PRL 112, 171603 (2014)
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PHYSICAL REVIEW LETTERS
Universal Thermal Corrections to Single Interval Entanglement Entropy for Two Dimensional Conformal Field Theories 1
John Cardy1 and Christopher P. Herzog2 Oxford University, Rudolf Peierls Centre for Theoretical Physics 1 Keble Road, Oxford OX1 3NP, United Kingdom and All Souls College, Oxford OX1 4AL, United Kingdom 2 C. N. Yang Institute for Theoretical Physics, Department of Physics and Astronomy Stony Brook University, Stony Brook, New York 11794, USA (Received 10 March 2014; published 30 April 2014) We consider single interval Rényi and entanglement entropies for a two dimensional conformal field theory on a circle at nonzero temperature. Assuming that the finite size of the system introduces a unique ground state with a nonzero mass gap, we calculate the leading corrections to the Rényi and entanglement entropy in a low temperature expansion. These corrections have a universal form for any two dimensional conformal field theory that depends only on the size of the mass gap and its degeneracy. We analyze the limits where the size of the interval becomes small and where it becomes close to the size of the spatial circle. DOI: 10.1103/PhysRevLett.112.171603
PACS numbers: 11.25.Hf, 03.65.Ud, 89.70.Cf
The idea of entanglement, that a local measurement on a quantum system may instantaneously affect the outcome of local measurements far away, is a central concept in quantum mechanics. As such, it underlies the study of quantum information, communication, and computation. However, entanglement also plays an increasing role in other areas of physics. To name three, measures of entanglement may detect exotic phase transitions in many-body systems lacking a local order parameter [1,2], such measures order quantum field theories under renormalization group flow [3,4], and entanglement is a key concept in the black hole information paradox (see, e.g., Refs. [5,6]). Important measures of entanglement for a many-body system in its ground state are the Rényi and entanglement entropies. To define these quantities, we first partition the Hilbert space into pieces A and complement A¯ ¼ B. Typically (and hereafter in this Letter), A and B correspond to spatial regions. Not all quantum systems may allow for such a partition. The reduced density matrix is defined as a partial trace of the full density matrix ρ over the degrees of freedom in B: ρA ≡ trB ρ:
(1)
The entanglement entropy is then the von Neumann entropy of the reduced density matrix: SE ≡ −trρA log ρA :
(2)
The Rényi entropies are Sn ≡
1 log trðρA Þn ; 1−n
where SE ¼ limn→1 Sn . 0031-9007=14=112(17)=171603(5)
(3)
The entanglement and Rényi entropies are well defined for excited and thermal states as well. One simply starts with the relevant excited state or thermal density matrix instead of the ground state density matrix ρ ¼ j0ih0j. For example, for thermal states, one would start with ρ¼
e−βH ; trðe−βH Þ
(4)
where β is the inverse temperature and H is the Hamiltonian. However, it is well known that for mixed states, the entanglement entropy is no longer a good measure of quantum entanglement. The entanglement entropy is contaminated by the thermal entropy of region A and, in fact, in the high temperature limit becomes dominated by it. It is, of course, never possible to take a strict zero temperature limit of real world systems. If we are to use entanglement entropy to measure quantum entanglement, we need to know how to subtract off nonzero temperature contributions to SE and Sn . Little has been said about these corrections in the literature thus far. Reference [7] studied entanglement entropy for a massive relativistic scalar field in 1 þ 1 dimensions. The authors provided numerical evidence that the corrections scale as e−βm in the limit βm ≫ 1, where m is the mass of the scalar field. They further conjectured that, for any a many-body system with a mass gap mgap , such corrections should scale as e−βmgap when βmgap ≫ 1. This conjecture more or less immediately follows from writing Eq. (4) as a Boltzmann sum over states. By assumption, the first excited state will come with contribution e−βmgap and give the dominant correction. Some possible but probably rare exceptions involve cases where, for some reason, the first excited state contribution
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to the entanglement entropy vanishes or where the degeneracy of states with energy E ¼ mgap itself has an exponential dependence on temperature. While knowledge of the e−βmgap dependence is a good starting point, it would be even better to know the coefficient. In this Letter, we calculate the coefficient for (1 þ 1) dimensional conformal field theories (CFTs) on a circle of circumference L in the case where A consists of a single interval of length l. The answer is
dimensional CFT. We then check that the corrections have expected behavior in the limits l ≪ L and l → L. Universal thermal corrections.—Starting from the thermal density matrix (4), we compute the first few terms in a low temperature expansion of the Rényi entropies. Introducing a complete set of states, we can write the thermal density matrix as a Boltzmann sum over states: ρ¼
