in a neutron diffraction experiment is the presence of an angular dependence of the amplitude hQ⊥ i, caused by a significant 4
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contribution to diffraction from one or more multipoles from table 1. In the available experimental data, only reflections with h + k + l even in the applied magnetic field were measured [9] and it is important to clarify the symmetry allowed multipoles contributing to these reflections. In the general case, the following situations should be distinguished in this context.
(3) The hidden order parameter breaks the four-fold symmetry and transforms as one of the two-dimensional − irreps, 05− , M+ 5 or M5 . Independent of the direction of the order parameter in the representation spaces (η1 , η2 ), the associated multipoles cannot directly contribute to the h + k + l even reflections. The presence of the magnetic field (Hz ), however, is essential in these cases and provides conditions for coupling multipoles transforming as 02+ and 04+ irreps from the table 1, as the secondary order parameters (δ) through the free-energy invariants of the form, δHz (η12 − η22 ) and δHz η1 η2 respectively. Thus, the cases when the experimentally observed multipoles are associated with the 02+ and 04+ irreps require careful consideration to decide whether they are directly representing hidden order parameter, or they are only coupled to the directly invisible primary order parameter with the symmetry of the two-dimensional representations.
(1) The hidden order parameter preserves the translational symmetry (the propagation vector is k = 0) and is associated with one of the 0 representations listed in table 1 (0i+ , i = 1–5). In this case, the appropriate multipoles contribute to the h + k + l even reflections and therefore these reflections carry information about angular anisotropy in hQ⊥ i. The presence of the magnetic field is not essential and introduces additional variables directly coupled to the induced dipole component. These variables are related to the five higher rank multipoles transforming as the 03+ representation (table 1)—labelled the naive model—and therefore their presence is always expected through the linear coupling to the ferromagnetic component, irrespective to the symmetry of the hidden order parameter. (2) The hidden order parameter transforms as a onedimensional parity-odd representation associated with k = 0 propagation vector (0i− , i = 1–4) or as any onedimensional irrep associated with M-point of symmetry (M± i , i = 1–4). In all these case, any signature of the hidden order parameter cannot be detected in the h + k + l even reflections. This conclusion comes from the fact that it is impossible to form an appropriate free-energy coupling invariant in the applied magnetic field between any of these representations and the irreps listed in table 1. As an explicit, illustrative example of our general symmetry argument, we examine electronic structure factors (2.1) for one of five candidates tested against neutron diffraction data in section 5. In the candidate with + M+ 1 ⊕ 03 symmetry, the two irreps represent symmetry of the hidden order parameter, M+ 1 , and the applied magnetic field, 03+ , respectively. The resultant magnetic space group in table 2 is P4/mn0 c0 , with the U site symmetry 4/m. This site symmetry allows multipoles to be complex, with Q0 = 0, ±4 for K 0 ≤ 7. We find the 0 electronic structure factor 9QK0 is allowed to be different from zero for two classes of reflections, ( K0 0 2hTQ0 i for h + k + l even K0 9Q0 = K 0 00 2ihTQ0 i for h + k + l odd.
Parity-odd cases in table 2 possess the diffraction pattern of the naive model, table 1, and are of no interest in an investigation of hidden order using neutron diffraction. However, they are of interest for x-ray diffraction, in which intensity of a Bragg spot is enhanced by tuning the primary energy to an atomic resonance. Parity-odd multipoles can be observed in this type of x-ray diffraction. By way of illustrations, consider 03− and M− 3 in table 2, noting that 422 is one of 11 enantiomorphic crystal classes. It may be shown that, for example, magneto-electric dipoles (anapoles) contribute to resonant diffraction at h + k + l even (h + k + l odd) for 03− (M− 3 ). Five candidates for hidden order tested in section 5 are parity-even cases in table 2, and U ions use sites that are centres of inversion symmetry. To ease the notation, in future parity-even cases are listed #1–#5. The five candidates possess multipoles associated with 03+ in table 1 (naive model) plus multipoles associated with one other space group irrep. Hence, #3 contains multipoles associated with 04+ in addition to those associated with 03+ . Multipoles over and above those in 03+ contribute to Bragg points with h + k + l even and h + k + l odd for 0-points and M-points, respectively. Recall that, 0-points and M-points correspond to ferromagnetic and antiferromagnetic motifs of dipole moments, respectively. #1 01+ ⊕ 03+ ; space group I4/m, U site symmetry 4/m.., and crystal class 4/m. Diffraction allowed for h + k + l even. 0 Multipoles can have an imaginary component with hTQK0 i00 6= 0, and Q0 = 0, ±4 for K 0 ≤ 7. #2 02+ ⊕ 03+ ; space group Fm0 m0 m, crystal class m0 m0 m (site symmetry mm0 m0 in the original tetragonal setting 4/mm0 m0 ). The extinction rule is h + k + l even, because 0 I-centring is present as in #1, and multipoles satisfy hTQK0 i =
Multipoles contributing to the first class of reflections are purely real. This fact, in combination with the conditions imposed by the 4/m site symmetry, restricts the allowed multipoles to those associated with the 03+ representation only, which are always permitted in the applied field. Thus, no additional multipoles which can be taken as a signature of the hidden order parameter are expected for the h + k + l even reflections, and all information about the hidden order parameter is contained in the h + k + l odd reflections.
0
(−1)p hTQK0 i∗ with Q0 = 2p = 0, ±2, ±4, ±6. Note that U ions occupy sites with the same symmetry in candidate #5, for which the primitive cell allows h + k + l odd. #3 04+ ⊕ 03+ ; space group Im0 m0 m, crystal class m0 m0 m, site symmetry m0 m0 m. Diffraction allowed for h + k + l even, 0 and hTQK0 i purely real with Q0 = 2p. 5
J. Phys.: Condens. Matter 26 (2014) 046003
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+ 0 0 #4 M+ 1 ⊕03 ; space group P4/mn c , site symmetry 4/m.., 0 0 and crystal class 4/mm m . Multipoles can be complex with allowed Q0 = 0, ±4. We find the electronic structure factor 0 9QK0 in (2.1) can be different from zero for two classes of reflections, 0 9QK0 0 9QK0
0 2hTQK0 i0 ; 0 2ihTQK0 i00 ;
= =
h + k + l even,
4.1. Russell–Saunders coupling 0
Let (J k T K k J) be a reduced matrix element in an atomic picture with Russell–Saunders (S–L) coupling. We define, 0
and
and coefficients t
h + k + l odd.
0 9QK0
=
p even, h + k + l even,
0 2ihTQK0 i00 ;
(4.3)
K0 Q0
in hT i are quantities to be determined. 0
0
0
0
is complex for Q0 6= 0, while t0K is purely real. Dependence on the magnitude of S appears in radial integrals hjn i with n even [19], normalized such that hj0 i = 1, 0 0 and all other hjn i = 0, for |S| = 0. The unit tensor W (0,K )K 0 0 in hTQK0 i is multiplied by [hjK 0 −1 i + hjK 0 +1 i], while W (1,b)K is multiplied by hjb i with b = K 0 ± 1. In the present work, 0 hjn is are included in reduced matrix elements of TQK0 , which are tabulated in [20] and reproduced here for convenience of the reader. For Russell–Saunders coupling and a single J-manifold K = K 0 ± 1, and K 0 is odd with maximum K 0 = 7 for equivalent f-electrons. Reduced matrix elements are purely real, and for f 2 with J = 4, S = 1 and L = 5, √ (J k T 1 k J) = (8/ 5)[hj0 i + (74/45)hj2 i], √ (J k T 3 k J) = −(52/15) (2/55)[hj2 i + (15/11)hj4 i], (4.4) √ (J k T 5 k J) = −(4/121) (195)[hj4 i + (248/39)hj6 i], √ (J k T 7 k J) = (136/99) (14/143)hj6 i.
0
0
K0 Q0
0
(−1)p hTQK0 i∗ with Q0 = 2p, and, 0
0
K i it follows that, tK From the property hTQK0 i∗ = (−1)Q hT−Q 0 Q0
Purely imaginary multipoles induced from 01+ (table 1) contribute at Bragg spots with h + k + l odd. + 0 0 #5 M+ 2 ⊕ 03 ; space group P42 /m.n m , crystal class 0 0 0 0 0 4/mm m , site symmetry mm m . Multipoles satisfy hTQK0 i = 9QK0 = 2hTQK0 i0 ;
0
K K k J), hTQK0 i = tQ 0 (J k T
and
p odd, h + k + l odd.
