Universality of quantum information in chaotic CFTs

Journal of High Energy Physics, Mar 2018

Nima Lashkari, Anatoly Dymarsky, Hong Liu

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Universality of quantum information in chaotic CFTs

HJE Universality of quantum information in chaotic CFTs Nima Lashkari 0 1 4 Anatoly Dymarsky 0 1 2 3 Hong Liu 0 1 4 0 77 Massachusetts Avenue , Cambridge, MA 02139 , U.S.A 1 Lexington , KY 40506 , U.S.A 2 Department of Physics and Astronomy, University of Kentucky , USA 3 Skolkovo Institute of Science and Technology, Skolkovo Innovation Center 4 Center for Theoretical Physics, Massachusetts Institute of Technology , USA We study the Eigenstate Thermalization Hypothesis (ETH) in chaotic conformal eld theories (CFTs) of arbitrary dimensions. Assuming local ETH, we compute the reduced density matrix of a ball-shaped subsystem of nite size in the in nite volume limit when the full system is an energy eigenstate. This reduced density matrix is close in trace distance to a density matrix, to which we refer as the ETH density matrix, that is independent of all the details of an eigenstate except its energy and charges under global symmetries. In two dimensions, the ETH density matrix is universal for all theories with the same value of central charge. We argue that the ETH density matrix is close in trace distance to the reduced density matrix of the (micro)canonical ensemble. We support the argument in higher dimensions by comparing the Von Neumann entropy of the ETH density matrix with the entropy of a black hole in holographic systems in the low temperature limit. Finally, we generalize our analysis to the coherent states with energy density that varies slowly in space, and show that locally such states are well described by the ETH density matrix. Conformal Field Theory; Integrable Hierarchies; Thermal Field Theory 1 Introduction and outline 2 ETH density matrix and thermal states 2.1 Di erent ensembles 2.2 Equivalence of reduced density matrices 3 Two dimensional CFTs 3.1 3.2 3.3 3.4 Universal reduced density matrix Renyi entropies Generalized Gibbs ensembles Matching with thermal density matrix in the in nite c limit 4 Higher dimensional CFTs 4.1 ETH density matrix 4.2 Entanglement entropy from ETH density matrix 4.3 Holographic theories 5 Local equilibrium 5.1 Time-dependent coherent states 5.2 Arbitrary initial states 6 Conclusions A Rindler space: a convenient conformal frame B Global descendants in two dimensions C Thermodynamically relevant quasi-primaries D One-point functions on a torus E Perturbative Renyi entropies F Twist operators G Failure of perturbation theory for GGE { i { 1 5 Introduction and outline Quantum information plays an increasingly important role in our understanding and characterization of quantum matter. The holographic duality together with the black hole information loss paradox give strong hints that quantum information is also likely to play a central role in our understanding of quantum gravity and the emergence of spacetime. In this paper, we discuss the quantum information properties of chaotic conformal eld theories (CFTs) expanding on the observations made in an earlier paper [1]. We provide evidence that the quantum information content of highly excited energy eigenstates of in conformal theories exhibit a great degree of universality. We de ne chaotic quantum eld theories (QFT) to be those satisfying a local version of the Eigenstate Thermalization Hypothesis (ETH) [1] (see [2, 3] for ETH in generic quantum systems including density matrix formulation [ 4, 5 ]). More explicitly, we say that a QFT on a homogenous compact space satis es local ETH if for a local operator Op (with p labeling di erent operators), hEajOpjEbi = Op(E) ab + pab; (1.1) function of E = Ea+Eb , and 2 pab where jEai is a highly excited energy eigenstate, the diagonal element Op(E) is a smooth e O(S(E)) where eS(E) is the density of states at energy E. If jEai has other quantum numbers associated with other global symmetries, Op(E) can also smoothly depend on those quantum numbers. To simplify the notation, we will suppress such dependence. In case of CFTs, de nition of ETH (1.1) will require additional clari cations which we explicitly described below. The high-energy eigenstates of a quantum many-body system are, in general, hard to access, and until now essentially all discussions of ETH have been limited to direct numerical diagonalizations (for instance see [6]). With the current computational resources, a direct numerical diagonalization approach to QFT seems unrealistic. In [1], we advocated that CFTs provide an exciting laboratory for exploring the implications of ETH and potentially even proving it. In a CFT, due to the state-operator correspondence, the energy eigenstates can be represented as local operators with de nite scaling dimensions, and (1.1) becomes a condition on the operator product expansion (OPE) coe cients. This opens up many powerful analytic tools for studying ETH. The previous studies of ETH in CFTs that have been inspiration for our work are [7{12]. More explicitly, consider a (d + 1)-dimensional CFT on a d-dimensional sphere S d with radius L. Since a primary operator and its descendants are algebraically related, the equation (1.1) written for CFTs should restrict only to the states jEai dual to primary operators [1]. In particular, for two-dimensional CFTs, jEai should correspond to Virasoro primary operators.1 Without loss of generality, we further restrict to scalar primary operators a of dimension ha = EaL. The energy density of the system in such a state is a = Ea Ld!d = ha 1In every two-dimensional CFT there is an in nite number of conserved charges associated to the KdV hierarchy [13]. As we will discuss later, for a Virasoro primary, all these charges are xed in terms of the conformal dimension, therefore Op(E) depends only on E. { 1 { where !d is the volume of a unit sphere Sd. For a CFT in a thermal state of temperature T , dT T d+1 where dT is the normalization of the two-point function of stress tensor (4.20). This motivates us to de ne the \thermal" length scale associated with jEai as In the thermodynamic limit with L ! 1, while keeping energy density a nite, and hence a nite T , the scaling dimension ha scales with L as T = a dT ha = dT !d 1 d+1 L T T 1 : d+1 : primaries a and by corresponding to jEai and hEbj, Applying the conformal transformation that maps the cylinder Sd Rt to Rd+1 (the radial quantization frame) the local ETH condition (1.1) translates into a statement about the OPE coe cient Cab multiplying the operator Op appearing in the expansion of two CpabL hp = Op(E) ab + pab ; h by(1)Op(1) a(0)i = Cpab ; a by = X CabOp : p p (1.3) (1.4) the OPE coe cient Capb must scale with ha ! 1 as We raise and lower the p index of Capb using the Zamolodchikov metric hOp(1)Op(0)i = dp. In the thermodynamic limit, under the assumption that (1.1) applies for any operator Op of dimension hp, which we keep xed as L becomes large, the equation (1.5) implies that hp hp Cpab = had+1 (dT !d) d+1 abfp(E) + Rpab : Here, the correction term Rpab = Lhp pab ha, and fp(E) = hTp Op(E) is a smooth dimensionless function of E. Since there are no other d e O(had+1 )+ dh+p1 log ha is exponentially small in dimensionfull parameters in the problem, fp(E) then has to be a constant, independent of E, i.e. hp hp Cpab = had+1 (dT !d) d+1 abfp + Rpab : We stress that the equation (1.7) encodes the following nontrivial implications of the local hp ETH. (i) Operators Op whose Cpaa grow slower than had+1 with ha cannot have a nonvanishing expectation value in the thermodynamic limit, while it is impossible for the OPE coe cient Capb to grow faster than had+1 as that would imply thermodynamic limit for such a theory does not exist. (ii) The spectrum of operators Op appearing in the OPE of a and ya is independent of speci c properties of a, and only depends on its scaling hp dimension (energy). Integrable systems are expected not to satisfy the local ETH. A simple example is a two-dimensional free massless boson on a spatial circle. This theory has heavy coherent { 2 { quantization Rd+1 conformal frame (c) The path-integral for ET H in the radial quantization frame. HJEP03(218)7 primary states ei j i with large dimension h = j j2=2 1. The OPE coe cient of this heavy state with a primary of dimension one, @ , explicitly violates (1.6) since it grows as we know that f@ on the right-hand-side of (1.6) is zero. Now consider a chaotic CFT in a highly excited energy eigenstate. We focus on the denoted as Bc. It was shown in [1] that the reduced density matrix reduced density matrix of a ball-shaped region B of size l inside Sd of size L and consider the thermodynamic limit L ! 1 with l kept xed. The complement of B inside Sd will be a(B) for the system in state jEai can be well approximated by a density matrix which we will refer as an ETH density matrix. ET H depends only on B and energy Ea TrBc jEaihEaj ETH(B; E), to jj a(B) ETH(B; E = Ea)jj e O(S(Ea)) ; where k k is the trace distance. In particular, it was shown that the ETH density matrix ETH(B; E) can be written as where Op denotes the family of operators which appear in the OPE of denotes their expectation values (1.1), and U is the unitary operator corresponding to a and ya, Op(E) the conformal transformation from the Rindler frame to the radial quantization frame; see gure 1(c). Equation (1.10) de nes a density matrix on B as being prepared via a Euclidean path-integral over Rd+1 with the speci ed boundary conditions \above" and \below" B within Sd of unit radius, and the sum of local operators on the right hand side of (1.10) inserted at the origin of Rd+1 (see gure 1). We will see later that the domain of convergence of this sum is xed by the conformal symmetry to be in nite. (1.8) (1.9) (1.10) { 3 { (1.11) (1.12) Expressing Op(E) in terms of constants fp of (1.7), we nd that (1.10) is an expansion in l T the high temperature limit l T convergent for any large but nite l= T . 