g 1 sin2Δ ðπl=LÞ δSn ¼ − n e−2πΔβ=L 1 − n n2Δ−1 sin2Δ ðπl=nLÞ þ oðe−2πΔβ=L Þ;
(5)
πl πl δSE ¼ 2gΔ 1 − cot e−2πΔβ=L þ oðe−2πΔβ=L Þ; (6) L L where Δ is the smallest scaling dimension among the set of operators including the stress tensor and all primaries not equal to the identity, and g is their degeneracy. In order for this result to hold, the CFT needs to have a unique ground state separated from the first excited state by a nonzero mass gap (induced by the finite volume of the system). The correction (6) was obtained from the n → 1 limit of Eq. (5). Given the form of these corrections, another possible use for them is the determination of Δ and g. From the literature [8–11], it is possible to deduce the correction (5) in two limiting cases–for free theories and for theories with AdS/CFT duals and large central charge. Using bosonization, Refs. [8,9] computed these corrections for massless free Dirac fermions. In this case, free fermionic operators with scaling dimension Δ ¼ 1=2 and g ¼ 4 yield the appropriate correction. Reference [11] considered CFTs with large central charge c and gravitational duals. In this case, the entanglement and Rényi entropies can be written as an asymptotic series in c where the leading term is OðcÞ. The authors computed the Oð1Þ corrections to the entanglement and Rényi entropies due to the stress tensor TðzÞ, which has Δ ¼ 2 and g ¼ 2. Although the authors do not write explicit formulas, it is trivial to extend their computation to operators of general conformal dimension. The computation involves a low temperature expansion of a Selberg-like zeta function. In all cases, their expression would match Eq. (5). For free CFTs more generally, the important observation is that this same Selberg-like zeta function appears in Ref. [10] where the authors compute the determinant of the Laplacian for a bc system on an arbitrary genus Riemann surface. From Ref. [10], one can then deduce that the same correction (5) holds for free CFTs [12]. A check of these statements was performed explicitly for a free complex boson and free fermions in Ref. [13]. In this Letter, we will show by explicit computation that the corrections (5) and (6) hold for a general (1 þ 1)
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X 1 jϕihϕje−βEϕ : trðe−βH Þ jϕi
(7)
For a CFT on a cylinder of circumference L, coordinatized by w ¼ x − it, the Hamiltonian is 2π c ~ L0 þ L0 − ; (8) H¼ L 12 where L0 and L~ 0 are the zeroth level left- and right-moving Virasoro generators and c is the central charge. We will assume in what follows that placing the CFT on a cylinder produces a unique ground state and gaps the theory. Let ψðwÞ ≠ 1 be a Virasoro primary operator and jψi ¼ limt→−∞ ψðwÞj0i the corresponding state where ~ ~ As the L0 jψi ¼ hjψi, L~ 0 jψi ¼ hjψi, and Δ ¼ h þ h. jψi are lowest weight states in the CFT, it follows—with one important exception—that the smallest nonzero Eϕ must correspond to a primary operator ψðwÞ and, moreover, that Eψ ¼ 2πðΔ − ðc=12ÞÞ=L. The one important exception is when Δ > 2 and two descendants of the identity ~ wÞ, ¯ operator, the stress tensor TðwÞ and its conjugate Tð give the dominant correction. (The identity operator itself can be thought of as yielding the leading zero temperature contribution to the Rényi entropies.) We are assuming Δ > 0, which may not be true for nonunitary CFTs, and no gravitational anomaly, i.e., cL ¼ cR . We divide up the spatial circle into regions A and complement B and compute the reduced density matrix ρA . Let us consider a low temperature expansion where ðj0ih0j þ jψihψje−2πΔβ=L þ Þ : 1 þ e−2πΔβ=L þ It follows then that ρ¼
tr½trB ðj0ih0j þ jψihψje−2πΔβ=L þ Þn ð1 þ e−2πΔβ=L þ Þn n ¼ trðtrB j0ih0jÞ 1þ
trðρA Þn ¼
(9)
(10)
tr½trB jψihψjðtrB j0ih0jÞn−1 −2πΔβ=L þ − 1 ne þ : trðtrB j0ih0jÞn (11) The first term gives the zero temperature contribution to the Rényi entropy for a finite system [14]
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1 − n2 L πl þ ; log trðtrB j0ih0jÞ ¼ c log sin π L 6n (12) n
trivially. A uniformizing map from n copies of a cylinder of circumference L to the plane is
ζ ðnÞ ¼
where l is the length of A. Up to a normalization issue we will come to shortly, the expression tr½trB jψihψjðtrB j0ih0jÞn−1 trðtrB j0ih0jÞn
(13)
is a two-point function of the operator ψðwÞ. By the state operator correspondence, we can interpret the state jψi as limt→−∞ ψðx; tÞj0i and similarly hψj ¼ limt→∞ h0jψðx; tÞ. In path integral language, the trace over an nth power of the reduced density matrix becomes a partition function on an n-sheeted copy of the cylinder, branched over the interval A. The expression (13) is then the two-point function hψðw2 ; w¯ 2 Þψðw1 ; w¯ 1 Þi on this multisheeted cylinder for a particular choice of w1 and w2 . By the Riemann-Hurwitz theorem, this multisheeted cylinder has genus zero. There exists then a uniformizing map which takes the multisheeted cylinder to the plane, and on the plane, the two-point function (13) can be computed