In line with #4, imaginary multipoles induced from 02+ (table 1) contribute at Bragg spots with h + k + l odd. 4. Neutron scattering amplitude
Ressouche et al [9] measured the component of hQ⊥ i parallel to the c-axis, hQ⊥,c i. Our coordinates in figure 1 are such that a dipole with spherical component p = 0 is parallel to the crystal c-axis, so hQ⊥,c i = hQ⊥,0 i. In hQ⊥,0 i, there is ˆ in terms of advantage in writing spherical harmonics YQK (S) Legendre polynomials of order l, Pl , and associated Legendre Q0 polynomials, Pl , that depend on θo only; see (2.3) and figure 1 [12]. From (2.2) we find, √ X 0 0 0 1/2 hQ⊥,0 i = − 3 [K /(2K +1)] hT0K i(PK 0+1 − PK 0−1 )
The dipole hT01 i and the magnetic moment h(L + 2S)c i are related, namely, hT01 i = h(L + 2S)c i/3 in the forward direction (0, 0, 0). A notable feature of f 2 in Russell–Saunders coupling is that the octupole (K 0 = 3) is independent of orbital angular momentum, which is true of (J k T 7 k J) in the general case, f n. An appendix contains explicit expressions for multipoles derived from a model wavefunction. The wavefunction in question encompasses the one examined by Ressouche et al [9].
K0
X 0 0 + ϒ(K 0 , Q0 )[hTQK0 i0 cos(Q0 φo ) + hTQK0 i00 sin(Q0 φo )] Q0
×
Q0 (PK 0+1 −[(K 0 +1)(K 0
+ Q )/K (K −Q 0
0
0
0
Q0 +1)]PK 0−1 )
5. Confrontation of experimental data and simulations
,
(4.1)
A simulation of the magnetic neutron scattering amplitude, hQ⊥ i, based on the naive model (section 3, appendix) has been confronted with available experimental data. Simulations use K 0 in (4.3) Russell–Saunders coupling, using the definition of tQ and reduced matrix elements (4.4). We find that the naive model does not give a satisfactory explanation of diffraction data. The quality of fit is not significantly different from use of the dipole moment alone (expression (4.1) with K 0 = 1), as can be seen in the results in figure 2. Other plausible candidates of the magnetic structure of URu2 Si2 in an applied magnetic field have been similarly tested. Constructions of our five candidates, labelled #1–#5, are the subjects of section 3. The quality of fits to data, set out in figure 3, is the same for all candidates, i.e., available neutron diffraction data are not able to distinguish between candidates for the magnetic structure and there is no useful
where K 0 is odd and Q0 > 0. The coefficient in the sum on Q0 in (4.1) is, 0
ϒ(K 0 , Q0 ) = 2(−1)Q [(K 0 − Q0 + 1)/(K 0 + 1)] × [(K 0 −Q0 −1)!(K 0 −Q0 )/(K 0 +Q0 − 1)!(K 0 +Q0 )]1/2 , (4.2) for Q0 < K 0 and, ϒ(K 0 , K 0 ) = −[2/(K 0 + 1)][1/(2K 0 )!]1/2 , for Q0 = K 0 . By definition, hQ⊥,0 i is proportional to sin2 θo and {2hQ⊥,0 i/sin2 θo } = h(L + 2S)c i for |S| = 0. The result (4.1) uses K = K 0 ± 1 in (2.1) and it is valid for either an itinerant or an atomic model of electronic structure. 6
J. Phys.: Condens. Matter 26 (2014) 046003
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Figure 2. Experimental (subscript obs) versus calculated (subscript
calc) values of hQ⊥,0 i2 , using the naive model (top panels) and the dipole approximation (bottom panels). Note that the naive model is identical to the candidates #4 and #5 in section 3, for h + k + l even. Sample temperature 2 K (left panels) and 25 K (right panels). The dipole approximation neglects all multipoles in the amplitude (4.1) apart from the dipole, hT01 i ∝ t01 in (4.3). Values of RF = P
100
2 2 s |(Q⊥,s )obs −(Q⊥,s )calc | P 2 (Q ) s ⊥,s obs
are given for each distribution.
information concerning a hidden order parameter. In figures 2 and 3, multipoles included in the fit to data are indicated in individual panels. Multipoles in candidates #1, #2 and #3 not in the naive model, top panel in figure 2, are indicated on each panel in figure 3, i.e., multipoles in a fit are the naive model plus those indicated. The experimental data present a normalized set of Qhkl = 2hQ⊥,0 i/sin2 θo values for 52 reflections. For fitting purposes, expression (4.1) for hQ⊥,0 i is reduced to the following form:
Figure 3. Experimental (obs) versus calculated (calc) intensities,
(5.1)
hQ⊥,0 i2 , for the candidates #1 (top), #2 and #3 (bottom) defined in section 3. Sample temperature 2 K (left panels) and 25 K (right K0 panels). RF is defined in the caption to figure 2. Quantities tQ 0 inferred from data are defined in (4.3). Multipoles included over and above the naive model—top panels in figure 2—are indicated, 0 K 0 00 0 e.g., imaginary components hTQK0 i00 ∝ (tQ 0 ) with Q = ±4 in top panels for candidate #1, 01+ ⊕ 03+ .
where the parameter vector t = {t } , vector Chkl consists of the numerical coefficients unique for each reflection and the corresponding component of the vector t. Numerical coefficients C0hkl are unique for each reflection, and h(L+2S)c i is the dipole magnetic moment, with {2hQ⊥,0 i/sin2 θo } = h(L + 2S)c i for |S| = 0. Radial integrals in (4.4) were calculated with expressions found in [19]. We implemented a simulated annealing algorithm to fit the experimental data. Simulated annealing has been demonstrated to be a powerful technique for finding a global minimum in complex multiple parameter optimization problems, including data mining [23–27]. The basic idea
behind it resembles crystal growth from melt. First the temperature of the system is set to be high, parameters are set to initial random values, and the energy of the system is calculated. Then a standard metropolis Monte Carlo [28] procedure is performed at this temperature: the parameters are randomly changed by small steps and the new energy is calculated. If the new energy is reduced, this parameter configuration is accepted, otherwise it is accepted with probability equal to the Boltzmann factor exp(−1E/kT). At the next simulated annealing step the system temperature is decreased, and the procedure repeated. While the temperature
Qhkl = C0hkl h(L + 2S)c i + Chkl · t, K0 Q0
7
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is high enough, the energy barriers can be overcome, and the parameter space can be explored by a random walk. When the temperature is decreasing slowly, this allows the algorithm to converge to the global minimum.
motif of dipole moments, an M-point) or is associated with the parity-odd multipoles. Cases when the multipoles specified in table 1 appear as secondary order parameters (case 3 in the section 3) cannot be unambiguously ruled out, because the corresponding coupling is expected to be small with the corresponding contribution to diffraction placed beyond available experimental sensitivity. An assumption that our experimental data are sensitive to secondary order parameters would imply that the primary candidates for the symmetry + of the parity-even, hidden order parameter are M+ 1 and M2 representations (candidates #4 and #5 in section 3). In the case of a parity-odd nature of the order parameter, our diffraction data does not set any restrictions and the symmetry can be represented by any out of the five one-dimensional, parity-odd representations set out in table 2. The first two candidates, + M+ 1 and M2 , can be identified in magnetic neutron diffraction by the appearance of Bragg spots with Miller indices h + k + l odd that violate the body-centring extinction rule and associated angular anisotropy, while parity-odd candidates can be explored by resonant x-ray Bragg diffraction.