1, it is enough to keep the rst few terms while in T ! 1 one has to sum the whole series, which should be In this paper, we rst give a general argument that the ETH density matrix (1.11) is close in trace distance to the reduced density matrix of a thermal state (there are subtleties that have non-zero thermal one-point functions by Atherm, we can also write (1.11) as All (quasi-)primary operators that are not in Atherm, and all the descendant elds drop out in the thermodynamic limit from the sum (1.12). We then discuss in detail the structure of the expansion (1.11) in the low temperature regime. Note that the reduced density matrix in the eigenstate is close to the ETH density matrix (1.11) (before we discard descendant elds) with exponential precision in S(E), as dictated by local ETH. However, the convergence of the ETH density matrix to the reduced thermal state is controlled with corrections that are polynomially supressed in S(E), as is the case anytime we compare quantities calculated in the microcanonical and the canonical ensembles. 1. In two dimensions (d = 1), the only Virasoro primary operator which has nonzero thermal value is the identity operator. Therefore, the ETH density matrix ETH(B; E) of (1.10) is solely expressed in terms of the Virasoro descendants of identity, i.e. Op(E) that are the polynomials of stress tensors and their derivatives. All fp's that correpond to the quasi-primaries in the Virasoro indentity block are xed by the Virasoro algebra, and hence are independent of any speci c properties of the 2d CFT except for the value of the central charge. The ETH density matrix in 2d is universal across all CFTs with the given value of central charge, thus we refer to it as the universal density matrix. We argue that if (1.1) holds for Virasoro primaries, the subsystem density matrix in the eigenstate is well approximated by the universal density matrix. Furthermore, we argue that the universal density matrix in the thermodynamic limit is close to the reduced Generalized Gibbs Ensemble (GGE) provided we can map all their conserved charges. That is to say 1 Z univ = trBc e H+Pi iQi + O(1= L); p (1.13) where the inverse temperature and the charges i are chosen such that the GGE has the same value of Qi charges as the universal density matrix. The conserved charges { 4 { Qi are the in nite set of Korteweg-de Vries integrals of motions in two-dimensional CFTs [13]. Due to the complexity of evaluating the expectation values of Qi in the GGE, we are not able to provide a direct support for (1.13) at this point. Note that CFT formulation of ETH does not require (1.13) to hold. The equation (1.13) should hold if we further assume that one can solve for i such that the GGE has the same values of charges Qi as the pure state. In the limit that the central charge c goes to 1, we show that all the i = 0 and the universal density matrix becomes close in trace distance to the standard Gibbs state. This is consistent with previous results of [7, 11].2 2. In higher dimensional CFTs, in general, the polynomials of the stress tensor do not exist in the spectrum as primary operators. Furthermore, the conformal symmetry is a lot less restrictive than 2d, and any primary operator can have nonzero Op(E). It is natural to expect, and we provide further support in section 2.2, that (1.11) sums into the standard thermal ensemble 1 Z ETH = trBc e H + O(1=pL) ; (1.14) where the inverse temperature is again chosen such that the thermal density matrix has the same energy E as the ETH density matrix. We provide support for (1.14) by computing the entanglement entropy of the ETH density matrix to the order (l= T )2(d+1) and matching the answer with the holographic entanglement entropy of the same subsystem as computed with the Ryu-Takayanagi formula in a black hole background. Note that up to this order, the entanglement entropy exhibits universality and depends only on the energy density and dT , the two-point function of stress tensor. That is why one can match the answer with holography. The plan of the paper is as follows. In section 2 we give a general discussion of the relation between the ETH density matrix and that of a thermal state. In section 3 we discuss the structure of the ETH density matrix for a two dimensional CFT in detail. In section 4 we study the subsystem ETH in CFTs of dimensions larger than two. In section 5 we consider states that have spatial and time dependence at scales much larger than the subsystem size and show that the same universal density matrix remains a good approximation to describe local physics. 2 ETH density matrix and thermal states We start with a brief discussion of various thermal ensembles for CFTs. The goal is to show that local ETH (1.1) implies that the expectation values of Op in eigenstates as de ned in (1.1) coincide with the thermal averages. This enables us to show that the reduced density matrix of an energy eigenstate is close in trace distance to those of various thermal ensembles. do not match. 2As we explain in detail in section 3 equivalence of ETH and the reduced Gibbs state does not imply to be much larger than the average level spacing that scales like exp( O(Ld)), but much smaller than the typical energy scales of interest. Here, N is the total number of states in the band. The density matrix of the canonical ensemble is denotes projection into the subspace of the Hilbert space with given f g. The grand canonical density matrix is de ned as grand( ; f g) = H Pi iQi ; Z f g = Tr e H Pi iQi Consider a QFT with a number of global symmetries living on a sphere. The microcanonical ensemble micro(E0; f g) is de ned as an equal-weight average over all energy eigenstates lying within a narrow band around E0 with a given set of quantum numbers f g under various global symmetries, micro(E0; f g) = 1 N E2(E0 X ;E0+ );given f g jE; f gihE; f gj : where Qi denote the complete set of commuting charges and f g denotes the collection of the corresponding chemical potentials. For a general quantum eld theory, in the thermodynamical limit, for a local operator O whose quantum numbers we keep xed as the volume goes to in nity, the microcanonical, canonical, and grand canonical averages are all equivalent by the standard arguments, prof g vided that one chooses and f g to give the average energy E0 and the average charges . For example, the micro-canonical and the canonical ensemble which average over rotationally-invariant states (i.e. with J 2 = 0 where J 2 denotes the Casimir operator of the rotation group) are equivalent to the grand canonical ensemble with the corresponding i = 0. The equivalence of ensembles in conformal eld theory is more intricate since the representations of a conformal group are in nite dimensional. Furthermore, the states which lie in the same representation of the conformal group in general do not have the same energy. Let us rst consider a CFT in d > 2. In this case, the conformal group is the higher dimensional Mobius transformations, and there are no new conserved charges beyond the generators of the conformal transformations. For convenience, let us introduce ^(m0i)cro(E0; fJ 2 = 0g) as the (un-normalized) microcanonical density matrix of scalar primaries with energies in a narrow band around E0, where one sums over only the energy eigenstates which are scalar primaries. Similarly we can de ne ^(mni)cro(E0; fJ 2 = 0g) to be the ensemble of states that descend at level n from primary states of energy E0. A state in the subspace de ned by ^(mni)cro(E0; fJ 2 = 0g) has energy approximately equal to E0 +O( Ln ). The standard microcanonical ensemble can then be expressed as micro(E0; fJ 2 = 0g) = (n) micro E0 O ; fJ 2 = 0g ; (2.4) 1 N X n { 6 { n L where N is the total number of states at energy E0 including both primaries and descendants. Now, we consider the thermodynamic limit that is L ! 