hψðw2 ; w¯ 2 Þψðw1 ; w¯ 1 Þin ¼
2πiw=L e − eiθ2 1=n : e2πiw=L − eiθ1
We choose θ1 and θ2 such that the map is branched over the interval A, which has end points on the cylinder such that θ2 − θ1 ¼ 2πl=L. Under this map, an operator inserted on the jth cylinder at t ¼ −∞ is located on the ζ ðnÞ plane ðnÞ at ζ −∞ ¼ eiðθ2 −θ1 Þ=nþ2πij=n , while an operator inserted at ðnÞ t ¼ ∞ is located at ζ ∞ ¼ e2πij=n . The correlation function on the ζ plane between two ðnÞ ðnÞ quasiprimaries inserted at points ζ 2 and ζ1 is hψðζ 2 ; ζ¯ 2 Þψðζ 1 ; ζ¯ 1 Þi ¼ ðnÞ
ðnÞ
ðnÞ
ðnÞ
1 ðnÞ 2h ðnÞ 2h~ ðζ 21 Þ ðζ¯ 21 Þ
dζ ðnÞ =dw 1 ζ ðnÞ : ¼ dζð1Þ =dw n ζð1Þ
ðnÞ
ðζ 21 Þ2h
(17)
We find then that hψðw2 ; w¯ 2 Þψðw1 ; w¯ 1 Þin ~ ¼ fðζ ðnÞ ; ζ ð1Þ ; hÞfðζ¯ ðnÞ ; ζ¯ ð1Þ ; hÞ; hψðw2 ; w¯ 2 Þψðw1 ; w¯ 1 Þi1 (18) where 1 x1 x2 h y2 − y1 2h : fðx; y; hÞ ≡ 2h y1 y2 x2 − x1 n
ðnÞ 2h~
ðζ¯ 21 Þ
;
(16)
The result (5) then follows immediately, and then Eq. (6) follows by taking the n → 1 limit of Eq. (5). The degeneracy factor g comes from the fact that we can choose an orthonormal basis for operators ψ i , i ¼ 1; …; g, with dimension Δ, where each ψ i contributes equally to Eq. (5). A numerical confirmation of Eq. (6) is presented in Fig. 1 for free fermions. We note in passing that a similar calculation was performed in Ref. [16]. In that Letter, the authors consider Rényi entropies in excited states of the CFT, for which they need to insert two operators ψðz; z¯ Þ on all of the n sheets of the cylinder. This makes the actual computation impractical for n > 2. In contrast, the leading correction term in our calculation comes from inserting only two operators and may, therefore, be carried out for arbitrary n. Limiting forms for the corrections.—In the limit l ≪ L, the correction (5) can be written as a power series in l=L, 4 1 þ n π 2 l2 l δSn ¼ gΔ þ O 4 e−2πΔβ=L 2 3n L L
(19)
If we take t2 ¼ ∞ and t1 ¼ −∞, the expression simplifies, hψð∞Þψð−∞Þin 1 sinðθ2 − θ1 =2Þ 2Δ ¼ : (20) hψð∞Þψð−∞Þi1 n2Δ sinðθ2 − θ1 =2nÞ
(15)
using the standard CFT normalization for two-point functions. Mapping back to the n-sheeted cover of the w plane, we find that
~ ðnÞ ðnÞ ðnÞ ðnÞ ððdζ 1 =dw1 ÞÞðdζ 2 =dw2 ÞÞh ððdζ¯ 1 =dw¯ 1 Þðdζ¯ 2 =dw¯ 2 ÞÞh
where ψðw1 ; w¯ 1 Þ and ψðw2 ; w¯ 2 Þ are inserted on just one of the n sheets. We would like to work with a normalization such that hψjψi ¼ 1 on the original w cylinder. To that end, we need to divide Eq. (16) by hψðw2 ; w¯ 2 Þψðw1 ; w¯ 1 Þi1. The following identity is useful:
(14)
þ oðe−2πΔβ=L Þ:
(21)
We can understand this form of the correction from a twist operator formulation of the Rényi entropy. Instead of
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The small l correction should then be
80
80
1 1 l2 4π 2 Δ −2πΔβ=L n 1− 2 ; g − 2 e 1−n n 12 L
LT
70 LT
S
60
S
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60 50 40
40
30 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
20
LT 0 0.0
(25)
0.2
0.4
0.6
0.8
1.0
L
FIG. 1 (color online). Thermal corrections to entanglement entropy δSE ¼ SE ðTÞ − SE ð0Þ plotted against l=L for a free, massless (1 þ 1) dimensional fermion. The points are numerically determined from a lattice model of size N ¼ 100 grid points using the method described in Refs. [9,15]. L is the size of the circle and l the size of the interval. From top to bottom, the points correspond to LT ¼ 0.15, 0.2, 0.3, 0.4, 0.5. The solid curve is the prediction (6). Inset: δSE plotted against LT for l ¼ L. The curve is the thermal entropy correction 4ð1 þ π=LTÞ.
working on the uniformized Riemann surface ζðnÞ , we work on the individual cylinders coordinatized by w but now with twist operators Pðl=2Þ and Pð−l=2Þ at the end points of the interval A. We can write down a couple of terms in the operator product expansion (OPE) of the twist operators [17–20]. ~ Pðl=2ÞPð−l=2Þ 2
¼ cn l−ðc=6Þð1−1=n Þ 1 1 2 ~ 1 − 2 l ðTð0Þ þ Tð0ÞÞ þ : × 1 þ þ 12 n (22) (A discussion of the normalization constants cn can be found in Ref. [21].) The leading ðl=LÞ2 correction (21) will then come from a three-point function involving the stress tensor and the two quasiprimary fields limt→∞ ψðx; tÞ and limt→−∞ ψðx; tÞ producing, via the operator state correspondence, the first excited state in the Boltzmann sum (9). Recall that the three-point function of the stress tensor with two quasiprimary fields ψðz; z¯ Þ of dimension Δ ¼ h þ h~ is hψðz2 ; z¯ 2 ÞTðz3 Þψðz1 ; z¯ 1 Þi ¼ h
z212 2 2 z31 z32
1 : jz12 j2Δ
(23)
If we now transform to the cylinder, z ¼ e2πiw=L , choosing z1 ¼ 0 and z2 ¼ ∞, we find that hψðw2 ; w¯ 2 ÞTðw3 Þψðw1 ; w¯ 1 Þi 4π 2 π2c ¼ − 2 hþ 2: hψðw2 ; w¯ 2 Þψðw1 ; w¯ 1 Þi 6L L
(24)