6. Discussion
Let us summarize results derived from our formulation of symmetry-protected hidden order at a continuous phase transition, and its application to magnetic neutron diffraction by tetragonal URu2 Si2 , illustrated in figure 1. A recent investigation of URu2 Si2 exploited a sophisticated neutron Bragg diffraction technique, two temperatures that straddle the onset of the enigmatic phase, and a strong magnetic field (9.6 T) parallel to the tetrad axis [9]. The principal conclusion therein, about the electronic triakontadipole (a rank 5 multipole also called a dotriacontapole) as the primary candidate for the hidden order parameter, is not confirmed in our present study. This result is graphically illustrated in figures 2 and 3. What is shown in these two figures is that, Bragg intensities that obey the body-centring extinction rule are adequately represented by the dipole approximation to magnetic scattering, with no discernible angular anisotropy in the distribution of Bragg spots. Our fresh approach takes advantage of magnetic symmetry to enumerate allowed multipoles, as well as coupling between them in the applied magnetic field. The method of reconstruction, of the magnetization distribution in the unit cell, used by Ressouche et al [9] is absolutely restricted to cases when multipoles representing the hidden order parameter have symmetry identical to the symmetry of the field-induced dipole. However, this situation is very unlikely, because, in this case, a non-zero ferromagnetic component is expected in zero magnetic field as a secondary order parameter, which has never been reported. The observed evolution of the magnetization distribution with temperature can be attributed to the trivial bilinear coupling between the induced dipole and the hidden order parameter [9]. It is crucial to understand that, this type of coupling is always allowed and does not affect the symmetry of the distribution and, therefore, cannot reveal the hidden order parameter. The approach used in the present communication is based on exact results for the magnetic neutron scattering amplitude, expressed with parity-even, time-odd electronic multipoles restricted by the symmetry of the hidden order parameter and the applied magnetic field. Results include multipoles, listed in table 1, that contribute to magnetic scattering but are not coupled to the field-induced dipole component and, in consequence, do not affect the symmetry of the magnetization distribution. Thus, the present approach has potentiality to reveal multipoles which depend on the symmetry of the hidden order parameter, and create an unambiguous signature of it. None of the candidates for hidden order, table 1, has been found to show a substantial increase of contribution to diffraction below the transition temperature, which strongly indicates that the hidden order parameter either has non-zero propagation vector (consistent with an antiferromagnetic
Acknowledgment
One of us (SWL) is grateful to Dr C Detlefs for his initial interest in the reported study. Appendix. Model wavefunction
We use Russell–Saunders coupling in an orientation to eligible multipoles in hQ⊥,0 i. Specifically, we examine values K 0 in (4.3) derived from the naive model introduced of tQ 0 0
in section 3, table 1. A purely real multipole hTQK0 i with Q0 = 0, ±4 satisfies symmetry elements in 4/mm0 m0 , and the extinction rule for the associated magnetic space group, I4/mm0 m0 , is h + k + l even. First-principle electronic structure calculations reported by Haule and Kotliar suggest that U4+ (5f2 ) atomic states in URu2 Si2 are adequately described by Russell–Saunders coupling and projections M = 0, ±4 [22]. In keeping with this finding, we adopt a singlet wavefunction, |ψi = a|4i + b| − 4i + c|0i,
(A.1)
where |Mi = |J = 4, Mi, and complex coefficients a, b and c are chosen to make the wavefunction properly normalized. The saturation magnetic moment associated with (A.1) is h(L + 2S)c i = (16/5)(|a|2 − |b|2 ). From the definition (4.3), √ 0 0 0 t0K = hψ|T0K |ψi/(J k T K k J) = [5/16 (2K 0 + 1)] × h(L + 2S)c i(444 − 4|K 0 0).
(A.2)
Evaluation of the Clebsch–Gordan coefficient in (A.2) shows 0 that all t0K for the naive model are of one sign, and diminish in size with increasing K 0 , e.g., t03 /t07 = 19.08. In addition, 0
0
0
K K t+4 = hψ|T+4 |ψi/(J k T K k J) √ = [1/ (2K 0 + 1)](bc∗ − a∗ c)(4044|K 0 4). (A.3) 8
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D D Khalyavin et al
5 and t7 have the same sign with According to (A.3), t+4 +4 5 7 t+4 /t+4 = 1.48. In terms of mixing parameters, α, β, ϕ, used by Ressouche et al [9],
[10] Lovesey S W et al 2005 Phys. Rep. 411 233 [11] Kusunose H and Harima H 2011 J. Phys. Soc. Japan 80 084702 [12] Edmonds A R 1960 Angular Momentum in Quantum Mechanics (Princeton, NJ: Princeton University Press) [13] Stokes H T, Hatch D M and Campbell B J 2007 ISOTROPYstokes.byu.edu/isotropy.html [14] Campbell B J, Stokes H T, Tanner D E and Hatch D M 2006 J. Appl. Crystallogr. 39 607 [15] Gallego S V et al 2012 J. Appl. Crystallogr. 45 1236 [16] Lovesey S W and Balcar E 2013 J. Phys. Soc. Japan 82 021008 [17] Villaume A et al 2008 Phys. Rev. B 78 012504 [18] Bourdarot F et al 2010 J. Phys. Soc. Japan 79 064719 [19] Brown P J 2004 Radial integrals for U4+ in section 4.4.5 of volume C International Tables of Crystallography (Dordrecht: Springer) [20] Osborn R et al 1991 Handbook on the Physics and Chemistry of Rare Earths vol 14 (Amsterdam: North-Holland) [21] Balcar E and Lovesey S W 1991 J. Phys.: Condens. Matter 3 7095 [22] Haule K and Kotliar G 2009 Nature Phys. 5 796 [23] Kirkpatrick S, Gelatt C D Jr and Vecchi M P 1983 Science 220 671 [24] Bertsimas D and Tsitsiklis J 1993 Stat. Sci. 8 10 [25] Barradas N P and Smith R 1999 J. Phys. D: Appl. Phys. 32 2964 [26] Munakata T and Nakamura Y 2001 Phys. Rev. B 64 046127 [27] Ireland J 2007 Solar Phys. 243 237 [28] Metropolis N et al 1953 J. Chem. Phys. 21 1087
(|a|2 − |b|2 ) = sin(2α) cos(β) sin(ϕ + ν), (A.4) √ (a∗ c − bc∗ ) = (1/ 2) sin(2α) cos(β) cos(ϕ + ν). 5 and Since (a∗ c − bc∗ ) is purely real, the quantities t+4 7 t+4 are purely real. In (A.4), ν is assumed to be known from independent calculations [11]. (For ν these authors use instead the symbol θ and find θ ≡ ν = 0.998 in a discussion of hexadecapole order, which is not visible in neutron diffraction, because rank 4 is excluded by a selection rule for states with the same values of S, L, and J.)
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
Palstra T T M et al 1985 Phys. Rev. Lett. 55 2727 Schlabitz W 1986 Z. Phys. B 62 171 Maple M B et al 1986 Phys. Rev. Lett. 56 185 Mydosh J A and Oppeneer P M 2011 Rev. Mod. Phys. 83 1301 Chandra P et al 2013 Nature 493 621 Cricchio F et al 2009 Phys. Rev. Lett. 103 107202 Kuramoto Y et al 2009 J. Phys. Soc. Japan 78 072001 Ikeda H et al 2012 Nature Phys. 8 528 Ressouche E et al 2012 Phys. Rev. Lett. 109 067202
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Symmetry-protected hidden order and magnetic neutron Bragg diffraction by URu2Si2
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 046003 (http://iopscience.iop.org/0953-8984/26/4/046003) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 132.239.1.231 This content was downloaded on 16/06/2017 at 21:14 Please note that terms and conditions apply.