1 with E0=Ld xed. In this limit, from (1.1) we have that for any n which does not scale with L hE0jOjE0i = D E0 O n L jOjE0 O n E L + O(L 1); = DE(n) 0 jOjE0(n)E + O(L 1) (2.5) where jE0i denotes a primary state while jE0(n)i denotes an n-th level descendant state of a primary state of approximate energy E0 exponentially with energy O( Ln ); see [1]. The density of states grows The contribution of states in (2.4) with n scaling as L or larger, is exponentially suppressed compared to the contribution of those with n = 0; hence we neglect such states. We conclude that in the thermodynamic limit for any local operator hE0jOjE0i = Tr O micro(E0; fJ 2 = 0g) + O(L 1 ) (2.6) and will also be the same as in the canonical and grand canonical ensembles. A CFT in d = 2 has an in nite number of conserved charges that commute with both L0 and L0. This is the KdV hierarchy of charges fQ2k+1; Q2k+1; k = 1; 2; corresponding microcanonical and canonical ensembles are denoted as g. Here, the micro(E0; fQ2k+1; Q2k+1g); canonical( ; fQ2k+1; Q2k+1g) (2.7) and the corresponding grand canonical ensemble is the so-called Generalized Gibbs Ensemble (GGE) GGE( ; f 2k+1; 2k+1g) = e (L0+L0) P k 2k+1Q2k+1 P k 2k+1Q2k+1 Z : (2.8) Again, micro(E0; fQ2k+1; Q2k+1g) contains descendant states. By descendants we are now referring to Virasoro descendants. Following the same arguments as above we conclude that E0; fQ2k+1; Q2k+1gjOjE0; fQ2k+1; Q2k+1g = Tr O micro(E0; fQ2k+1; Q2k+1g) : (2.9) The same holds also for the canonical ensemble and the GGE, provided we assume an appropriate growth of the density of states as a function of Q. 2.2 Equivalence of reduced density matrices We now present a general argument showing that given (2.6), the reduced density matrix for a region B of jE0ihE0j, and the ETH density matrix ETH are close in trace distance to the reduced state of the subsystem B of a thermal state (the two-dimensional case is di erent and will be discussed in more depth in section 4). The argument works for any of the three ensembles mentioned earlier. { 7 { The reduced density matrix of a region B is a map from the observables living on B to the expectation values. In conformal eld theory, if B is a topologically-trivial region the set of local operators on B provide a basis for all operators in B. One can compute the expectation value of a k-point function of operators local in the subsystem B in a reduced state such as or ETH by successively applying OPEs to reduce the k-point function to a one-point function. This is possible because neither nor ETH have any operator insertions in their corresponding Euclidean path-integrals that limits the domain of the convergence of OPEs on the subsystem. Consider any two reduced density matrices and whose Euclidean path-integral de nitions do not involve any operator insertions that limits the subsystem OPE. We will now show that = if and only if they have the same expectation value for all the local operators. The proof is a simple application of the Pinsker inequality: HJEP03(218)7 k (S( k ) + S( k )) = Tr (( )(K K )) (2.10) 2 k x 1 2 p p;q K = X lhp (d 1) Z fp(x)Op(x)+X lhp+hq 2(d 1) Z p x;y fp;q(x; y)Op(x)Oq(y)+ (2.11) where K and K denote the modular operators for and , respectively. The modular operators of both and can be expanded as where p sums over the set of all local operators. We can use the OPEs of operators in conformal eld theory to reduce the expression above to an in nite sum over local operators K = X lhp (d 1) Z x f~p(x)Op(x) : From (2.10) it then follows that if all the one-point functions of local operators match then the density matrices are the same. Now, imagine that the two density matrices have matching one-point functions of local operators up to precision 1: Tr (( )Op) = O ; (p) Then, from the analysis above, we claim that the relative entropy is order , which implies that the density matrices are close. One might worry that the sum over in nite terms (the coe cient of ) can diverge. In this case the relative entropy will diverge which implies that and have support on unequal subspaces in the Hilbert space. However, in a continuum eld theory we believe that all nitely excited energy density matrices are full rank.3 In our case, we are comparing ETH with the reduced state of a thermal density matrix. From (2.6), the one-point functions of local operators in these two states match up to volume suppressed corrections 1=L. We thus conclude that the states are close in trace distance up to volume suppressed corrections. { 8 { (2.12) (2.13) In this section, we explore the structure of ETH (1.11) for a general two-dimensional CFT. We show that it is universal across all CFTs of the same central charge. That is to say that the density matrix is comprised of only the polynomials of the stress tensor and the derivative operator, and thus does not depend on any speci c structure of a CFT other than the central charge. The ETH density matrix ( ETH) enables us to compute the Renyi and entanglement entropies for primary energy eigenstate. In next section, we will compare these quantities with those of a generalized Gibbs ensemble. 3.1 Universal reduced density matrix . The energy density is eigenstate j i of energy E. We take the subsystem B to be an interval of length 2l. We will work with a Euclidean time and it is convenient to use complex coordinates w = t + i with 2 [0; 2 L]. In radial quantization, with z = e L , j i and h j are mapped to operators w y(+1) of dimension h = EL, and B is on the unit circle between 0 and 0 In the thermodynamic limit we take L ! 1 with l and We de ne the thermal length as xed, and thus h / L2 ! 1. = E 3If a density matrix is not full rank it means that the state where it was reduced from can be killed by a local operator with support only on the subsystem, that is the projector to the eigenvector with eigenvalue zero. This violates the \separating" property of the states of a von Neumann algebra. In the algebraic formulation of quantum eld theory, the states are often chosen to be cyclic and separating [14]. { 9 { A convenient conformal frame to study the reduced density matrix of B is the Rindler frame in which the subsystem is mapped to the negative half-line see gure 2: ! = z qz q 1 ; is the two dimension version of the map written introduced in [1]; see appendix A. The key observation of [1] is that in the thermodynamical limit, where we take L ! 1 and keep replaced by their OPEs, and the reduced density matrix for region B in the Rindler frame can be written as4 ~ = (! ; ! ) (!+; !+) = X X p m;n 0 (! !+)hp+m(! is inserted at ! = 1 which we have suppressed. The expression (3.4) can be further simpli ed with the following two observations: 1. The ratios of the OPE coe cients Cp;;p;m;n Cp;;p;0;0 is nite (see also appendix B for explicit expressions). Thus, in the thermodynamic limit the operators with spatial derivatives are 1=L suppressed as they are multiplied with extra powers of (! the terms with m = n = 0. !+)m(! !+)n ! 0 for m; n > 0. We can keep only 2. From (1.5) the OPE coe cient for quasi-primary Op;p is given by Cp;;p = L(hp+hp) dp;p Op;p(E) where we have now allowed an arbitrary normalization factor dp;p for two-point function of Op;p. We then have (! !+)hp (! !+)hp Cp;p = ihp hp (2l)hp+hp Op;p(E) dp;p where we have used that in the thermodynamic limit 2 sin 0L = 2 0L = 2l. Local ETH implies that Op;p is, up to corrections suppressed in L, the same as the onepoint function in the canonical ensemble. The thermal one-point functions of quasiprimaries which are outside the identity Virasoro block vanish in the L ! 1 limit as 4We use tilde to denote density matrices in ! coordinates: ~ = U y U where U is the unitary that implements the conformal transformation. (3.5) (3.6) (3.7) they can be mapped to one-point functions on a complex plane.5 This implies that the contribution of any operator outside of the identity Virasoro block vanishes. We thus conclude that ~ ' X (p;p)2Viraosoro identitiy block ihp hp (2l)hp+hp dp;p Op;p(E)OpOp : (3.8) The Virasoro algebra xes the dimensions of the operators in the above sum to positive integers. We can organize the sum (3.8) in terms of quasi-primaries of dimension k and k constructed from the holomorphic (anti-holomorphic) stress tensor and its derivatives. More explicitly, Op in (3.8) are given by Tk( )'s which can be schematically written as6 Tk ( ) = X k1+k2=k L1Tk ( ) = 0: and satis es the quasi-primary constraint (Ln denote the Virasoro operators) At any positive integer k there are several linearly independent Tk quasi-primary constraint, which are labeled by index . We show in appendix C, for k even (odd) only one (none) of them survives the thermodynamic limit which is the one with the ( ) that solve the above T k term in it. We take = 0 to be the surviving quasi-primary at each level. The same holds for the anti-holomorphic OPE coe cients. Then (3.8) becomes where ~ ' X k;k2N ik k (2l)k+k d2kd2k Ok;k(E)T2(k0)T2(k0) Ok;k(E) = h jT2k T2k j i; (0) (0) hT2(k0)(z)T2(k0)i = jdzj24kk : with T k (T (T : : : (T T ))). The rst few T2(k0) are computed in appendix C: Operator T2(k0) is a polynomial of order k in holomorphic stress tensor T that starts (3.9) (3.10) (3.11) (3.12) T2 T6 3(42c + 67) @2(T T ) (22c + 41) 4 3c(2c 1)(5c + 22)(7c + 68) 5In fact, one can compute the one-point function of primaries on a torus with the modular parameter =L 1, and see that the nite-size corrections are exponentially suppressed in volume, see appendix D. 6The expression below should be understood as summing over di erent ways the derivatives are distributed among T 's. We thus nd that HJEP03(218)7 (3.14) (3.15) (3.16) (3.17) expectation value in j i. across all two-dimensional CFTs. 3.2 Renyi entropies The set of thermodynamically relevant observables are those with non-vanishing expectation value in j i. From the local ETH we know that this set does not include any operator outside of the Virasoro identity block. The translation-invariance of j i further implies that among the operators in the identity block only quasi-primaries have a chance of having a non-zero expectation value, because the descendants of quasi-primaries have the derivative operator which are suppressed by 1=L. The quasi-primaries of dimension k can be organized in the orthonormal basis introduced in appendix C. Since only T2k appear in the universal density matrix ~, they are the only quasi-primaries with non-vanishing To conclude this subsection we stress that the reduced density matrix (3.16) is universal Renyi entropies are invariant under unitary transformations. Hence, we can directly compute them in the Rindler conformal frame. The n-th Renyi entropy of a spinless quasiprimary state (h = h) is given by the Euclidean path-integral over an n-sheeted complex plane with 2n operators inserted at q and q 1 on each sheet.7 This manifold is topologically a Riemann sphere, and can be uniformized to one sheet using the map z = !1=n. Then, For large h, we have where we have used (3.2) and all the other terms in T2(k0) are suppressed in h: h jT 1+m=2 j i h m=2 1: ~ ' X k;k2N ik k 2l p 2 T 2(k+k) ck+k d2kd2k log n 4nh hQjn=01 (zj;n; zj;n) (zj0;n; zj0;n)i ! h (z0;1; z0;1) (z00;1; z00;1)in sin( Ll ) ! n sin( nlL ) + 1 1 n log hQjn=01 j (zj;n; zj;n) j (zj0;n; zj0;n)i ! Qjn=01h j (zj;n; zj;n) j (zj0;n; zj0;n)i where zj;n = ei(2 j+l=L)=n and zj0;n = ei(2 j l=L)=n. Using the universal OPE of in the thermodynamic limit we nd 1 j n + X X j 1 j<l n j j j j 1 j<l<q n j TK1 TK2 l l TK1 TK2 TK3 l l q q + · · · are universal. In appendix F we compute Renyi entropies perturbatively in subsystem size up to order O (l= T )8 and nd The authors of [10] computed the Renyi entropies above to the eighth order in the large c limit. The equation (3.18) is consistent with their result. 