which matches with Eq. (21). The g is the degeneracy factor. The 1=ð1 − nÞ comes from the definition of the Rényi entropy. The n is the number of different sheets where we can compute this three-point function. The ð1 − 1=n2 Þl2 =12 comes from the OPE of the twist operators. Finally, we get contributions from three-point ~ wÞ ¯ and a Boltzmann factor. functions involving TðwÞ and Tð Note that the Schwarzian derivative contribution to the three-point function (24) does not contribute to Eq. (25) but instead gets incorporated into the leading zero temperature part of the Rényi entropies. In the limt l → L, the correction (5) becomes n l 2Δ −2πΔβ=L δSn ¼ −g þO 1− e þoðe−2πΔβ=L Þ: 1−n L This expression traces its origin to the normalization of the thermal density matrix, i.e., e−2πΔβ=L in the denominator of Eq. (9), and leads to a peculiar singularity in the limit n → 1. Interestingly, the analytic continuation n → 1 is sensitive to the order of limits β → ∞ and l → L. Above, we have taken β → ∞ first. If we instead first take l → L, and then take n → 1, the coefficient of e−2πΔβ=L now becomes singular in the limit β → ∞. Let us see how this result comes about in greater detail. The nth Rényi entropy can be computed from the partition function of an n-sheeted copy of the torus, glued along the interval 0 to l. On each sheet of the Riemann surface, we insert the identity operator Id ¼ PP−1 along a spatial circle where P shifts up a sheet and P−1 shifts down. Then, we move P−1 so that it overlaps and partially cancels with the interval 0 to l, leaving P−1 along the interval l to L − l. In the limit l → L, we can replace the sewing prescription along the interval l to L − l with the OPE of twist operators that sit at l and L − l. The effect of the remaining P−1 around the spatial circle is to glue the n tori of length β into a single torus of length nβ. Using the OPE (22), we find that trðρA Þn ¼ cnn ðL − lÞð−c=6Þðn−1=nÞ
trðρn Þ þ : ðtrρÞn
(27)
We will return to (L − l)-dependent corrections in a moment. The leading (L − l) dependence yields the universal single interval Rényi entropy for a CFT on a line while the remaining ratio of thermal density matrices gives the thermal Rényi entropy for the whole system. The n → 1 limit of the thermal Rényi entropy will yield the ordinary thermal entropy. At low temperature for our system, the thermal Rényi entropy, can be expanded,
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trðρn Þ 1 þ ge−2πΔnβ=L þ : n ¼ ðtrρÞ ð1 þ ge−2πΔβ=L þ Þn
(28)
From the n → 1 limit of Eq. (27), we may compute the entanglement entropy for the interval A: c L−l 2πΔβ −2πΔβ=L þg 1þ e þ SE ¼ log 3 ϵ L þ OððL − lÞ2 ; ðL − lÞ2Δ Þ:
(29)
Note that the Oðe−2πΔβ=L Þ correction matches well the result for free fermions, shown in the inset of Fig. 1. We expect an OðL − lÞ2 correction to Eq. (29) from expanding the universal leading log sinðπl=LÞ piece of the entanglement and Rényi entropies on a circle. In terms of the OPE (22), this correction comes from the one-point function of the stress tensor on a cylinder. Note that this correction is temperature independent and does not come with a e−2πΔβ=L Boltzmann factor. The leading correction that does come with a e−2πΔβ=L suppression should scale as OðL − lÞ2Δ . This correction will come from a ψ 2 term in the OPE of the two twist operators [19]. By dimensional analysis, the coefficient of ψ 2 in the OPE contains the ðL − lÞ2Δ dependence while the Boltzmann suppression comes from two-point functions of the ψ operators separated by a distance β on the torus of length nβ and width L. Discussion.—For pure states, the Rényi entropy of a region and its complement are equal, Sn ðAÞ ¼ Sn ðBÞ, as follows from a Schmidt decomposition of the Hilbert space (see, for example, Ref. [15]). However, for thermal states, this symmetry is broken. Indeed for our corrections (5), δSn ðlÞ ≠ δSn ðL − lÞ. However, the Rényi entropies and our corrections do have the symmetry Sn ðlÞ ¼ Sn ðnL − lÞ. To see this symmetry, imagine taking one of the twist operators around a spatial cycle on the torus (n − 1) times. Then, on the interval ð0; lÞ we have a permutation Pnn ¼ 1, and on the complementary interval a permutation Pn−1 n , which is equivalent to Pn . [For the correction (5), note that one should take the modulus of the ratio of sines.] The corrections (5) and (6) are very similar in spirit to universal corrections to two interval entanglement entropy on the plane [17–19]. For two intervals of length l1 and l2 whose centers are separated by a distance r, an OPE argument predicts a leading correction to the Rényi entropies of the form l1 l2 Δ δSn ¼ Cn 2 2 ; (30) nr where Δ is again the lowest conformal scaling dimension among the primary operators and Cn is known and calculable. It would be interesting to see if similar formulas can be found for CFTs in higher dimension.
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We thank K. Balasubramanian, P. Calabrese, T. Faulkner, T. Hartman, S. Hartnoll, M. Headrick, M. Kulaxizi, A. Parnachev, T. Nishioka, L. Takhtajan, and E. Tonni for discussion. We thank the Kavli Institute for Theoretical Physics, Santa Barbara, where this research was carried out, for its hospitality. This work was supported by the National Science Foundation (NSF) under Grant No. PHY11-25915. J. C. was supported in part by the Simons Foundation, and C. P. H. by the NSF under Grants No. PHY08-44827 and No. PHY13-16617. C. P. H. thanks the Sloan Foundation for partial support.