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 046003 (9pp)
doi:10.1088/0953-8984/26/4/046003
Symmetry-protected hidden order and magnetic neutron Bragg diffraction by URu2Si2 D D Khalyavin1 , S W Lovesey1,2 , A N Dobrynin2 , E Ressouche3 , R Ballou4 and J Flouquet3 1 2 3 4
ISIS Facility, STFC Oxfordshire OX11 0QX, UK Diamond Light Source Ltd, Oxfordshire OX11 0DE, UK SPSMS, UMR-E CEA/UJF-Grenoble 1, INAC, Grenoble F-38054, France Institut N´eel, CNRS and Universit´e Joseph Fourier, BP 166, Grenoble F-38042, France
E-mail: [email protected] Received 1 November 2013, revised 25 November 2013 Accepted for publication 28 November 2013 Published 6 January 2014 Abstract
We investigate how the order parameter of a continuous phase transition can be protected from view by symmetry in a magnetic crystal. The symmetry in question forbids atomic displacements and formation of magnetic dipoles, rendering the order parameter invisible in standard x-ray and magnetic neutron Bragg diffraction. Analysis of the allowed magnetic space-groups reveals exact properties of the hidden order parameter. We demonstrate that Bragg spots forbidden by the chemical structure can unveil magnetic hidden order. The method is applied to URu2 Si2 , which has been thoroughly investigated in the past few decades using all manner of experimental techniques. Starting from the established chemical structure of URu2 Si2 , we have performed a critical analysis of available data for magnetic neutron Bragg diffraction. Keywords: hidden order, URu2 Si2 , neutron diffraction (Some figures may appear in colour only in the online journal)
1. Introduction
candidates for the hidden order that have been aired, at one time or another, include electronic multipoles with rank 2 (time-even, quadrupole), 3 (time-odd, octupole) and 5 (time-odd, triakontadipole) [6–8]. Symmetry-protected hidden order of a continuous phase transition does not feature in the vast literature on URu2 Si2 , to the best of our knowledge. We start from the premise, derived from many experimental observations, that, in the absence of an applied magnetic field, a hidden order parameter (i) does not allow any coupling to atomic displacements, (ii) it does not allow a non-zero magnetic dipole moment at U sites and (iii) it does not induce any molecular fields on Ru and Si sites. Requirement (i) follows from the observation that the phase transition does not induce a modulation of the chemical structure, (ii) follows from the absence of evidence for an ordering of magnetic dipoles, and (iii) follows from the absence of NMR signals on ligand nuclei.
Specific heat data for the uranium compound URu2 Si2 shows it undergoes two mean-field-like phase transitions, one at ≈1.0 K and a second at To ≈ 17.5 K [1–3]. Bulk superconductivity occurs at the low temperature transition, while the physical property of the transition at To is a puzzle to this day; for a review of developments in the period 1985–2013 see, for example, Mydosh and Oppeneer [4], and Chandra et al [5]. There is no consensus on the order parameter at To , because neither a modulation of the chemical structure or an ordering of magnetic dipoles is visible. In the absence of a visible, conventional order parameter it is customary to use the term hidden order for the enigmatic phase appearing at To . It is not decided whether 5f-electrons in URu2 Si2 should be treated as localized or itinerant, and 0953-8984/14/046003+09$33.00
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c 2014 IOP Publishing Ltd Printed in the UK
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By its very nature, symmetry formulated hidden order is independent of a specific model of the correlated electrons, with concomitant conclusions valid for either localized or itinerant electrons, in S–L or j–j coupling schemes. Moreover, candidate magnetic space-groups for URu2 Si2 can be applied to an interpretation of data gathered on the material using any experimental technique. We choose the technique of magnetic neutron Bragg diffraction to illustrate symmetry protected, hidden order in URu2 Si2 . For an experiment of this type, the quantity of interest is a unit-cell structure factor for uranium ions, constructed from a knowledge of the symmetry of sites used by U ions and translation symmetry of U ions in a unit cell. Symmetry of the chemical structure, tetragonal I4/mmm-type, includes body-centring that imposes the extinction rule h + k + l even on Bragg spots, where h, k, l are integer Miller indices. Magnetic Bragg spots that satisfy h + k + l odd are also allowed and, not surprisingly, they identify the symmetry of the hidden order parameter. A recent investigation of URu2 Si2 at 2 and 25 K, exploiting a sophisticated neutron Bragg diffraction technique, inspired the present study [9]. A single crystal of URu2 Si2 was placed in a strong magnetic field (9.6 T) aligned with the tetrad axis of the crystal (c-axis). We re-visit the published diffraction data, and confront them with exact expressions for the neutron diffraction amplitude compatible with allowed magnetic space-groups. We prove, beyond reasonable doubt, that available magnetic neutron Bragg diffraction data do not answer the question of hidden order in URu2 Si2 , and propose an experiment with the potential to settle the question. Chemical structure, definition of atomic multipoles and electronic structure factors for magnetic neutron diffraction are subjects of section 2. Thereafter, in section 3, we formulate symmetry-protected hidden order, and derive five candidate magnetic space-groups for investigation by neutron diffraction. A general expression for the scattering amplitude of neutrons is given in section 4, with pertinent examples discussed in an appendix. Section 5 contains our findings from tests of our five candidates, derived in section 3, against data gathered by Ressouche et al [9] on URu2 Si2 in a magnetic field parallel to the c-axis. A discussion of results is given in section 6.
Figure 1. (Left panel) Tetragonal, I4/mmm-type chemical structure
of URu2 Si2 . (Right panel) Orthonormal coordinate system and the polar angles, equation (2.3), defining the scattering vector S = (h, k, l)—shown for the (1, 1, 1) plane—used in calculations.
Angular brackets h· · ·i in a multipole denote an expectation value, or time average, of the enclosed operator, a spherical tensor with projections Q0 that obey −K 0 ≤ Q0 ≤ K 0 . 0 Parity-even multipoles engaged in neutron diffraction, hTQK0 i, 0
0
0
K i and a possess a complex conjugate hTQK0 i∗ = (−1)Q hT−Q 0 0
time signature (−1)K [10]. Unit-cell structure factors for magnetic diffraction are derived from [10], X 0 0 9QK0 = exp(id · S)hTQK0 id , (2.1) d
where the sum is over U ions at positions d in the unit cell, and the Bragg wavevector S = (h, k, l) is depicted in figure 1. The scattering amplitude for magnetic neutron Bragg diffraction is a scalar product of an electronic dipole hQ⊥ i and the neutron spin. The operator Q⊥ is a sum of electron spin and linear momentum field operators, and we use a universal expression [10], XX ˆ K00 (KQK 0 Q0 |1p). (2.2) hQ⊥,p i = (4π )1/2 YQK (S)9 Q KQ K 0 Q0 0 ˆ In (2.2), 9QK0 is the electronic structure factor (2.1), YQK (S) 0 0 is a spherical harmonic, and (KQK Q |1p) a Clebsch–Gordan ˆ is coefficient [12]. The unit vector for spherical harmonics, S, derived from the Bragg wavevector S, namely,
2. Material properties and multipoles
Sˆ = (h, k, l)/|(h, k, l)|
The chemical structure of paramagnetic URu2 Si2 is represented by a centrosymmetric, tetragonal space group I4/mmm ˚ c = 9.538 A ˚ at (#139), with lattice constants a = 4.112 A, 25 K [9]. Body-centring imposes an extinction rule h + k + l even, where h, k, l are integer Miller indices. Uranium ions use sites 2a that have point group symmetry 4/mmm (D4h ). Figure 1 shows the chemical structure and our orthonormal coordinates. Electronic spin and orbital degrees of freedom in the ground-state of a material are often expressed in terms of multipoles with discrete symmetries. Multipoles observed by magnetic neutron diffraction are parity-even and time-odd. We elect to use spherical multipoles, and these have a rank K 0 that is an odd integer.
= (sin θo cos φo , sin θo sin φo , cos θo ),
(2.3)
with polar angles θo and φo shown in figure 1. A specific example of the amplitude (2.2) appears in section 4, and confrontations with data are reported in section 5. 0 A multipole is a sum of unit tensors W (a,b)K , where a = 0 or 1 labels spin degrees of freedom, and b = 0, 1, . . . , 2l (l = 3 for f-electrons) labels orbital degrees of freedom. Multipoles 0 hTQK0 i engaged in neutron diffraction are a sum of three unit tensors, W (0,K )K for the purely orbital contribution, and 0 W (1,b)K with b = K 0 ± 1 [10] for the mixed spin–orbital contribution, about which we have more to say here and in section 4. 0
2
0
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Table 1. Parity-even irreducible representations conjugated with the 4/mmm10 site symmetry and the multipoles transformed by them, i.e., all allowed multipoles for neutron diffraction. The third column shows the corresponding space group I4/mmm10 representations associated with 0(k = 0) and M (k = 1, 1, 1) points. Multipole rank is odd with a maximum K 0 = 7. Real, 0 , and imaginary, 00 , parts of a 0 0 0 multipole are defined with the phase convention hTQK0 i = hTQK0 i0 + ihTQK0 i00 . The third row, 03+ , defines our naive model, cf appendix. For Eg , f1 and f2 are two orthogonal basis functions.