3.3 Generalized Gibbs ensembles In this section, we explore the relation between the ETH density matrix ETH computed in the last section with that of a Generalized Gibbs Ensemble (GGE). The comparison of observables in these two states can be used to study distinguishibility of the corresponding density matrices. Due to the complexity of computing the value of observables in a GGE, our comparison is, so far, incomplete. We hope this discussion can set the stage for future investigations of the properties of GGE. Two-dimensional CFTs have an in nite number of conserved charges, which are the KdV hierarchy of charges fQ2k 1; Q2k 1; k = 1; 2; g constructed from the polynomials of stress tensor [13, 15] 7Due to the Zn symmetry of this correlator one can alternatively compute it using a 4-point function with twist operators in a Zn orbifold theory. This is done in appendix F. with the rst few local currents given by On a cylinder of circumference 2 L the rst two charges are J2 = T; J4 = (T T ); J6 = (T (T T )) Q1 = Q3 = 1 L 1 L3 L0 c 24 1 n=1 2 X L nLn + L 2 0 c + 2 12 L0 + c(5c + 22) ! 2880 , with all the eigenvalues fq2k 1; q2k 1; k = 1; 2; g dimension h = EL. For example, the charges associated to Q1 and Q3 are xed in terms of only the conformal c 24 q1 = L 1 h ; q3 = L 3 h 2 c + 2 12 h + for the eigenstate j i prepared from matrix of the GGE In what follows we assume that the hypothesis of local ETH (1.1) holds for any sufciently excited Virasoro primary . As we discussed in section 2, we expect that ETH to be close in trace distance to the reduced density 1 X k=1 GGE = Z 1 exp 2k 1Q2k 1 + 2k 1Q2k 1 ! ; where the chemical potentials f 2k 1; 2k 1g are chosen to match the set of charges fq2k 1; q2k 1g of j i. If correct, (3.24) would provide a non-trivial consistency check of the local ETH hypothesis. In the thermodynamic limit the KdV charges of a Virasoro primary are easy to compute: L q1 = 2 ; L q3 = (2 )2; L ; q2k 1 = (2 )k; where is the energy density. To proceed further, we assume that the central charge c is large. In the c ! 1 limit, all 2k 1 except 1 = vanish; thus we recover the standard Gibbs ensemble [11, 16]. To see this, note that in the large c limit (see the next subsection for a derivation) 1 Z Tr J2ke H = 2 c k The two-dimensional thermal energy density is Matching this with the energy density in the eigenstate, using (1.3) and the de nition of dT in (3.2), we nd ( ) = c (3.21) (3.22) (3.24) (3.25) (3.26) (3.27) (3.28) HJEP03(218)7 In the next subsection, we use this change of parameters to compare the one-point functions in the energy eigenstate in (3.14) with those of the thermal state in (3.26) in the c ! 1 limit and they match exactly. The reduction of the conventional Gibbs ensemble e H =Z is only matching the ETH density matrix in the in nite central charge limit. The necessity to modify it when the central charge is nite is suggested by the non-zero values of KdV charges (3.25). Historically, rst indication that the excited primary state is locally di erent from thermal state came from the comparison of entanglement and thermal entropies in [10], although it should be noted that such a discrepancy by itself does not immediately preclude the corresponding density matrices to be trace-distance close [ 1, 5 ]. A direct comparison of local observables unambiguously showing that ETH density matrix can not match the canonical one was soon performed in [11], with more analysis probing nite 1=c corrections in an attempt to match ETH density matrix with the GGE one following in [12]. In this paper we further investigate this question. The main unresolved challenge here is to compute the expectation value of KdV currents in the GGE at nite c. Despite the fact that for larger k corresponding 2k 1 are suppressed by the increasingly negative powers of c, we nd a strong indication that one cannot perform a perturbative analysis by truncating (3.24) to nite number of 's even for the next to the leading order in the 1=c expansion (see appendix G). Hence to complete the check, one needs a truly non-perturbative expression for (3.24) both in terms of powers and numbers of included 's. We leave this task for a c 1, the universal density matrix in (3.16) simpli es and expofuture investigation. In the limit ha nentiates (see appendix C) This is because at large c ~ = e(DaT +DaT ); a2 = (2 L)2 11a6 70 4! 9a8 140 34a10 1925 8! ' k!ck 6! ; and we have used the change of parameters in (3.28). Note that in order to properly de ne the operator ~ one has to smooth out the exponent on a circle of radius around z = 0 where the operator is inserted: h ~ i = heHr= DaT +DaT i 3.4 Matching with thermal density matrix in the in nite c limit In two dimensions, the thermal cylinder is conformally at, therefore the expectation value of any operator that is outside of the Virasoro identity block vanishes in the Gibbs state. The translation-invariance further restricts the set of observables with non-vanishing thermal one-point function to the quasi-primaries. Below, we show that at large c the thermal expectation value of T2k scales as ck, whereas the expectation value of other quasi-primaries of the same conformal dimension scale with lower powers of c. (3.29) (3.30) (3.31) (3.32) where in the second line the normal ordering is imposed by a Cauchy integral. In a thermal state HJEP03(218)7 appears at dimension six: The normal-ordering is imposed by 2 c k (3.33) (3.36) (3.37) At large central charge, the multi-point thermal correlators are dominated by the disconnected piece: tr( T (x1) T (xk)) = tr( T (x))k (1 + O(1=c)) = (1 + O(1=c)) : (3.35) Plugging this in the right hand side of (3.34), and performing the Cauchy integral we obtain Now, consider a quasi-primary that is not Tk. The rst non-trivial such quasi-primary operator (T k) = (T (T the OPE: The current J2k is a polynomial of order k in stress tensor, where the normal-ordered (T T ))) is de ned by isolating the distance independent term in (AB)(!) = lim (A(!1)B(!) divergent terms) = !1!! A(!1)B(!); tr( (T T )(!)) = tr( (T (T T ))(!)) = tr( T (!1)T (!)) d!1d!2 !1)(! !2) tr( T (!1)T (!2)T (!)): (3.34) tr( (T k)) = (1 + O(1=c)) : 2 c k 9 (3.38) where we have used the fact that the thermal state is translation-invariant in space and time; hence, the disconnected piece of the expectation value on the right hand side is zero. The same conclusion applies to all other quasi-primary operators in the Virasoro identity block that are not T2k, as they also can be considered as multi-trace operators with at least one factor containing derivatives. If we rede ne the stress tensor in the large c limit according to T~ = T =c, the expectation value of T2k become order one, while the expectation value of any other quasi-primary in the Virasoro identity block is suppressed by negative powers of c. Thus, the only operators with non-vanishing expectation values in this limit are T2k. The quasi-primary operators T2k and the KdV charges J2k are both polynomials of order k in T and start with (T k). The derivative terms are di erent, however, as we just discussed the derivative terms are suppressed in large central charge. Therefore, 2 c This is the same answer as the one-point functions in the eigen-state (3.14) after we replace 2 = 3 2T =3. Since there are no other thermodynamically-relevant observables we have found that all the one-point functions of the eigenestate matches of those of the Gibbs state in the large central charge limit. Thus, in the large c limit we have proved that the universal density matrix of a Virasoro primary eigenstate is indistinguishable from that of the Gibbs state. It is interesting to compare the Renyi entropies in the thermal state with the eigenstate in the large c limit. We can take a large c limit in the low temperature expansion in (3.18) the perturbation theory of small x= T : HJEP03(218)7 It is clear that the Renyi entropies for n > 1 do not acquire thermal values given by = log log where in the second line we have used the change of parameters in (3.28). This is in contrast with the entanglement entropies of the states that match to the eighth order that we have computed. In the large c limit, one can in fact compute the dominant c piece of the entanglement entropy of the eigenstate non-perturbatively for nite values of l= T . In section 3.2, we computed the Renyi entropies directly by constructing the partition function that represents tr( 2) and uniformizing it. An alternative method to compute the Renyi entropy of the eigenstate is computing the four-point function of twist operators with orbifold theory; see (F.1) of Appenix F. The assumption of local ETH tells us that only the Virasoro identity block contributes to the correlator G4(z; z) = h n (1) n(z; z) n(1) n(0)i; where z = eix=L. The leading c piece of the contribution of the Virasoro identity block to the four point function above in the large c limit was found by solving the monodromy (3.40) (3.41) (3.42) n in an (3.43) equation for n near n = 1 in [7]: log G4(z; z) ' log c(1 n) i r h 6 24 The entanglement entropy computed this way from the identity block in the large c limit matches the entanglement entropy in the Gibbs state for any l= . Note that here we are working in the limit where h c 1, which in the language of [7] translates to 1. In our approach the assumption of local ETH guarantees that only the Virasoro identity block dominates. However, the authors of [7] assumed a sparse spectrum of low-dimension operators to truncate to the identity block. 