[1] T. J. Osborne and M. A. Nielsen, Phys. Rev. A 66, 032110 (2002). [2] G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett. 90, 227902 (2003). [3] H. Casini and M. Huerta, J. Phys. A 40, 7031 (2007). [4] H. Casini and M. Huerta, Phys. Rev. D 85, 125016 (2012). [5] L. Bombelli, R. K. Koul, J. Lee, and R. D. Sorkin, Phys. Rev. D 34, 373 (1986). [6] M. Srednicki, Phys. Rev. Lett. 71, 666 (1993). [7] C. P. Herzog and M. Spillane, Phys. Rev. D 87, 025012 (2013). [8] T. Azeyanagi, T. Nishioka, and T. Takayanagi, Phys. Rev. D 77, 064005 (2008). [9] C. P. Herzog and T. Nishioka, J. High Energy Phys. 03 (2013) 077. [10] A. McIntyre and L. A. Takhtajan, Geom. Funct. Anal. 16, 1291 (2006). [11] T. Barrella, X. Dong, S. A. Hartnoll, and V. L. Martin, J. High Energy Phys. 09 (2013) 109. [12] The authors of Ref. [10] demonstrate their formula only for a bc system with integer scaling dimension, but it is likely to hold more generally; L. Takhtajan (private communication). [13] S. Datta and J. R. David, arXiv:1311.1218. [14] P. Calabrese and J. L. Cardy, J. Stat. Mech. (2004) P06002. [15] I. Peschel and V. Eisler, J. Phys. A 42, 504003 (2009). [16] F. C. Alcaraz, M. I. Berganza, and G. Sierra, Phys. Rev. Lett. 106, 201601 (2011). [17] M. Headrick, Phys. Rev. D 82, 126010 (2010). [18] P. Calabrese, J. Cardy, and E. Tonni, J. Stat. Mech. (2009) P11001. [19] P. Calabrese, J. Cardy, and E. Tonni, J. Stat. Mech. (2011) P01021. [20] We take there to be 2n twist operators, two on each cylindrical sheet of the Riemann surface. Note that the coefficient of the stress tensor in the OPE and the exponent of the leading l are fixed by the conformal dimension of the twist operators, h ¼ h~ ¼ ðc=24Þð1 − 1=n2 Þ [14]. Recall that the stress tensor TðzÞ will appear with coefficient 2h=c in the OPE of two primary operators, provided that the primaries are normalized such that the two-point function takes the standard form (15). Here there is instead an overall multiplicative factor cn. [21] P. Calabrese and J. Cardy, J. Phys. A 42, 504005 (2009).
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Universal Thermal Corrections to Single Interval Entanglement Entropy for Two Dimensional Conformal Field Theories 1
John Cardy1 and Christopher P. Herzog2 Oxford University, Rudolf Peierls Centre for Theoretical Physics 1 Keble Road, Oxford OX1 3NP, United Kingdom and All Souls College, Oxford OX1 4AL, United Kingdom 2 C. N. Yang Institute for Theoretical Physics, Department of Physics and Astronomy Stony Brook University, Stony Brook, New York 11794, USA (Received 10 March 2014; published 30 April 2014) We consider single interval Rényi and entanglement entropies for a two dimensional conformal field theory on a circle at nonzero temperature. Assuming that the finite size of the system introduces a unique ground state with a nonzero mass gap, we calculate the leading corrections to the Rényi and entanglement entropy in a low temperature expansion. These corrections have a universal form for any two dimensional conformal field theory that depends only on the size of the mass gap and its degeneracy. We analyze the limits where the size of the interval becomes small and where it becomes close to the size of the spatial circle. DOI: 10.1103/PhysRevLett.112.171603
PACS numbers: 11.25.Hf, 03.65.Ud, 89.70.Cf
The idea of entanglement, that a local measurement on a quantum system may instantaneously affect the outcome of local measurements far away, is a central concept in quantum mechanics. As such, it underlies the study of quantum information, communication, and computation. However, entanglement also plays an increasing role in other areas of physics. To name three, measures of entanglement may detect exotic phase transitions in many-body systems lacking a local order parameter [1,2], such measures order quantum field theories under renormalization group flow [3,4], and entanglement is a key concept in the black hole information paradox (see, e.g., Refs. [5,6]). Important measures of entanglement for a many-body system in its ground state are the Rényi and entanglement entropies. To define these quantities, we first partition the Hilbert space into pieces A and complement A¯ ¼ B. Typically (and hereafter in this Letter), A and B correspond to spatial regions. Not all quantum systems may allow for such a partition. The reduced density matrix is defined as a partial trace of the full density matrix ρ over the degrees of freedom in B: ρA ≡ trB ρ:
(1)
The entanglement entropy is then the von Neumann entropy of the reduced density matrix: SE ≡ −trρA log ρA :
(2)
The Rényi entropies are Sn ≡
1 log trðρA Þn ; 1−n
where SE ¼ limn→1 Sn . 0031-9007=14=112(17)=171603(5)
(3)
The entanglement and Rényi entropies are well defined for excited and thermal states as well. One simply starts with the relevant excited state or thermal density matrix instead of the ground state density matrix ρ ¼ j0ih0j. For example, for thermal states, one would start with ρ¼
e−βH ; trðe−βH Þ
(4)