Site symmetry irrep
Multipoles
Space group irrep
A1g (4/mmm)
hT45 i00 , hT47 i00 hT23 i00 , hT25 i00 , hT27 i00 , hT67 i00 hT01 i, hT03 i, hT05 i, hT45 i0 , hT07 i, hT47 i0 hT23 i0 , hT25 i0 , hT27 i0 , hT67 i0 f1 → hT11 i0 , hT13 i0 , hT33 i0 , hT15 i0 , hT35 i0 , hT55 i0 , hT17 i0 , hT37 i0 , hT57 i0 , hT77 i0 f2 → hT11 i00 hT13 i00 , hT33 i00 , hT15 i00 , hT35 i00 , hT55 i00 , hT17 i00 , hT37 i00 , hT57 i0 , hT77 i0
01+ , M+ 1
B1g (40 /mmm0 ) A2g (4/mm0 m0 ) B2g (40 /mm0 m) Eg (η1 , 0; m0y m0z mz ) (η1 , η2 η1 = η2 ; m0z m0−xy mxy ) (η1 , η2 η1 6= η2 ; 20z /m0z )
At present there is no direct evidence that a localized 5f2 configuration for U ions is appropriate [4, 7]. Indeed, at low temperatures an itinerant f-electron character seems to prevail [5, 6, 8]. Even so, many multipolar theories using a crystal-field potential of hidden order have been proposed [4, 7, 11]. On the other hand, Cricchio et al [6] studied hidden order in URu2 Si2 using density-function theory for itinerant electrons, and analysed the solution in terms of unit tensors. For the order parameter they proposed a ferromagnetic motif of multipoles K 0 = 5 with W (1,6)5 the dominant player. 0 Contributions to diffraction by W (1,b)K with rank K 0 = 5 (triakontadipole) and K 0 = 7 (octaeicosahecatontapole) merit (1,b)K 0 arises from further comment. First, the unit tensor WP the contribution to Q⊥ proportional to exp(iR · S)s, a mixture of spin, s, and orbital, R, electron variables. The simulation of URu2 Si2 by Cricchio et al [6] uses an itinerant model of electrons with j–j coupling, and the j = 5/2 manifold of states is found to be the most important in determining ground-state electronic properties. Later work of a similar nature [8] confirms that energy bands crossing the Fermi level have mainly j = 5/2, with the hidden order a time-odd, triakontadipole in a nematic motif. States with j = 5/2 do not generate multipoles beyond K 0 = 5, whereas Russell–Saunders coupling of f-electrons permits K 0 = 7 (a theory of magnetic neutron scattering by ions described by j–j coupling is given [21]).
02+ , M+ 2 03+ , M+ 3 04+ , M+ 4 05+ , M+ 5
3.1. Local (or site) order parameters
Elements of symmetry that exist when URu2 Si2 is placed in a magnetic field are an essential set of references and, for the moment, we set aside the question of hidden order. (Magnetic space-groups are specified in terms of the Miller and Love and Belov–Neronova–Smirnova notations [13–15].) At this elementary level of investigation, magnetic properties of U ions are prescribed by symmetry found in the point group of a U site in the grey group (I4/mmm10 ) and the five parity-even irreps 0i+ with i = 1–5. Results are found in table 1; entries are all the parity-even and time-odd multipoles that are allowed to contribute in magnetic neutron diffraction. To explain the making of entries consider a particular case 02+ . The corresponding site symmetry is 40 /mmm0 of sites (2a) in the space group I40 /mmm0 derived from 02+ conjugated with 4/mmm10 . Entries in table 1 are parity-even multipoles that 0 satisfy elements of symmetry in 40 /mmm0 , namely, ImhTQK0 i ≡ 0
hTQK0 i00 with Q0 = ±2 and ±6 (examples of the steps involved in imposing elements of symmetry on a spherical multipole can be found in [16]). Multipoles listed in table 1 represent local (or site) order parameters either primary, if they drive the transition, or secondary, through a coupling to the primary one via the applied magnetic field. Global distortions induced from the local order parameters are determined by the space group representations subducing the site irreps and incorporating the translational cosets. A magnetic field parallel to the c-axis of I4/mmm has symmetry 03+ . In consequence, multipoles listed in table 1 for 03+ conjugated with 4/mmm10 are those that may be observed, irrespective of hidden order, and they define a naive model. Experimental observation of multipoles not present in the naive model will identify the hidden order, e.g., evidence of multipoles in the second line of entries, 40 /mmm0 , identify a hidden order parameter with 02+ symmetry.
3. Symmetry-protected hidden order
We assume that the phase transition with a hidden order parameter is continuous. Subsequent discussions are independent of a specific model of electronic correlations. The one and only reference to our knowledge about the amplitude for neutron diffraction, (2.2), is a restriction on the rank of multipoles, K 0 ; the rank is odd, with a maximum K 0 = 7 appropriate for actinide ions, and multipoles are time-odd (magnetic). A unit-cell structure factor for diffraction by uranium ions is constructed from a knowledge of the symmetry of sites used by U ions and translation symmetry of U ions in a unit cell.
3.2. Hidden order
To discuss possible symmetries of the system in the hidden order phase, the information about the translational symmetry 3
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Table 2. Irreducible representations of the I4/mmm10 space group with zero subduction frequencies, equation (3.1), for the mechanical and
magnetic reducible representations on the U, Ru and Si sites. The corresponding magnetic space-groups without and in a magnetic field parallel to the c-axis are given as well, along with the site symmetries for occupied atomic sites. Parity-odd candidates have identical neutron diffraction amplitudes that coincide with the naive model, and U ions use sites deprived of a centre of inversion symmetry. On the other hand, U sites are centres of inversion symmetry in parity-even candidates that feature in our discussion and section 5. Irrep. 01+ 01+ ⊕ 03+ 02+ 02+ ⊕ 03+ 04+ 04+ ⊕ 03+ 03− 03− ⊕ 03+ 04− 04− ⊕ 03+ M+ 1 + M+ 1 ⊕ 03 + M2 + M+ 2 ⊕ 03 − M2 + M− 2 ⊕ 03 M− 3 + M− 3 ⊕ 03 − M4 + M− 4 ⊕ 03
Magnetic space group
U (site symmetry)
Ru (site symmetry)
Si (site symmetry)
I4/mmm
4/mmm
4mm
I4/m
4/m..
¯ 4m2 ¯ 4..
4..
I4 /mmm
4 /mmm
4 m2
40 mm0
Fm0 m0 m
m0 m0 m
m0 m0 2
I4 /mm m
4 /mm m
20 20 2 4¯ 0 m0 2
Im0 m0 m
m0 m0 m
I4/m0 mm
4/m0 mm
0
0
0
0
0
0
0 0
0
I4 /m mm ¯ 0 m0 I 42 0
0
0
0
PI 4/mmm
4/mmm
P4/mn0 c0
4/m..
PI 42 /mmc
4 /mmm
P42 /mn0 m0
m.m0 m0
PI 42 /nnm P42 /nm c
40 /m0 m0 m ¯ 0 20 4m
PI 4/nmm
4/m0 mm
0 0
0 0
0
420 20
PI 42 /nmc
4 /m mm ¯ 0 m0 42
P42 /nn0 m0
4 mm 4.. 40 mm0
¯ 4.. ¯40 m20
0
0
m0 m0 2
2.m0 m0 4mm
2.20 20 ¯ 4m2
P4/nn c
0
40 m0 m
2.20 20 ¯ 4m2
4 /m mm ¯ 0 m0 42 0
0
m0 m0 2 4¯ 0 m20
420 20
I42 2 0
¯0
0
4.. 40 mm0
¯ 4.. 4¯ 0 m0 2
2.m0 m0 40 m0 m
2m m . ¯ 4m2 0
0
change across the transition is necessary. At the present time, reliable information is not available and only indirect measurements [17, 18] indicate the M-point of symmetry (k = 1, 1, 1) as the most probable candidates for the propagation vector of the hidden order parameter (k = 1, 1, 1 ≡ 0, 0, 1, because reciprocal and real lattices are I-centred and translation by 1, 1, 0 is allowed). Let us point out the specific property of this point, namely, the eight out of ten irreducible representations of the I4/mmm10 space group associated with this propagation vector are one-dimensional (M± i , i = 1–4). A timeodd order parameter transforming accordingly to one of these representations does not allow coupling to any symmetry breaking physical quantity like atomic displacement, for instance, measurable in high-precision diffraction experiments. This can stimulate a hidden nature of the associated phase if the order parameter also does not generate a magnetic dipole on the U position and internal fields on the Ru and Si sites. This peculiarity of the M-point is also relevant to the 0-point (k = 0) whose eight irreps (0i± , i = 1–4) are one-dimensional as well and, therefore, we will restrict our consideration to these two propagation vectors. Any other special/general point or line/plane of symmetry involves only multidimensional irreps which imply coupling to physical quantities detectable in conventional diffraction and spectroscopic measurements.