4 Higher dimensional CFTs In this section, we rst discuss the general structure of the ETH density matrix in higher dimensions, and then compute the entanglement entropy to the leading nontrivial order in l= T expansion. We compare the result to the holographic entanglement entropy computed using the Ryu-Takayanagi formula at this order and nd agreement. The intuition is that even though our CFT computation does not assume large N or strong coupling, at this order the answer is universal because it depends only on dT that is the normalization of the two-point function of stress tensor. To match the entanglement entropies we have to set the coe cient dT to be (4.28), as is required in a holographic CFT. This provides a consistency check of the local ETH. 4.1 ETH density matrix We observed that in two dimensions assuming local ETH implies that only the polynomials of stress tensor propagate in the thermodynamic limit of OPE. Here, we consider density matrices in primary energy eigenstates of higher-dimensional CFTs satisfying local ETH. A generalization of the map introduced in (3.3) (see appendix A) maps the radial quantization frame to the Rindler frame. In Rindler coordinates, the subsystem B is mapped to the negative half-space X1 < 0, and the operators that create and annihilate the state are, respectively, at X and X+. Since Xi>2 = 0 we can use the two-dimensional complex coordinates to describe their location: X0 +iX1 = e i 0 = 1=q and X0+ +iX1+ = ei 0 = q.8 The distance between the two operators in these coordinates is 2 sin 0 ' 2l=L. The operator product expansion in the thermodynamic limit l=L ! 0 becomes (X+) (X ) h (X+) (X ) i 1 p ' X Cp;n^ ~n hp j j Opn^(X ) = X fpn^(l= T )hp Opn^(X ) (4.1) 1 p where X+ = 2 sin 0n^ +X , n^ is the unit vector in the X0 directions, and we have dropped the descendant elds because their contribution is 1=L suppressed. The operator Opn^ is a 8Note that compared to the two-dimensional map the location of ! and !+ are swapped. primary with spin with its indices contracted with n^ according to and nally fp is de ned by Opn^ = (n^ 1 n^ 2 traces) (Op) 1 2 Cp;n^ = h (1)Opn^(1) (0)i hOpn^(1)Opn^(0)ih (1) (0)i ; f pn^ = (2 T =L)hp Cp dp;n^ = hOpn^(1)Opn^(0)i: n^ dp;n^ = h jOp j i (2 T )hp It is customary to de ne a coe cient dp that is independent of n^ in the following way: h(Op) 1 m (x )(Op0 ) 1 m (0)i = dp pp0 jxj 2hp I 1 m; 1 m ; where the tensor I 1 m; 1 m is xed by conformal symmetry [17, 18]. Every CFT has a stress tensor that is a primary of dimensions d + 1. The energy density in primary state j ai is dT; where !d is the volume of the unit sphere Sd. As an example, consider the term in the OPE expansion (4.1) that corresponds to stress tensor (2l)(d+1) (n^ n^ =(d + 1)) T = (n^ n^ =(d + 1)) T ; = E Ld!d = ha where T is the length associated with the energy density, and dT is the central charge de ned by the two-point function of stress tensor: h To obtain the density matrix in the thermodynamic limit we have to study the OPE in (4.1) in more detail. From the equivalence of the microcanonical ensemble and the thermal ensemble we expect the coe cient fp ' (2 T )hp dp tr( T Op) to have the interpretation of a thermal one-point function up to volume suppressed corrections, where the thermal state is chosen to have the same energy density as the eigenstate j i. In two dimensions, we saw that thermal one-point functions vanish which let to a truncation of the OPE to only the Virasoro identity block. However, in higher than two dimensions thermal one-point functions do not vanish, and fp are, potentially, non-zero. One way to obtain universality in higher dimensions is by restricting the class of higher dimensional theories we study; for instance the holographic theories. In holographic CFTs the thermal one-point function of conformal primaries are 1=N suppressed except for operators constructed from the stress tensor. Tn large N CFTs resemble two-dimensional CFTs in the sense that they have multi-trace operators T m in their spectrum that are primaries of conformal dimension m(d + 1), up to 1=N corrections. In holographic theories the thermal correlator is essentially classical, that is to say the thermal variance of the the \universal" density matrix.9 Therefore, from local ETH and the equivalence of ensembles one expects CT m hm which implies that they survive the thermodynamic limit and contribute to Atherm. In holographic theories, T m are in Atherm and one needs to include them in the sum in the de nition of Entanglement entropy from ETH density matrix As opposed as two-dimensional case, the ETH density matrix in (4.1) is not universal. That is to say that at nite central charge we only know one operator in the set of thermodynamically relevant operators Atherm. If we try to repeat our low temperature analysis of the ETH density matrix in d > 2 we need to make further assumptions about the spectrum of the theory. Let us assume that there are no relevant primary operators in the set Atherm. In other words, we are assuming that fp = 0 for all operators Op in (4.1) with hp < d. Then, to the rst non-trivial order the ETH density matrix is tr( ~ * ) ' 1 + d + 1 d 2l T d+1 n^ n^ T + ! + : (4.10) In a CFT the operator T = 0 in at space. Now, we can compute the entanglement entropy of the ETH density matrix at this order and compare it with the reduced density matrix of the Gibbs state. Renyi entropies are unitarily invariant, and it is more convenient 9At nite central charge the only primaries one can construct from T are large spin operators of type (T T )n;l (T m)(n1;l1) (nm;lm) = ((T T )n1;l1 Tn2;l2 ) hence one expects their OPE coe Tn;lm ). However, every derivative su ers a 1=L suppression and cients to scale, at best, as hm rather than hm+(2n+l)=(d+1) that is required to survive the thermodynamic limit. An explicit calculation of the OPE coe cients C expectation [9]. This calculation is done assuming that the spin is largest parameter. However, for our case of interest we want the conformal dimension of the operator to be much larger than its spin which is much larger than one. It is plausible that in our limit of interest these operators survive the thermodynamic limit and contribute to Atherm. We thank Liam Fitzpatrick and Sasha Zhiboedov for pointing this out to us. [T T ]n;l con rms this to compute the entanglement entropy in Rindler coordinates. The vacuum-subtracted Renyi entropy in primary state j i is given by where the subscript (2 n) refers to the angle around the boundary of B: X0 = X1 = 0 in Rindler space. We denote the generator of rotation around this hypersurface as @ : X0 + iX1 = !; !=! = e2i : We are interested in entanglement entropy which is found from the n ! 1 limit of i(2 n) i(2 ) n!1 n i(2 n) " hQjn=1 j j i(2 n) # n!1 : i(2 n) = tr e 2 nH P( ) n!1 = 2 hH h i(2 ) i(2 ) = 2 !d 1 ld+1 d(d+2) = d(d+2) (4.11) (4.12) (4.13) (4.14) l T d+1 (4.15) (4.16) Our calculation closely follows the method used in [19], and uses the Hamiltonian language: where P is the path-ordering operator in the Euclidean space. The rst term in (4.13) is the change in the expectation value of the vacuum modular operator H: h i(2 ) 2 d=2 This is the so-called rst law of entanglement entropy; for small variation of density matrix S = 2 H, where H is the generator of Euclidean rotation in the direction. The second term in (4.13) is the relative entropy of the eigenstate with respect to the vacuum reduced to the subsystem B: S( k ). The task is to compute the relative entropy above perturbatively in powers of l= T . Since 's approach each other pairwise in Rindler space, one can use the at space OPE. At the next-to-leading order the entanglement entropy is n 1 j=1 n X G0n0(2 j)5 3 n!1 Gn (2 j) = hT (0)T (2 j)i(2 n); where the index 0 signi es the X0 in Rindler coordinates. We follow the method advocated in [19] to analytically continue the expression above in n: 2⇡ n 0 ⌧ · · · (a) C s 2⇡ n 0 ⌧ · · · (b) C s where s is the complexi ed angle. The contour C is deformed to run over ( 1 + i(2 n ); 1 + i(2 n )) and (1 + i ; 1 + i ); see gure 4: HJEP03(218)7 The analytic continuation is the choice to set e2 in = 1 in the denominator. Gn ( is + ) Gn ( is + 2 n ) es+i 1 es+2 in i 1 (4.17) n!1 = @ntr e 2 nH T (0)T (s + i ) n!1 = 2 tr e 2 H HT (0)T (s + i ) tr(e 2 H HT (0)T (s+i )) tr(e 2 H