where β is the inverse temperature and H is the Hamiltonian. However, it is well known that for mixed states, the entanglement entropy is no longer a good measure of quantum entanglement. The entanglement entropy is contaminated by the thermal entropy of region A and, in fact, in the high temperature limit becomes dominated by it. It is, of course, never possible to take a strict zero temperature limit of real world systems. If we are to use entanglement entropy to measure quantum entanglement, we need to know how to subtract off nonzero temperature contributions to SE and Sn . Little has been said about these corrections in the literature thus far. Reference [7] studied entanglement entropy for a massive relativistic scalar field in 1 þ 1 dimensions. The authors provided numerical evidence that the corrections scale as e−βm in the limit βm ≫ 1, where m is the mass of the scalar field. They further conjectured that, for any a many-body system with a mass gap mgap , such corrections should scale as e−βmgap when βmgap ≫ 1. This conjecture more or less immediately follows from writing Eq. (4) as a Boltzmann sum over states. By assumption, the first excited state will come with contribution e−βmgap and give the dominant correction. Some possible but probably rare exceptions involve cases where, for some reason, the first excited state contribution
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to the entanglement entropy vanishes or where the degeneracy of states with energy E ¼ mgap itself has an exponential dependence on temperature. While knowledge of the e−βmgap dependence is a good starting point, it would be even better to know the coefficient. In this Letter, we calculate the coefficient for (1 þ 1) dimensional conformal field theories (CFTs) on a circle of circumference L in the case where A consists of a single interval of length l. The answer is
dimensional CFT. We then check that the corrections have expected behavior in the limits l ≪ L and l → L. Universal thermal corrections.—Starting from the thermal density matrix (4), we compute the first few terms in a low temperature expansion of the Rényi entropies. Introducing a complete set of states, we can write the thermal density matrix as a Boltzmann sum over states: ρ¼
g 1 sin2Δ ðπl=LÞ δSn ¼ − n e−2πΔβ=L 1 − n n2Δ−1 sin2Δ ðπl=nLÞ þ oðe−2πΔβ=L Þ;
(5)
πl πl δSE ¼ 2gΔ 1 − cot e−2πΔβ=L þ oðe−2πΔβ=L Þ; (6) L L where Δ is the smallest scaling dimension among the set of operators including the stress tensor and all primaries not equal to the identity, and g is their degeneracy. In order for this result to hold, the CFT needs to have a unique ground state separated from the first excited state by a nonzero mass gap (induced by the finite volume of the system). The correction (6) was obtained from the n → 1 limit of Eq. (5). Given the form of these corrections, another possible use for them is the determination of Δ and g. From the literature [8–11], it is possible to deduce the correction (5) in two limiting cases–for free theories and for theories with AdS/CFT duals and large central charge. Using bosonization, Refs. [8,9] computed these corrections for massless free Dirac fermions. In this case, free fermionic operators with scaling dimension Δ ¼ 1=2 and g ¼ 4 yield the appropriate correction. Reference [11] considered CFTs with large central charge c and gravitational duals. In this case, the entanglement and Rényi entropies can be written as an asymptotic series in c where the leading term is OðcÞ. The authors computed the Oð1Þ corrections to the entanglement and Rényi entropies due to the stress tensor TðzÞ, which has Δ ¼ 2 and g ¼ 2. Although the authors do not write explicit formulas, it is trivial to extend their computation to operators of general conformal dimension. The computation involves a low temperature expansion of a Selberg-like zeta function. In all cases, their expression would match Eq. (5). For free CFTs more generally, the important observation is that this same Selberg-like zeta function appears in Ref. [10] where the authors compute the determinant of the Laplacian for a bc system on an arbitrary genus Riemann surface. From Ref. [10], one can then deduce that the same correction (5) holds for free CFTs [12]. A check of these statements was performed explicitly for a free complex boson and free fermions in Ref. [13]. In this Letter, we will show by explicit computation that the corrections (5) and (6) hold for a general (1 þ 1)
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X 1 jϕihϕje−βEϕ : trðe−βH Þ jϕi
(7)
For a CFT on a cylinder of circumference L, coordinatized by w ¼ x − it, the Hamiltonian is 2π c ~ L0 þ L0 − ; (8) H¼ L 12 where L0 and L~ 0 are the zeroth level left- and right-moving Virasoro generators and c is the central charge. We will assume in what follows that placing the CFT on a cylinder produces a unique ground state and gaps the theory. Let ψðwÞ ≠ 1 be a Virasoro primary operator and jψi ¼ limt→−∞ ψðwÞj0i the corresponding state where ~ ~ As the L0 jψi ¼ hjψi, L~ 0 jψi ¼ hjψi, and Δ ¼ h þ h. jψi are lowest weight states in the CFT, it follows—with one important exception—that the smallest nonzero Eϕ must correspond to a primary operator ψðwÞ and, moreover, that Eψ ¼ 2πðΔ − ðc=12ÞÞ=L. The one important exception is when Δ > 2 and two descendants of the identity ~ wÞ, ¯ operator, the stress tensor TðwÞ and its conjugate Tð give the dominant correction. (The identity operator itself can be thought of as yielding the leading zero temperature contribution to the Rényi entropies.) We are assuming Δ > 0, which may not be true for nonunitary CFTs, and no gravitational anomaly, i.e., cL ¼ cR . We divide up the spatial circle into regions A and complement B and compute the reduced density matrix ρA . Let us consider a low temperature expansion where ðj0ih0j þ jψihψje−2πΔβ=L þ Þ : 1 þ e−2πΔβ=L þ It follows then that ρ¼