2m0 m0 .
0
4mm
¯ 4.. ¯40 m20
40 mm0
2.20 20
2.m0 m0
4..
In consequence, an order parameter which (i) does not allow any coupling to atomic displacements, (ii) does not allow non-zero magnetic dipole moment at U sites and (iii) does not induce internal fields on the Ru and Si sites is ‘symmetry protected’ from a direct measurements using standard techniques. These symmetry conditions can be rigorously worked out from subduction frequencies, mν , of the irreps for the appropriate reducible representations on the U, Ru and Si sites, namely, X mν = (1/n(g)) χ (g)χ ∗ν (g), (3.1) g
where n(g) is the number of elements in the space group, χ (g) and χ ν (g) are characters of the element, g, in the reducible and irreducible representations, respectively. The irreps with zero frequencies are listed in table 2, along with the corresponding magnetic space-groups. All of them are one-dimensional and represent preferable candidates for the symmetry of the hidden order parameter. We take advantage of the experimental evidence of the continuous nature of the phase transition at To implying a single irreducible representation responsible for the symmetry breaking. The signature of the hidden order parameter in a neutron diffraction experiment is the presence of an angular dependence of the amplitude hQ⊥ i, caused by a significant 4
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contribution to diffraction from one or more multipoles from table 1. In the available experimental data, only reflections with h + k + l even in the applied magnetic field were measured [9] and it is important to clarify the symmetry allowed multipoles contributing to these reflections. In the general case, the following situations should be distinguished in this context.
(3) The hidden order parameter breaks the four-fold symmetry and transforms as one of the two-dimensional − irreps, 05− , M+ 5 or M5 . Independent of the direction of the order parameter in the representation spaces (η1 , η2 ), the associated multipoles cannot directly contribute to the h + k + l even reflections. The presence of the magnetic field (Hz ), however, is essential in these cases and provides conditions for coupling multipoles transforming as 02+ and 04+ irreps from the table 1, as the secondary order parameters (δ) through the free-energy invariants of the form, δHz (η12 − η22 ) and δHz η1 η2 respectively. Thus, the cases when the experimentally observed multipoles are associated with the 02+ and 04+ irreps require careful consideration to decide whether they are directly representing hidden order parameter, or they are only coupled to the directly invisible primary order parameter with the symmetry of the two-dimensional representations.
(1) The hidden order parameter preserves the translational symmetry (the propagation vector is k = 0) and is associated with one of the 0 representations listed in table 1 (0i+ , i = 1–5). In this case, the appropriate multipoles contribute to the h + k + l even reflections and therefore these reflections carry information about angular anisotropy in hQ⊥ i. The presence of the magnetic field is not essential and introduces additional variables directly coupled to the induced dipole component. These variables are related to the five higher rank multipoles transforming as the 03+ representation (table 1)—labelled the naive model—and therefore their presence is always expected through the linear coupling to the ferromagnetic component, irrespective to the symmetry of the hidden order parameter. (2) The hidden order parameter transforms as a onedimensional parity-odd representation associated with k = 0 propagation vector (0i− , i = 1–4) or as any onedimensional irrep associated with M-point of symmetry (M± i , i = 1–4). In all these case, any signature of the hidden order parameter cannot be detected in the h + k + l even reflections. This conclusion comes from the fact that it is impossible to form an appropriate free-energy coupling invariant in the applied magnetic field between any of these representations and the irreps listed in table 1. As an explicit, illustrative example of our general symmetry argument, we examine electronic structure factors (2.1) for one of five candidates tested against neutron diffraction data in section 5. In the candidate with + M+ 1 ⊕ 03 symmetry, the two irreps represent symmetry of the hidden order parameter, M+ 1 , and the applied magnetic field, 03+ , respectively. The resultant magnetic space group in table 2 is P4/mn0 c0 , with the U site symmetry 4/m. This site symmetry allows multipoles to be complex, with Q0 = 0, ±4 for K 0 ≤ 7. We find the 0 electronic structure factor 9QK0 is allowed to be different from zero for two classes of reflections, ( K0 0 2hTQ0 i for h + k + l even K0 9Q0 = K 0 00 2ihTQ0 i for h + k + l odd.
Parity-odd cases in table 2 possess the diffraction pattern of the naive model, table 1, and are of no interest in an investigation of hidden order using neutron diffraction. However, they are of interest for x-ray diffraction, in which intensity of a Bragg spot is enhanced by tuning the primary energy to an atomic resonance. Parity-odd multipoles can be observed in this type of x-ray diffraction. By way of illustrations, consider 03− and M− 3 in table 2, noting that 422 is one of 11 enantiomorphic crystal classes. It may be shown that, for example, magneto-electric dipoles (anapoles) contribute to resonant diffraction at h + k + l even (h + k + l odd) for 03− (M− 3 ). Five candidates for hidden order tested in section 5 are parity-even cases in table 2, and U ions use sites that are centres of inversion symmetry. To ease the notation, in future parity-even cases are listed #1–#5. The five candidates possess multipoles associated with 03+ in table 1 (naive model) plus multipoles associated with one other space group irrep. Hence, #3 contains multipoles associated with 04+ in addition to those associated with 03+ . Multipoles over and above those in 03+ contribute to Bragg points with h + k + l even and h + k + l odd for 0-points and M-points, respectively. Recall that, 0-points and M-points correspond to ferromagnetic and antiferromagnetic motifs of dipole moments, respectively. #1 01+ ⊕ 03+ ; space group I4/m, U site symmetry 4/m.., and crystal class 4/m. Diffraction allowed for h + k + l even. 0 Multipoles can have an imaginary component with hTQK0 i00 6= 0, and Q0 = 0, ±4 for K 0 ≤ 7. #2 02+ ⊕ 03+ ; space group Fm0 m0 m, crystal class m0 m0 m (site symmetry mm0 m0 in the original tetragonal setting 4/mm0 m0 ). The extinction rule is h + k + l even, because 0 I-centring is present as in #1, and multipoles satisfy hTQK0 i =
Multipoles contributing to the first class of reflections are purely real. This fact, in combination with the conditions imposed by the 4/m site symmetry, restricts the allowed multipoles to those associated with the 03+ representation only, which are always permitted in the applied field. Thus, no additional multipoles which can be taken as a signature of the hidden order parameter are expected for the h + k + l even reflections, and all information about the hidden order parameter is contained in the h + k + l odd reflections.
0
(−1)p hTQK0 i∗ with Q0 = 2p = 0, ±2, ±4, ±6. Note that U ions occupy sites with the same symmetry in candidate #5, for which the primitive cell allows h + k + l odd. #3 04+ ⊕ 03+ ; space group Im0 m0 m, crystal class m0 m0 m, site symmetry m0 m0 m. Diffraction allowed for h + k + l even, 0 and hTQK0 i purely real with Q0 = 2p. 5
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+ 0 0 #4 M+ 1 ⊕03 ; space group P4/mn c , site symmetry 4/m.., 0 0 and crystal class 4/mm m . Multipoles can be complex with allowed Q0 = 0, ±4. We find the electronic structure factor 0 9QK0 in (2.1) can be different from zero for two classes of reflections, 0 9QK0 0 9QK0
0 2hTQK0 i0 ; 0 2ihTQK0 i00 ;
= =
h + k + l even,
4.1. Russell–Saunders coupling 0
Let (J k T K k J) be a reduced matrix element in an atomic picture with Russell–Saunders (S–L) coupling. We define, 0
and
and coefficients t
h + k + l odd.