HT (s i )T (0)) es+i 1 es i 1 and n!1 = i the KMS condition Putting this back in A(n) gives The second term can be further simpli ed using the commutator [H; T (s)] = i dTds and tr(e 2 H HT (s i )T (0)) = tr(e 2 H (T (s i )H [H; T (s i )])T (0)) = tr(e 2 H HT (0)T (s + 2 i i )) + i tr(e 2 H T (s i )T (0)) d ds 1 es i G 1 ( is + ) es+i 1 ds G 1 ( is es i 1 ) tr(e 2 H T (s i )T (0)) (4.18) The term in the rst line vanishes since there are no poles in the region encircled by the contour integration. Using integration by parts we can write the second term as Z 1 1 4 sinh2((s i )=2) h T (Xs)T (X0)i (4.19) d 1 ds ds where X0 = (1; is=2; ) and Xs = (1; is=2; 0; ) in Rindler coordinates. Therefore, ds dT 1 (2 sinh(s~=2))2(d+2) S ; where s~ = s i and v); d + 1 HJEP03(218)7 Then, Therefore, One can perform the integral explicitly h 2u u juj2 dCddT d + 1 Cd = (d + 2) 2F1 [2(2 + d); 2 + d; 3 + d; 1] = Cd = ( 1)d 1 (2 sinh(s~=2))2(d+2) : (d + 1) CddT 2d 2l T 2(d + 1)2 (d + 3) (d)dT 2 (5 + 2d) 2(d+1) (2)S = 4c l T 4 2 (d + 3)2 (d + 2) (5 + 2d) 2l T 2(d+1) 2 = ds2 = L2 z2 (4.20) (4.21) (4.22) (4.23) Note that here dT = hT00T00i(d + 1)=d, and in d = 1 we have dT = c=(2 2) therefore which is the same as the result we found in two dimensions. In d > 2 we do not know the entanglement entropy in the reduced state of the Gibbs ensemble, lT , however, if the theory is holographic we can compare the result with the prediction of the Ryu-Takayanagi formula. Next, we show that the above result can be reproduced using a gravitational calculation in a black hole background. 4.3 Holographic theories Consider the thermal state of a holographic CFT in at space dual to the planar black hole dz2 f (z) f (z) = 1 (4.24) zd+1 zd+1 : h ds2 = L2 z2 (dz2 + g (z; x )dx dx ); It is convenient to switch to the Fe erman-Graham coordinates to compute the entanglement entropy perturbatively in l= l=zh: g (z; x ) = + azd+1T + a2z2(d+1)(n1T T + n2 T T ) + (4.26) where a = (d1+61)GLd , n1 = 1=2 and n2 = coordinates with = z2=L2 (dimensionless) is 81d . The bulk Ricci tensor written in these R L2R Perturbatively in l we nd that the vacuum subtracted entropy is [20]10 2 g00 + 2 g (g0 )2 g0 g g0 + (d 2)g0 + g g g0 d g : Here, zh is related to the thermal wavelength zh = (d+1) . The entanglement entropy of the 4 reduced state on a ball of radius l is the area of an extremal surface in the bulk anchoring on the boundary of the subsystem: S( T ; l) = LdSd 1 Z l 4G 0 dr r d 1 s zd 1 + f (z) This is exactly the answer we found in the eld theory in (4.22) for the entanglement entropy of the universal density matrix in arbitrary dimension d. If the local ETH hypothesis is correct in holographic CFTs, the reduced density matrix in any energy eigenstate is well approximated by the ETH density matrix (4.10). According 10Note that there is a typo in equation (3.55) of that paper. T020l2(d+1) d+1 dT Ld : l T 2(d+1) (4.25) HJEP03(218)7 (4.27) (4.28) (4.29) d 2 1 2 = T dT = 1 4 g g00 + (g )2(g0 )2 2 !d 1T00ld+1 d(d + 2) = l T where we have used T00 = ddT+1 and !d = 2((d(d++11))==22) . The rst term is simply the rst law of entanglement entropy. The quantity 8LdG is related to the two-point function of stress tensor as: Plugging this back in (4.27) gives d + 2 d (d + 2) (d+1)=2 ((d + 1)=2) 8 G (d + 1)2 (d + 3) (d)dT (2d + 5) 2l T 2(d+1) to holography, the gravity dual of a heavy energy eigenstate is a black hole of the same energy density. Therefore, if the local ETH holds the entanglement entropy of the ETH density matrix should match the entanglement entropy computed holographically in the dual black hole geometry. In this section, we checked that in the same temperature limit l= T 1, indeed, the local ETH hypothesis passes this consistency check. 5 Local equilibrium Up to this point we were only concerned with the eigenstate thermalization hypothesis. We showed that the reduced density matrix of small subsystems in energy eigenstates are universal. Energy eigenstates are highly ne tuned and that their time-evolution is given by just an overall phase. Intuitively, we expect the density matrix of small subsystems to be only a function of energy not only in translationally-invariant energy eigenstates but also in all states that have spatial and time dependence over scalecs much larger than the size of the subsystem. In this section, we establish that this is indeed the case by studying the reduced density matrices in two classes of time-dependent states: \coherent" states, and arbitrary superpositions of N eS(E)=2 energy eigenstates. Time-dependent coherent states We de ne \coherent states" j (~s)i via a Euclidean path-integral with a local operator inserted at ~s inside the unit ball in the radial quantization frame: j (~s)i = es P (0)e s P j i (5.1) as a real parameter. around S primary state: We can use the rotational symmetry of the unit ball to bring the operator insertion to the point (r = e ; 1 = ) and i = 0 for all i > 1. Coherent states include a superposition of many energy eigenstates, and hence evolve non-trivially in time. Mapping to the Rindler space the operators that create and annihilate the state go to, respectively, Y and Y+ : cos( 0 + ) cosh cos( 0 + ) cosh sin 0 sinh cos cos 0 cosh Y i>1 = 0 (5.2) where 0 it and we have analytically continued to the real time to keep track of the time evolution of the state. The analytic continuation in time is achieved by treating The parameter 0 controls the width and angular dependence of the energy pro le d at time t = 0. To see that we compute the energy density in this spinless HJEP03(218)7 ha ha j ; 0 (t)iCyl = Ld+1!d (cos( ) cosh )(cos( ) cosh +) = Ld+1!d ( cos t coth 0 cos( ) csch 0)2 +sin2 t (d+1)=2 sinh2 d+1 # 2 Then, the distance between the operator insertions is T ( 0; ; t) = (t)L jY+ Y j2 = At t = 0 the energy density around Sd has its peak value coth2( 0=2) at the point ( ; 0; ). In the thermodynamic limit of small subsystem l=L 1 the energy density is constant over the subsystem (t; 2 B) = Ld+1!d d+1(t) (1 + O(1=L)) 2(t) = (cos t coth 0 cos csch 0)2 + sin2 t The \local" length scale associated to the energy density is and the density matrix becomes X which shows that the reduced density matrix is universal with T multiplied by (t). That is to say at any time t the reduced density matrix is in equilibrium with a time-dependent thermal wavelength (t). 5.2 Arbitrary initial states An arbitrary CFT state in the Schrodinger picture expanded in the energy eigen-basis is N a=1 j (t)i = X eihat=Lcaj ai The reduced density matrix on a ball-shaped region in this state is a partial trace over the complement region BR (t) = trBRc j (t)ih (t)j = X cacb eit(ha hb)trBRc j aih bj ab Now, it is straightforward to see X a k BR (t) jcaj2 uni(E = Ea)k supa6=bk abk X cacb j a6=b 1 (d+1) ; 1 !ddT 4l2 L2 (t) X e S(E)=2(X jcaj)2 N a=1 N e S(E)=2 (5.9) (5.3) (5.4) (5.5) (5.6) (5.7) (5.8) for some = O(1). Therefore, as long as the number of superposed energy eigenstates N does not scale with entropy the reduced density matrix is well-approximated with a classical mixture of universal density matrices: which does not evolve in time. If the state has h (t)jHj (t)i = E0 and h (t)jH2 E02j (t)i = E0 then the density matrix is approximately Z dE p(E) uni(E) uni(E0) + 2 (5.10) (5.11) Quenching an energy eigenstate with a local operator of energy order one is an example of a state that necessarily includes a large number of energy eigenstates. 6 Conclusions In this work, we continue the study of the Eigenstate Thermalization Hypothesis (ETH) in the context of Conformal Field Theories initiated in [1]. In that paper, we formulated the subsystem ETH in CFTs as a statement about the smooth dependence of the reduced density matrix of an energy eigenstate on energy. We proved that if ETH is satis ed at the level of individual local operators (local ETH ), the subsystem ETH follows. In [1] it was shown that the ETH density matrix exhibits a great degree of universality provided that the subsystem in question is small compared to the total volume. When the subsystem is small in comparison to the inverse e ective temperature, the ETH density matrix admits a perturbative expansion in terms of the light primary operators (1.12). In 2d CFTs the statement of ETH implies that no operator outside of the Virasoro descendants of identity contributes to the OPE of any two heavy Virasoro primaries. As a result the ETH density matrix exhibits a greater degree of universality, depending only on the e ective temperature and the central charge, but on other detail of the underlying theory (3.16). In section 2 of the paper we provided an argument based on the equivalence of ensembles, modi ed for the case of CFTs, to argue that the ETH density matrix for a small subsystem is trace-distance close to other thermal ensembles, the reduced canonical and the microcanonical ones. This general argument is further supported by the calculation and comparison of the eigenstate entanglement entropy with the holographic one in section 4. In case of two dimensions, because of the additional conservation laws, the canonical ensemble must be substituted by the grand canonical ensemble that includes an in nite number of conserved KdV charges | the Generalized Gibbs Ensemble. A new representation of the ETH density matrix and its equivalence with the thermal one in the limit of in nite central charge is demonstrated in section 3. There we also calculate the von Neumann and the Renyi entropies for the eigenstate and discuss the nite c case. Finally, in section 5 we discuss the reduced density matrix of time-dependent coherent states and show that their reduced density matrix on a small subsystem is well-described by the universal ETH density matrix with time-dependent e ective temperature. Acknowledgments We would like to thank John Cardy, Liam Fitzpatrick, Thomas Hartman, Matthew Headrick, Tarun Grover, Mark Srednicki, Matthew Walters and Sasha Zhiboedov for valuable discussions. The research of NL is supported in part by funds provided by MIT-Skoltech Initiative. AD is supported by NSF grant PHY-1720374. This work is supported by the O ce of High Energy Physics of U.S. Department of Energy under grant Contract Number DE-SC0012567. A Rindler space: a convenient conformal frame Consider a (d + 1)-dimensional CFT in radial quantization with a ball-shaped subsystem of angular size 0 on Sd at r = 1. According to the operator/state correspondence the density matrix in the subsystem is given by a path-integral over the (d + 1)-dimensional space with two operators inserted, at r = and y at r = 1= with open at the location of the subsystem. The initial metric in the radial quantization is ! 