tr½trB ðj0ih0j þ jψihψje−2πΔβ=L þ Þn ð1 þ e−2πΔβ=L þ Þn n ¼ trðtrB j0ih0jÞ 1þ
trðρA Þn ¼
(9)
(10)
tr½trB jψihψjðtrB j0ih0jÞn−1 −2πΔβ=L þ − 1 ne þ : trðtrB j0ih0jÞn (11) The first term gives the zero temperature contribution to the Rényi entropy for a finite system [14]
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1 − n2 L πl þ ; log trðtrB j0ih0jÞ ¼ c log sin π L 6n (12) n
trivially. A uniformizing map from n copies of a cylinder of circumference L to the plane is
ζ ðnÞ ¼
where l is the length of A. Up to a normalization issue we will come to shortly, the expression tr½trB jψihψjðtrB j0ih0jÞn−1 trðtrB j0ih0jÞn
(13)
is a two-point function of the operator ψðwÞ. By the state operator correspondence, we can interpret the state jψi as limt→−∞ ψðx; tÞj0i and similarly hψj ¼ limt→∞ h0jψðx; tÞ. In path integral language, the trace over an nth power of the reduced density matrix becomes a partition function on an n-sheeted copy of the cylinder, branched over the interval A. The expression (13) is then the two-point function hψðw2 ; w¯ 2 Þψðw1 ; w¯ 1 Þi on this multisheeted cylinder for a particular choice of w1 and w2 . By the Riemann-Hurwitz theorem, this multisheeted cylinder has genus zero. There exists then a uniformizing map which takes the multisheeted cylinder to the plane, and on the plane, the two-point function (13) can be computed
hψðw2 ; w¯ 2 Þψðw1 ; w¯ 1 Þin ¼
2πiw=L e − eiθ2 1=n : e2πiw=L − eiθ1
We choose θ1 and θ2 such that the map is branched over the interval A, which has end points on the cylinder such that θ2 − θ1 ¼ 2πl=L. Under this map, an operator inserted on the jth cylinder at t ¼ −∞ is located on the ζ ðnÞ plane ðnÞ at ζ −∞ ¼ eiðθ2 −θ1 Þ=nþ2πij=n , while an operator inserted at ðnÞ t ¼ ∞ is located at ζ ∞ ¼ e2πij=n . The correlation function on the ζ plane between two ðnÞ ðnÞ quasiprimaries inserted at points ζ 2 and ζ1 is hψðζ 2 ; ζ¯ 2 Þψðζ 1 ; ζ¯ 1 Þi ¼ ðnÞ
ðnÞ
ðnÞ
ðnÞ
1 ðnÞ 2h ðnÞ 2h~ ðζ 21 Þ ðζ¯ 21 Þ
dζ ðnÞ =dw 1 ζ ðnÞ : ¼ dζð1Þ =dw n ζð1Þ
ðnÞ
ðζ 21 Þ2h
(17)
We find then that hψðw2 ; w¯ 2 Þψðw1 ; w¯ 1 Þin ~ ¼ fðζ ðnÞ ; ζ ð1Þ ; hÞfðζ¯ ðnÞ ; ζ¯ ð1Þ ; hÞ; hψðw2 ; w¯ 2 Þψðw1 ; w¯ 1 Þi1 (18) where 1 x1 x2 h y2 − y1 2h : fðx; y; hÞ ≡ 2h y1 y2 x2 − x1 n
ðnÞ 2h~
ðζ¯ 21 Þ
;
(16)
The result (5) then follows immediately, and then Eq. (6) follows by taking the n → 1 limit of Eq. (5). The degeneracy factor g comes from the fact that we can choose an orthonormal basis for operators ψ i , i ¼ 1; …; g, with dimension Δ, where each ψ i contributes equally to Eq. (5). A numerical confirmation of Eq. (6) is presented in Fig. 1 for free fermions. We note in passing that a similar calculation was performed in Ref. [16]. In that Letter, the authors consider Rényi entropies in excited states of the CFT, for which they need to insert two operators ψðz; z¯ Þ on all of the n sheets of the cylinder. This makes the actual computation impractical for n > 2. In contrast, the leading correction term in our calculation comes from inserting only two operators and may, therefore, be carried out for arbitrary n. Limiting forms for the corrections.—In the limit l ≪ L, the correction (5) can be written as a power series in l=L, 4 1 þ n π 2 l2 l δSn ¼ gΔ þ O 4 e−2πΔβ=L 2 3n L L
(19)
If we take t2 ¼ ∞ and t1 ¼ −∞, the expression simplifies, hψð∞Þψð−∞Þin 1 sinðθ2 − θ1 =2Þ 2Δ ¼ : (20) hψð∞Þψð−∞Þi1 n2Δ sinðθ2 − θ1 =2nÞ
(15)
using the standard CFT normalization for two-point functions. Mapping back to the n-sheeted cover of the w plane, we find that
~ ðnÞ ðnÞ ðnÞ ðnÞ ððdζ 1 =dw1 ÞÞðdζ 2 =dw2 ÞÞh ððdζ¯ 1 =dw¯ 1 Þðdζ¯ 2 =dw¯ 2 ÞÞh
where ψðw1 ; w¯ 1 Þ and ψðw2 ; w¯ 2 Þ are inserted on just one of the n sheets. We would like to work with a normalization such that hψjψi ¼ 1 on the original w cylinder. To that end, we need to divide Eq. (16) by hψðw2 ; w¯ 2 Þψðw1 ; w¯ 1 Þi1. The following identity is useful:
(14)
þ oðe−2πΔβ=L Þ:
(21)
We can understand this form of the correction from a twist operator formulation of the Rényi entropy. Instead of
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The small l correction should then be
80
80
1 1 l2 4π 2 Δ −2πΔβ=L n 1− 2 ; g − 2 e 1−n n 12 L
LT
70 LT
S
60
S
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60 50 40
40
30 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
20
LT 0 0.0
(25)
0.2
0.4
0.6
0.8
1.0
L
FIG. 1 (color online). Thermal corrections to entanglement entropy δSE ¼ SE ðTÞ − SE ð0Þ plotted against l=L for a free, massless (1 þ 1) dimensional fermion. The points are numerically determined from a lattice model of size N ¼ 100 grid points using the method described in Refs. [9,15]. L is the size of the circle and l the size of the interval. From top to bottom, the points correspond to LT ¼ 0.15, 0.2, 0.3, 0.4, 0.5. The solid curve is the prediction (6). Inset: δSE plotted against LT for l ¼ L. The curve is the thermal entropy correction 4ð1 þ π=LTÞ.