0 9QK0
=
p even, h + k + l even,
0 2ihTQK0 i00 ;
(4.3)
K0 Q0
in hT i are quantities to be determined. 0
0
0
0
is complex for Q0 6= 0, while t0K is purely real. Dependence on the magnitude of S appears in radial integrals hjn i with n even [19], normalized such that hj0 i = 1, 0 0 and all other hjn i = 0, for |S| = 0. The unit tensor W (0,K )K 0 0 in hTQK0 i is multiplied by [hjK 0 −1 i + hjK 0 +1 i], while W (1,b)K is multiplied by hjb i with b = K 0 ± 1. In the present work, 0 hjn is are included in reduced matrix elements of TQK0 , which are tabulated in [20] and reproduced here for convenience of the reader. For Russell–Saunders coupling and a single J-manifold K = K 0 ± 1, and K 0 is odd with maximum K 0 = 7 for equivalent f-electrons. Reduced matrix elements are purely real, and for f 2 with J = 4, S = 1 and L = 5, √ (J k T 1 k J) = (8/ 5)[hj0 i + (74/45)hj2 i], √ (J k T 3 k J) = −(52/15) (2/55)[hj2 i + (15/11)hj4 i], (4.4) √ (J k T 5 k J) = −(4/121) (195)[hj4 i + (248/39)hj6 i], √ (J k T 7 k J) = (136/99) (14/143)hj6 i.
0
0
K0 Q0
0
(−1)p hTQK0 i∗ with Q0 = 2p, and, 0
0
K i it follows that, tK From the property hTQK0 i∗ = (−1)Q hT−Q 0 Q0
Purely imaginary multipoles induced from 01+ (table 1) contribute at Bragg spots with h + k + l odd. + 0 0 #5 M+ 2 ⊕ 03 ; space group P42 /m.n m , crystal class 0 0 0 0 0 4/mm m , site symmetry mm m . Multipoles satisfy hTQK0 i = 9QK0 = 2hTQK0 i0 ;
0
K K k J), hTQK0 i = tQ 0 (J k T
and
p odd, h + k + l odd.
In line with #4, imaginary multipoles induced from 02+ (table 1) contribute at Bragg spots with h + k + l odd. 4. Neutron scattering amplitude
Ressouche et al [9] measured the component of hQ⊥ i parallel to the c-axis, hQ⊥,c i. Our coordinates in figure 1 are such that a dipole with spherical component p = 0 is parallel to the crystal c-axis, so hQ⊥,c i = hQ⊥,0 i. In hQ⊥,0 i, there is ˆ in terms of advantage in writing spherical harmonics YQK (S) Legendre polynomials of order l, Pl , and associated Legendre Q0 polynomials, Pl , that depend on θo only; see (2.3) and figure 1 [12]. From (2.2) we find, √ X 0 0 0 1/2 hQ⊥,0 i = − 3 [K /(2K +1)] hT0K i(PK 0+1 − PK 0−1 )
The dipole hT01 i and the magnetic moment h(L + 2S)c i are related, namely, hT01 i = h(L + 2S)c i/3 in the forward direction (0, 0, 0). A notable feature of f 2 in Russell–Saunders coupling is that the octupole (K 0 = 3) is independent of orbital angular momentum, which is true of (J k T 7 k J) in the general case, f n. An appendix contains explicit expressions for multipoles derived from a model wavefunction. The wavefunction in question encompasses the one examined by Ressouche et al [9].
K0
X 0 0 + ϒ(K 0 , Q0 )[hTQK0 i0 cos(Q0 φo ) + hTQK0 i00 sin(Q0 φo )] Q0
×
Q0 (PK 0+1 −[(K 0 +1)(K 0
+ Q )/K (K −Q 0
0
0
0
Q0 +1)]PK 0−1 )
5. Confrontation of experimental data and simulations
,
(4.1)
A simulation of the magnetic neutron scattering amplitude, hQ⊥ i, based on the naive model (section 3, appendix) has been confronted with available experimental data. Simulations use K 0 in (4.3) Russell–Saunders coupling, using the definition of tQ and reduced matrix elements (4.4). We find that the naive model does not give a satisfactory explanation of diffraction data. The quality of fit is not significantly different from use of the dipole moment alone (expression (4.1) with K 0 = 1), as can be seen in the results in figure 2. Other plausible candidates of the magnetic structure of URu2 Si2 in an applied magnetic field have been similarly tested. Constructions of our five candidates, labelled #1–#5, are the subjects of section 3. The quality of fits to data, set out in figure 3, is the same for all candidates, i.e., available neutron diffraction data are not able to distinguish between candidates for the magnetic structure and there is no useful
where K 0 is odd and Q0 > 0. The coefficient in the sum on Q0 in (4.1) is, 0
ϒ(K 0 , Q0 ) = 2(−1)Q [(K 0 − Q0 + 1)/(K 0 + 1)] × [(K 0 −Q0 −1)!(K 0 −Q0 )/(K 0 +Q0 − 1)!(K 0 +Q0 )]1/2 , (4.2) for Q0 < K 0 and, ϒ(K 0 , K 0 ) = −[2/(K 0 + 1)][1/(2K 0 )!]1/2 , for Q0 = K 0 . By definition, hQ⊥,0 i is proportional to sin2 θo and {2hQ⊥,0 i/sin2 θo } = h(L + 2S)c i for |S| = 0. The result (4.1) uses K = K 0 ± 1 in (2.1) and it is valid for either an itinerant or an atomic model of electronic structure. 6
J. Phys.: Condens. Matter 26 (2014) 046003
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Figure 2. Experimental (subscript obs) versus calculated (subscript
calc) values of hQ⊥,0 i2 , using the naive model (top panels) and the dipole approximation (bottom panels). Note that the naive model is identical to the candidates #4 and #5 in section 3, for h + k + l even. Sample temperature 2 K (left panels) and 25 K (right panels). The dipole approximation neglects all multipoles in the amplitude (4.1) apart from the dipole, hT01 i ∝ t01 in (4.3). Values of RF = P
100
2 2 s |(Q⊥,s )obs −(Q⊥,s )calc | P 2 (Q ) s ⊥,s obs
are given for each distribution.
information concerning a hidden order parameter. In figures 2 and 3, multipoles included in the fit to data are indicated in individual panels. Multipoles in candidates #1, #2 and #3 not in the naive model, top panel in figure 2, are indicated on each panel in figure 3, i.e., multipoles in a fit are the naive model plus those indicated. The experimental data present a normalized set of Qhkl = 2hQ⊥,0 i/sin2 θo values for 52 reflections. For fitting purposes, expression (4.1) for hQ⊥,0 i is reduced to the following form:
Figure 3. Experimental (obs) versus calculated (calc) intensities,
(5.1)
hQ⊥,0 i2 , for the candidates #1 (top), #2 and #3 (bottom) defined in section 3. Sample temperature 2 K (left panels) and 25 K (right K0 panels). RF is defined in the caption to figure 2. Quantities tQ 0 inferred from data are defined in (4.3). Multipoles included over and above the naive model—top panels in figure 2—are indicated, 0 K 0 00 0 e.g., imaginary components hTQK0 i00 ∝ (tQ 0 ) with Q = ±4 in top panels for candidate #1, 01+ ⊕ 03+ .
where the parameter vector t = {t } , vector Chkl consists of the numerical coefficients unique for each reflection and the corresponding component of the vector t. Numerical coefficients C0hkl are unique for each reflection, and h(L+2S)c i is the dipole magnetic moment, with {2hQ⊥,0 i/sin2 θo } = h(L + 2S)c i for |S| = 0. Radial integrals in (4.4) were calculated with expressions found in [19]. We implemented a simulated annealing algorithm to fit the experimental data. Simulated annealing has been demonstrated to be a powerful technique for finding a global minimum in complex multiple parameter optimization problems, including data mining [23–27]. The basic idea
behind it resembles crystal growth from melt. First the temperature of the system is set to be high, parameters are set to initial random values, and the energy of the system is calculated. Then a standard metropolis Monte Carlo [28] procedure is performed at this temperature: the parameters are randomly changed by small steps and the new energy is calculated. If the new energy is reduced, this parameter configuration is accepted, otherwise it is accepted with probability equal to the Boltzmann factor exp(−1E/kT). At the next simulated annealing step the system temperature is decreased, and the procedure repeated. While the temperature
Qhkl = C0hkl h(L + 2S)c i + Chkl · t, K0 Q0
7
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is high enough, the energy barriers can be overcome, and the parameter space can be explored by a random walk. When the temperature is decreasing slowly, this allows the algorithm to converge to the global minimum.