0, and a cut ds2 = dr2 + r2d 2d with ( 1; d) the coordinates on Sd. We perform the following conformal transformation X X X ; ; ; LV 2 ds2 = (X)2dXidXi (X) = X0 V = (1 2X1 + X V+ 8L X): 1 X = ( sin 0; cos 0; 0 ; 0); (X ) = (2 sin 0) 1; (X+) = 2(2 sin 0) 1: 1) 1 + r2 + 2r cos 1 2Lr sin 1 sin 2 1 + r2 + 2r cos 1 2Lr sin 1 sin 2 1 + r2 + 2r cos 1 cos i+1 = sin d = 1 1 1 X0 Xi Xd 2X1 + X 2X1 + X 2X1 + X 2Lr sin 1 cos 2 1 + r2 + 2r cos 1 = (1 X X)=2 2X1 + X X ; 1 1 2 d > i > 1 L = cot( 0=2); that maps the subsystem at r = 0 and 1 0 to the negative half-space, i.e. (0; X1 < 0; 0 0). Here L is the radius of Sd in units where R is set to one. The new metric in the X-coordinates that we call Rindler frame is given by In these coordinates the path-integral without operator insertions prepares the Rindler density matrix in vacuum. The operators and y are now inserted at X and X+ re(A.1) (A.2) (A.3) Under this map a conformal primary transforms according to Therefore, h (r = 0) i (X) ij = (X(r = 0)) h h (X(r = 0) i ij h (1= ) ( ) iradial = (2 sin 0)2hh (X+) (X ) iRind In the thermodynamic limit 0 1 the distance between and X j = 2 sin 0 1, and we use the OPE to obtain y goes to zero: jX+ h (1= ) ( ) iradial = 2h X C p (2 sin 0)hp hOp(X0) i : p HJEP03(218)7 B Global descendants in two dimensions Consider the OPE of two quasi-primaries (z; z) (0; 0) h (z; z) (0; 0)i p p where p are quasi-primaries and a j C p p = C(j; hp + j C(j; 2hp + j 1 h p pi h j pj i; 1) 1) in CF T2 X a j j;j p a j p = C(j; h) = C(j; hp + j C(j; 2hp + j 1) 1) (h + 1) (j + 1) (h j + 1) p In the thermodynamic limit z = l=L, h and L go to in nity with T L= h kept xed we have a j p z j ! 0 8j > 0: Therefore, all the derivative terms are subleading, and we have (z; z) (0; 0) h (z; z) (0; 0)i X C p p zhp zhp p + O(1=L): This argument generalizes to higher dimensions. Consider a primary Op and its rst descendant. Then, the OPE coe cients are the same order (B.1) (B.2) (B.3) (B.4) COp = dOp h (1)@Op(1) (0)i = 2hp(2hp 1)hp = O(h0 ) (B.5) however, by in the OPE of s, the derivative term has an extra power of l=L and is hence more suppressed. correponding operator insertion of operator In this appendix, we expand the reduced state on an interval of length 2k in a highly excited primary energy eigenstate, and nd the quasi-primaries that contribute to the universal density matrix, that are T2k in (3.11). Consider a primary energy eigenstate j ai and its a. In Rindler coordinates, the density matrix is created by the a(z; z) a(0) h a(z; z) a(0)i X X p fk;kg Capafk;kgzhp+K zhp+K L fkgL fkgOp (C.1) in the Euclidean path-integral. Here, fkg = fk1 klg, K = k1 + with L going to in nity in the thermodynamic limit. The OPE coe cient Capa;fk;kg (growing + kl, and z = x=L with ha) competes with the vanishing coe cient (x=L)hp+K . To determine what operator survive the thermodynamic limit in (C.1) we need to investigate the growth of this OPE coe cient with ha. It is convenient to de ne the OPE coe cient with lowered indices [21] Capa;fk;kg = X fk0;k0g M 1 pfkgfk0g M 1 p;fkgfk0g Caa;pfk0gfk0g Caa;pfk0gfk0g = L fk0gL fk0gh a(1) a(1)Op(y)i y=0 : (C.2) The matrix M is the Kac matrix de ned by Mfkg;fk0g(hp; c) = hhpjLfkgL fk0gjhpi, and We only need to consider Caa;pfkgfkg. The di erential operator z2ha ha(k 1)(z k +! k L kh a(1) a(1)Op(y)i = Capa (z;!)!(1;1) lim = Capa(ha(k 1)+hp) ' Capaha(k 1) : At order K we are comparing OPE coe cients of operators of the form Lk1 Lk2 with k1 + + kl = K. From (C.3) it is clear that the OPE coe cient of operators with L 1 does not grow fast enough with ha and they drop out of the thermodynamic limit, which is consistent with the result in appendix B. We only need to consider the case with ki > 1. Then, Caa;pfk1; ;klgfk1; kmg hl+m: a For even K the OPE coe cient of the quasi-primary that includes LK2=2 wins over other terms. When K is odd none of the OPE coe cients are large enough to compete with (x=L)K+K . Therefore, the sum over fk0; k0g in (C.2) only has one term, and Capafk;kg = Capabp;fk;kghaK=2+K=2 bp;fk;kg = M 1 f2; ;2gfkg 1 f2; ;2gfkg M (C.3) Lkl Op (C.4) (C.5) where K and K are both even. Note that in two dimensions Caa = 0 for all non-identity Virasoro primaries p. Therefore, a(ei 0 ; e i 0 ) a(e i 0 ; ei 0 ) h a(e i 0 ; ei ) a(e i 0 ; ei 0 )i 1 d2m 0 (0) X (2pha sin 0)2m X m2N 2l 2 X k1+ kl=2m T [M 0 fkg X T2m k1+ kl=2m 2m cm d2m [M h:c: 1]f2 2gfk1 klgL k1 L kl p X bfkg(2 ha sin 0)K L fkgA h:c: 1 1]f2 2gfk1 klgL k1 1 L kl A h:c: (C.6) hT2(m0)(1)T2(m0)(0)i. The rst few T2(k0) are where in the last two lines we have de ned an operator T2m (0) with the norm d2m = T2 T6 T8 70c+29 L2 3 70c+29 3(42c+67) L 4L 2 70c+29 6(10c+13) L 6 6 630c2 +3471c 557 L 4L2 2 + (5844 1512c)L 5L 3 5c(210c+661) 251 5c(210c+661) 251 27(c(42c+265) 167)L2 4 5c(210c+661) 251 24(c(150c+569)+67)L 6L 2 6(5c(126c+463) 543)L 8 5c(210c+661) 251 5c(210c+661) 251 25c(462c+3067)+3767 12(11650c+15341) 25c(462c+3067)+3767 18 L 8L 2 36(4358 3225c)L 7L 3 + 25c(462c+3067)+3767 36(c(1650c+16783) 8405)L 6L 4 + 25c(462c+3067)+3767 (31032c+220236)L2 5 25c(462c+3067)+3767 9(45c(154c+1873)+25133)L2 4L 2 + 25c(462c+3067)+3767 48(5115c+1081) 25c(462c+3067)+3767 6 L 4L3 2 30(5115c+1081)L2 3L2 2 25c(462c+3067)+3767 504(c(300c+1693)+266)L 10 25c(462c+3067)+3767 924(90c+259)L 5L 3L 2 25c(462c+3067)+3767 18(5115c+1081)L 4L2 3 25c(462c+3067)+3767 (L n 2 )(!) = L3 2(!) = (T (T T ))(!); 1 2 { 31 { d8 = d10 = Note that (C.7) (C.8) (C.9) d4 = 10 d6 = 3c(2c 1)(5c+22)(7c+68) 4(70c+29) 3c(2c 1)(3c+46)(5c+3)(5c+22)(7c+68) 10c(210c+661) 502 15c(2c 1)(3c+46)(5c+3)(5c+22)(7c+68)(11c+232) 4(25c(462c+3067)+3767) L2 3 + L 4L 2 + L 2L 4 (!) = 3 2 10 An alternative way to construct the quasi-primary operators T2k is by choosing the basis where the Kac matrix is diagonal. In this basis, it is evident that the only quasiprimaries that include the term Lm2 = T m(0) propagate. Here, T m = (T (T (T T ))). We can choose our operator basis such that at even order K only one quasi-primary includes LK2=2 which becomes our operator of interest T2(k0). Below, we describe how to construct it at any even order K. 1. Consider an arbitrary superposition of L fkg with no L 2; 2 : 2. Choose afkg such that this state is annihilated by L1. The result is the most generic Pfkg6=(2; ;2) akL fkg(0). quasi-primary with no L 2; 2 . 1 d2m 1 m! c 2 m T2 T6 T8 27 2 3. Find an arbitrary superposition state with L 2; 2 that is perpendicular to the state above, and demand that it is killed by L1. The resulting state is TK(0). We end this appendix by consider the quasi-primaries T2k in the limit h 1. In this limit, the expressions for the rst T2k simplify to exp a2 = 1; X 0<m2N a2m a4 = X m2N 2l 3 10 4l2 2 T T 2m L 2m a6 = ! 11 70 Therefore, the holomorphic part of the density matrix operator becomes It is convenient to write the universal density matrix in an exponentiated form in this limit: 5 L 4L2 2 60 m 1 m! T2(m0) = X m2N 4 2l2 m 1 3 2 m! T2m (0) 6L 4L3 2 L 10 : (C.10) (C.11) a8 = a10 = { 32 { One-point functions on a torus The nite temperature expectation value of a primary operator at nite volume in twodimensions is a one-point function on a torus with modular parameter and are the periodicities of the spatial and time circles, respectively. Modular invariance related the one-point function at high temperatures to low temperatures = iL where L Therefore, for ~ = L we have hOpi 1= = ( 1)hp hp hp hp hOpi : tr( Op) = (LT )2hp tr( ~O) The parameter q = e2 i at = iLT becomes q = e 2 LT and small at large LT . Therefore, we can expand the one-point function perturbatively in small q: hOpiq = X C(h;h)(h;h) q p h;h h c241 qh c241 1 (q) (q) 1 N=0 X qN HN;h;p: The coe cients HN are found using a recursive relation with the rst term H0;h;p = 1 [22]. At large LT only the lowest dimension primary of dimension ( ; ) contributes tr( ~Op) ' p Ph;h C(h;h)(h;h) e 2 LT (h+h c=12) Ph;h e 2 LT (h+h c=12) ' Cp ; e 2 LT ( + ) : This conclude our estimate of the size of one-point function probes in the thermodynamic tr( Op) = (T L)2hp e 2 LT ( + ) p C ; : 1 the thermal one-point functions are exponentially (D.1) (D.2) (D.3) (D.4) (D.5) HJEP03(218)7 limit As expected in the limit LT suppressed. E Perturbative Renyi entropies In this appendix, we compute the Renyi entropies of the universal density matrix via a direct calculation of tr( n). We take the subsystem to