working on the uniformized Riemann surface ζðnÞ , we work on the individual cylinders coordinatized by w but now with twist operators Pðl=2Þ and Pð−l=2Þ at the end points of the interval A. We can write down a couple of terms in the operator product expansion (OPE) of the twist operators [17–20]. ~ Pðl=2ÞPð−l=2Þ 2
¼ cn l−ðc=6Þð1−1=n Þ 1 1 2 ~ 1 − 2 l ðTð0Þ þ Tð0ÞÞ þ : × 1 þ þ 12 n (22) (A discussion of the normalization constants cn can be found in Ref. [21].) The leading ðl=LÞ2 correction (21) will then come from a three-point function involving the stress tensor and the two quasiprimary fields limt→∞ ψðx; tÞ and limt→−∞ ψðx; tÞ producing, via the operator state correspondence, the first excited state in the Boltzmann sum (9). Recall that the three-point function of the stress tensor with two quasiprimary fields ψðz; z¯ Þ of dimension Δ ¼ h þ h~ is hψðz2 ; z¯ 2 ÞTðz3 Þψðz1 ; z¯ 1 Þi ¼ h
z212 2 2 z31 z32
1 : jz12 j2Δ
(23)
If we now transform to the cylinder, z ¼ e2πiw=L , choosing z1 ¼ 0 and z2 ¼ ∞, we find that hψðw2 ; w¯ 2 ÞTðw3 Þψðw1 ; w¯ 1 Þi 4π 2 π2c ¼ − 2 hþ 2: hψðw2 ; w¯ 2 Þψðw1 ; w¯ 1 Þi 6L L
(24)
which matches with Eq. (21). The g is the degeneracy factor. The 1=ð1 − nÞ comes from the definition of the Rényi entropy. The n is the number of different sheets where we can compute this three-point function. The ð1 − 1=n2 Þl2 =12 comes from the OPE of the twist operators. Finally, we get contributions from three-point ~ wÞ ¯ and a Boltzmann factor. functions involving TðwÞ and Tð Note that the Schwarzian derivative contribution to the three-point function (24) does not contribute to Eq. (25) but instead gets incorporated into the leading zero temperature part of the Rényi entropies. In the limt l → L, the correction (5) becomes n l 2Δ −2πΔβ=L δSn ¼ −g þO 1− e þoðe−2πΔβ=L Þ: 1−n L This expression traces its origin to the normalization of the thermal density matrix, i.e., e−2πΔβ=L in the denominator of Eq. (9), and leads to a peculiar singularity in the limit n → 1. Interestingly, the analytic continuation n → 1 is sensitive to the order of limits β → ∞ and l → L. Above, we have taken β → ∞ first. If we instead first take l → L, and then take n → 1, the coefficient of e−2πΔβ=L now becomes singular in the limit β → ∞. Let us see how this result comes about in greater detail. The nth Rényi entropy can be computed from the partition function of an n-sheeted copy of the torus, glued along the interval 0 to l. On each sheet of the Riemann surface, we insert the identity operator Id ¼ PP−1 along a spatial circle where P shifts up a sheet and P−1 shifts down. Then, we move P−1 so that it overlaps and partially cancels with the interval 0 to l, leaving P−1 along the interval l to L − l. In the limit l → L, we can replace the sewing prescription along the interval l to L − l with the OPE of twist operators that sit at l and L − l. The effect of the remaining P−1 around the spatial circle is to glue the n tori of length β into a single torus of length nβ. Using the OPE (22), we find that trðρA Þn ¼ cnn ðL − lÞð−c=6Þðn−1=nÞ
trðρn Þ þ : ðtrρÞn
(27)
We will return to (L − l)-dependent corrections in a moment. The leading (L − l) dependence yields the universal single interval Rényi entropy for a CFT on a line while the remaining ratio of thermal density matrices gives the thermal Rényi entropy for the whole system. The n → 1 limit of the thermal Rényi entropy will yield the ordinary thermal entropy. At low temperature for our system, the thermal Rényi entropy, can be expanded,
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trðρn Þ 1 þ ge−2πΔnβ=L þ : n ¼ ðtrρÞ ð1 þ ge−2πΔβ=L þ Þn
(28)
From the n → 1 limit of Eq. (27), we may compute the entanglement entropy for the interval A: c L−l 2πΔβ −2πΔβ=L þg 1þ e þ SE ¼ log 3 ϵ L þ OððL − lÞ2 ; ðL − lÞ2Δ Þ:
(29)
Note that the Oðe−2πΔβ=L Þ correction matches well the result for free fermions, shown in the inset of Fig. 1. We expect an OðL − lÞ2 correction to Eq. (29) from expanding the universal leading log sinðπl=LÞ piece of the entanglement and Rényi entropies on a circle. In terms of the OPE (22), this correction comes from the one-point function of the stress tensor on a cylinder. Note that this correction is temperature independent and does not come with a e−2πΔβ=L Boltzmann factor. The leading correction that does come with a e−2πΔβ=L suppression should scale as OðL − lÞ2Δ . This correction will come from a ψ 2 term in the OPE of the two twist operators [19]. By dimensional analysis, the coefficient of ψ 2 in the OPE contains the ðL − lÞ2Δ dependence while the Boltzmann suppression comes from two-point functions of the ψ operators separated by a distance β on the torus of length nβ and width L. Discussion.—For pure states, the Rényi entropy of a region and its complement are equal, Sn ðAÞ ¼ Sn ðBÞ, as follows from a Schmidt decomposition of the Hilbert space (see, for example, Ref. [15]). However, for thermal states, this symmetry is broken. Indeed for our corrections (5), δSn ðlÞ ≠ δSn ðL − lÞ. However, the Rényi entropies and our corrections do have the symmetry Sn ðlÞ ¼ Sn ðnL − lÞ. To see this symmetry, imagine taking one of the twist operators around a spatial cycle on the torus (n − 1) times. Then, on the interval ð0; lÞ we have a permutation Pnn ¼ 1, and on the complementary interval a permutation Pn−1 n , which is equivalent to Pn . [For the correction (5), note that one should take the modulus of the ratio of sines.] The corrections (5) and (6) are very similar in spirit to universal corrections to two interval entanglement entropy on the plane [17–19]. For two intervals of length l1 and l2 whose centers are separated by a distance r, an OPE argument predicts a leading correction to the Rényi entropies of the form l1 l2 Δ δSn ¼ Cn 2 2 ; (30) nr where Δ is again the lowest conformal scaling dimension among the primary operators and Cn is known and calculable. It would be interesting to see if similar formulas can be found for CFTs in higher dimension.
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We thank K. Balasubramanian, P. Calabrese, T. Faulkner, T. Hartman, S. Hartnoll, M. Headrick, M. Kulaxizi, A. Parnachev, T. Nishioka, L. Takhtajan, and E. Tonni for discussion. We thank the Kavli Institute for Theoretical Physics, Santa Barbara, where this research was carried out, for its hospitality. This work was supported by the National Science Foundation (NSF) under Grant No. PHY11-25915. J. C. was supported in part by the Simons Foundation, and C. P. H. by the NSF under Grants No. PHY08-44827 and No. PHY13-16617. C. P. H. thanks the Sloan Foundation for partial support.
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