motif of dipole moments, an M-point) or is associated with the parity-odd multipoles. Cases when the multipoles specified in table 1 appear as secondary order parameters (case 3 in the section 3) cannot be unambiguously ruled out, because the corresponding coupling is expected to be small with the corresponding contribution to diffraction placed beyond available experimental sensitivity. An assumption that our experimental data are sensitive to secondary order parameters would imply that the primary candidates for the symmetry + of the parity-even, hidden order parameter are M+ 1 and M2 representations (candidates #4 and #5 in section 3). In the case of a parity-odd nature of the order parameter, our diffraction data does not set any restrictions and the symmetry can be represented by any out of the five one-dimensional, parity-odd representations set out in table 2. The first two candidates, + M+ 1 and M2 , can be identified in magnetic neutron diffraction by the appearance of Bragg spots with Miller indices h + k + l odd that violate the body-centring extinction rule and associated angular anisotropy, while parity-odd candidates can be explored by resonant x-ray Bragg diffraction.
6. Discussion
Let us summarize results derived from our formulation of symmetry-protected hidden order at a continuous phase transition, and its application to magnetic neutron diffraction by tetragonal URu2 Si2 , illustrated in figure 1. A recent investigation of URu2 Si2 exploited a sophisticated neutron Bragg diffraction technique, two temperatures that straddle the onset of the enigmatic phase, and a strong magnetic field (9.6 T) parallel to the tetrad axis [9]. The principal conclusion therein, about the electronic triakontadipole (a rank 5 multipole also called a dotriacontapole) as the primary candidate for the hidden order parameter, is not confirmed in our present study. This result is graphically illustrated in figures 2 and 3. What is shown in these two figures is that, Bragg intensities that obey the body-centring extinction rule are adequately represented by the dipole approximation to magnetic scattering, with no discernible angular anisotropy in the distribution of Bragg spots. Our fresh approach takes advantage of magnetic symmetry to enumerate allowed multipoles, as well as coupling between them in the applied magnetic field. The method of reconstruction, of the magnetization distribution in the unit cell, used by Ressouche et al [9] is absolutely restricted to cases when multipoles representing the hidden order parameter have symmetry identical to the symmetry of the field-induced dipole. However, this situation is very unlikely, because, in this case, a non-zero ferromagnetic component is expected in zero magnetic field as a secondary order parameter, which has never been reported. The observed evolution of the magnetization distribution with temperature can be attributed to the trivial bilinear coupling between the induced dipole and the hidden order parameter [9]. It is crucial to understand that, this type of coupling is always allowed and does not affect the symmetry of the distribution and, therefore, cannot reveal the hidden order parameter. The approach used in the present communication is based on exact results for the magnetic neutron scattering amplitude, expressed with parity-even, time-odd electronic multipoles restricted by the symmetry of the hidden order parameter and the applied magnetic field. Results include multipoles, listed in table 1, that contribute to magnetic scattering but are not coupled to the field-induced dipole component and, in consequence, do not affect the symmetry of the magnetization distribution. Thus, the present approach has potentiality to reveal multipoles which depend on the symmetry of the hidden order parameter, and create an unambiguous signature of it. None of the candidates for hidden order, table 1, has been found to show a substantial increase of contribution to diffraction below the transition temperature, which strongly indicates that the hidden order parameter either has non-zero propagation vector (consistent with an antiferromagnetic
Acknowledgment
One of us (SWL) is grateful to Dr C Detlefs for his initial interest in the reported study. Appendix. Model wavefunction
We use Russell–Saunders coupling in an orientation to eligible multipoles in hQ⊥,0 i. Specifically, we examine values K 0 in (4.3) derived from the naive model introduced of tQ 0 0
in section 3, table 1. A purely real multipole hTQK0 i with Q0 = 0, ±4 satisfies symmetry elements in 4/mm0 m0 , and the extinction rule for the associated magnetic space group, I4/mm0 m0 , is h + k + l even. First-principle electronic structure calculations reported by Haule and Kotliar suggest that U4+ (5f2 ) atomic states in URu2 Si2 are adequately described by Russell–Saunders coupling and projections M = 0, ±4 [22]. In keeping with this finding, we adopt a singlet wavefunction, |ψi = a|4i + b| − 4i + c|0i,
(A.1)
where |Mi = |J = 4, Mi, and complex coefficients a, b and c are chosen to make the wavefunction properly normalized. The saturation magnetic moment associated with (A.1) is h(L + 2S)c i = (16/5)(|a|2 − |b|2 ). From the definition (4.3), √ 0 0 0 t0K = hψ|T0K |ψi/(J k T K k J) = [5/16 (2K 0 + 1)] × h(L + 2S)c i(444 − 4|K 0 0).
(A.2)
Evaluation of the Clebsch–Gordan coefficient in (A.2) shows 0 that all t0K for the naive model are of one sign, and diminish in size with increasing K 0 , e.g., t03 /t07 = 19.08. In addition, 0
0
0
K K t+4 = hψ|T+4 |ψi/(J k T K k J) √ = [1/ (2K 0 + 1)](bc∗ − a∗ c)(4044|K 0 4). (A.3) 8
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5 and t7 have the same sign with According to (A.3), t+4 +4 5 7 t+4 /t+4 = 1.48. In terms of mixing parameters, α, β, ϕ, used by Ressouche et al [9],
[10] Lovesey S W et al 2005 Phys. Rep. 411 233 [11] Kusunose H and Harima H 2011 J. Phys. Soc. Japan 80 084702 [12] Edmonds A R 1960 Angular Momentum in Quantum Mechanics (Princeton, NJ: Princeton University Press) [13] Stokes H T, Hatch D M and Campbell B J 2007 ISOTROPYstokes.byu.edu/isotropy.html [14] Campbell B J, Stokes H T, Tanner D E and Hatch D M 2006 J. Appl. Crystallogr. 39 607 [15] Gallego S V et al 2012 J. Appl. Crystallogr. 45 1236 [16] Lovesey S W and Balcar E 2013 J. Phys. Soc. Japan 82 021008 [17] Villaume A et al 2008 Phys. Rev. B 78 012504 [18] Bourdarot F et al 2010 J. Phys. Soc. Japan 79 064719 [19] Brown P J 2004 Radial integrals for U4+ in section 4.4.5 of volume C International Tables of Crystallography (Dordrecht: Springer) [20] Osborn R et al 1991 Handbook on the Physics and Chemistry of Rare Earths vol 14 (Amsterdam: North-Holland) [21] Balcar E and Lovesey S W 1991 J. Phys.: Condens. Matter 3 7095 [22] Haule K and Kotliar G 2009 Nature Phys. 5 796 [23] Kirkpatrick S, Gelatt C D Jr and Vecchi M P 1983 Science 220 671 [24] Bertsimas D and Tsitsiklis J 1993 Stat. Sci. 8 10 [25] Barradas N P and Smith R 1999 J. Phys. D: Appl. Phys. 32 2964 [26] Munakata T and Nakamura Y 2001 Phys. Rev. B 64 046127 [27] Ireland J 2007 Solar Phys. 243 237 [28] Metropolis N et al 1953 J. Chem. Phys. 21 1087
(|a|2 − |b|2 ) = sin(2α) cos(β) sin(ϕ + ν), (A.4) √ (a∗ c − bc∗ ) = (1/ 2) sin(2α) cos(β) cos(ϕ + ν). 5 and Since (a∗ c − bc∗ ) is purely real, the quantities t+4 7 t+4 are purely real. In (A.4), ν is assumed to be known from independent calculations [11]. (For ν these authors use instead the symbol θ and find θ ≡ ν = 0.998 in a discussion of hexadecapole order, which is not visible in neutron diffraction, because rank 4 is excluded by a selection rule for states with the same values of S, L, and J.)
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