have size 2x, and the length scale associated with the energy density in to be T . The trace of n is computed by sewing n copies of the path-integrals that prepares (the path-integral in Rindler space with the operator (3.16) on each copy). Therefore, the vacuum subtracted Renyi entropy of is 1 1 log (n + 1)c 12n j=1 KjKj 2xpc 2 n T Kj+Kj e2 ij(Kj Kj)=n TKj (e2 ij=n)TKj (e 2 ij=n) E : dKj dKj We expand the above expression in powers of 2x= T and consider the rst few terms. The rst term corresponds to (Kj ; Kj ) = (0; 0) for all j except for K0 and K0. This term is X n 1 n X 2 l=1 K1K2=0;2;4 K1K2=0;2;4 n X X 2 l=1 K;K 2xpc 2 n T 2xpc 2 2 n T X 1 l<m<q n 1 K1K2K3=2 X X 1 equal to one by the normalization of two point functions. The rst non-trivial term appears at j = 2 and (2x= T )4: K1+K1+K2+K2 e2 il=n(K2 K2) d2K1dK2 2 h(TK1TK1)(1)(TK2TK2)(e2 il=n) i dK dK 2K+2K sin( l=n) 2(K+K) c(x= T )4 (n2 1)(n2 +11) 90n3 : At j = 3 we have 6-point functions of a (3-point functions of TK ) D TK1(e2 il=n)TK2(e2 im=n)TK3(e2 iq=n)TK1(e 2 il=n)TK2(e 2 im=n)TK3(e 2 iq=n)E 2xpc 2 n T Pi3=1(Ki+Ki) e2 i(l K1+m K2+q K3)=n d2K1d2K2dK3 2 2xpc T 6 2CT T T sl2ms2mqsql 2 = (2x= T ) 6 c (n2 1)(n4 4)(n2 +47) 32 3 2835n5 where Ki = Ki Ki and slm = sin( (l m)=n). We have used the summation identities in [23]. It is important to note that up to the order (l= T )6 the density matrix depends only on the energy density of the pure state. Therefore, to the sixth order we nd Sn( ; x) = 12n (1 + n)c (4 12n2 n2)(n2 + 47) 144n4 The next non-trivial one-point function h jT4j i contributes to the entanglement entropy at order (l= T )8. In the next appendix, we result above to the sixth order and compute the eighth-order term using the twist operator method. F Twist operators The correlation function (E.1) that appears in the calculation of the Renyi entropy of the universal density matrix is Zn symmetric. That is to say that it is invariant under z ! e2 i=nz. An alternative way to compute this correlator is by employing twist operators in a Zn-orbifold theory. Here, we use the orbifold theory to reproduce the result of the last subsection and extend it to the eighth order in subsystem size. In the orbifold theory, the vaccum-subtracted Renyi entropy in terms of the four-point function below 1 1 Sn( ; x) = log G4(z; z); G4(z; z) = h n(1) n(z) n(1) n(0)i h (1) (0)inh n(z) n(0)i (E.1) (F.1) where z = eix=L. The quasi-primaries of the orbifold theory take the form Qin=1 O(i), where Oi is the primary on the ith copy. Local ETH implies that this correlator is dominated by the Virasoro identity block. Below we use perturbation theory to compute Renyi entropies order by order in 2x= T . to z6 are The quasi-primaries that contribute to the Virasoro identity block at even orders up T (j) T (i)T (j)(i 6= j); T4 T (i)T (j)T (l)(i 6= j 6= l 6= i); T (i)T (j)T (l)T (m)(6=); T4(j)T4(l)(j 6= l); T6(j)T (l)(j 6= l); T8(j) T (i)T (j) (l)(6=) T4 T4 (j)T2(l)(j 6= l); T6 where the symbol 6= means that all pairs of indices are unequal. These operators are listed in [23]. The correlator factorizes into the holomorphic and anti-holomorphic parts G4(z; z) = jF (z; n; c)j2 where the vacuum conformal block F is only a function of cross ratio z, Renyi index n and CTn(j)n CTn(j)n (1 z)2 + CTn(j)Tn (l) CTn(j)nT (l) +CTn4 n CT4n(j)n (1 z)4 (j) CTn(j)Tn (l)T (q) CTn(j)nT (l)T (q) +2CT4n n (j)T (l) CTn4 n (j)T (l) +CTn6 n CT6n(j)n (1 z)6 (j) + CTnT TnT CTnT TnT +3CTnT Tn4 CTnT Tn4 +CT4nT4n CTn4T4n +2CT6nT n CTn6Tn +CT8n n CTn8 n (1 z)8 order z2 order z4 order z6 order z8 central charge c. F (z) = 1+ X ordered bT4T4 = where Pordered runs over all indices of the operator as 1 j1 < j2 < < jk large h we have CTkn1 nTkm = hk1+ km , and de ne bTk1 Tkm = Pordered CTnk1 n Tkm . These sums are computed in [10]: ; bT4 = 2 1 2 288n3 ; bT6 = 2 1 3 10368n5 ; bT8 = 2 1 4 497664n7 2 1 12n 2 1 5c(n+1)(n 1)2 +2n2 +22 2 1 2 5c(n+1)(n 1)2 +4n2 +44 2 1 3 5c(n+1)(n 1)2 +6n2 +66 1440cn3 17280cn5 622080cn7 +70c n 2 1 3 11n3 7n2 11n+55 +8 n 2 1 n2 +11 157n4 298n2 +381 (F.2) (F.3) n. At (n 2) n2 1 35c2(n+1)2(n 1)3 +42c n4 +10n2 11 16(n+2) n2 +47 362880c2n5 175c2(n+1)3(n 1)4 +350c n 2 1 2 n2 +11 175c3(n+1)3(n 1)4 +420c2 n2 1 2 n2 +11 4c 59n5 +121n4 +3170n3 +6550n2 6829n 11711 bT T T = bT T T4 = bT T T T = (n 2) n2 1 14515200c2n7 (n 3)(n 2) n2 1 87091200c3n7 128(n+2) n4 +50n2 111 +192(n+2)(n+3) n2 +119 Performing the Zn sums over trigonometric functions we nd F (z) = 1+a2h(1 z)2 +a4h2(1 z)4 +a6h3(1 z)6 + a2 = a6 = a8 = 12n 10368n3 + 3 3n+3 n 497664n7 a4 = 288n2 + 720n3c (n2 1)2 (n2 1)(n2 +11) 1)3 (n2 1)2(n2 +11) (n2 1)(4 n2)(n2 +47) 8640n4c 2 1 4 n4 +9n2 22 22680n5c2 2 1 3 207360cn7 (n 2)(n 1)(n+1)(59n6 +136n5 +3191n4 +6640n3 7279n2 12536n 7491) 21772800c2n7 (n 3)(n 2)(n 1)(n+1)(n+2)(n+3) n2 +119 453600c3n7 +bT4T4 (F.4) : (F.5) Squaring the above vacuum block we nd s8(n; c) = The entanglement entropy is Sn( ; x) = (2x= T ) 2 (1+n)c (n2 +11) 12n2 (2x= T ) 12n 144n4 13n6 +1647n4 33927n2 +58213 5184(5c+22)n6 S1( ; x) = (2x= T ) 8 1 + Note again that up to the order (l= T )6 all the contributions to the entanglement entropy come from T and TiTj and TiTjTk. That is because bT2k 1)2. Therefore, up to this order the one-point function of h jT4j i does not appear. However, at the eighth order in l= T there is a term in bT4T4 and bT4T T that are proportional to the rst power of (n 1) and hence contribute to the entanglement entropy. Failure of perturbation theory for GGE In this appendix, we expand the GGE in small KdV chemical potential in a perturbative expansion. We show that demanding that the one-point functions of GGE to match those of the eigenstate is inconsistent in perturbation theory. All orders of chemical potential contribute to the one- rst correction in 1=c, and one needs a non-pertubative expression for one-point functions of GGE to compare with the eigenstate. We choose the following simplifying notation 1 Z 1 Z tr e tr e H A = hAi H iQiA = hAi ; i (G.3) (G.4) HJEP03(218)7 where repeated indices are summed over. Then, assuming a perturbative expansion for the GGE we have hAi ; i = hAi ihA~ Q~ii + i2 j hA~Q~iQ~ji + O( i j k) Taking A to be the KdV current J2k we have hT i ; i = hT i hJ2ki ; i = hJ2ki ihT~ Q~ii + i2 j hT~Q~iQ~ji + O( i j k) ihJ~2k Q~ii + i2 j hJ~2kQ~iQ~ji + O( i j k) In (G.3) it is understood that the index i = 2m 1 is summed over, and m runs rst term in the series above hJ2ki expansion is a valid perturbation theory if chemical potentials are suppressed at large c by 2m 1 charge hJ2kQ~2m 1i = O(ck+m 1). The rst order term gives us the condition (m) > ~ c (m). Since the disconnected piece of hJ2kQ~2m 1i is zero, at large central ~ m 1, and from the second order term we nd (m) > m. In order to match this with the energy eigenstate we should solve for i such that ck at large c. The above hT ik; i = hJ2ki ; i: If i are suppressed by powers of c, we can try to impose the above condition by setting 1 X m=2 2m 1 khT ik 1hT~Q~2m 1i hJ~2kQ~2m 1i = hJ2ki hT ik + O(ck 2) The coe cient of 2m 1 in the left hand side of (G.5) is O(ck+m 1), hence the each term in the sum on the left is scales at bet as ck 1; while on the right hand side we have terms that are order ck 1 . The only option is to take = m. According to the perturbation expansion (G.3) this means that the higher orders terms in contribute to the same order in c. In order to make sense of the perturbation theory we should be able to truncate the sum on the left to a nite number of terms. Say we keep the coe cients 2m 1 c (m) with for (m) = m for m C and (m) < m for m > C, where C is a nite number. Then, we have C unknowns ( 2m 1 for m C) that should satisfy an in nite number of equations at the rt order in 1=c in (G.5). We take this over-constrained system of equations as an indication that the question of nding a GGE with the same one-point functions as the energy eigenstate is non-perturbative in nature. Below, we develop the perturbation theory in small chemical potential further, even though it does not shed light on our study of ETH. In the remainder of this appendix, we compute some of the one-point function of J4 and T in an example of a GGE with only 3 turned on. The conserved currents are T (!) and (T T )(!) = T4(!) + 130 @!2 T (!) on the thermal cylinder of circumference . Under a conformal transformation z = f (!) the currents change according to (T T )(!) = T4(!) + T (!) = f 02T (f ) + Schw(f ) 12 3 (5c + 22) f 000 f 0 130 @!2 f 02T (f ) + 12 3 f 00 2 2 f 0 Schw(f ) f 02T (f ) + Schw(f ) Schw(f ) Mapping the thermal cylinder to the complex plane by z = e2 != we nd (see [24]) T (!) = (T T )(!) = D2 = 2 2 3 10 z4T4(z) + D2T (z) + 5(c 10) z2 : c(5c + 22) 2880 From this it is immediately clear that on the complex plane After some straightforward algebra we nd hT~(0)Q~3i = hT~(0)Q~3Q~3i = z2T (z) z4T4(z) + D2T (z) : z hT ( 1) z4T4(z) + D2T (z) i = 2 3 c(5c + 22) 720 zz0 hT ( 1) z4T4(z) + D2T (z) z04T4(z0) + D2T (z0) i 2 4 2 2 T~(z) = J~4(z) = 3 Z 1 dz 6 Z 1 dzdz0 6 c(5c + 22)(7c + 74) 8640 (G.6) (G.8) (G.9) hT (1)T (1)T4(0)i = hT4(1)T4(1)T4(0)i = c(5c + 22) 10 c(5c + 22)(5c + 64) After some algebra we nd that the expectation value of currents in the GGE in the and for the KdV current hJ~4(0)Q~3i = hJ~4(0)Q~3Q~3i = 2 2 3 c(5c + 22)(7c + 74) 60480 6 c(5c + 22) 5c + 22 2 10 (5c + 43) ! (G.10) + O( 3= 9) (G.12) hT (1)T (1)T (0)i = c; hT4(1)T (1)T4(0)i = 2c(5c + 22) small chemical potential limit is given by tr( ; T (0)) = tr( ; (T T )(0)) = From which we obtain tr( GGEH) = L tr( GGEQ3) = L 2 2 2 2 2 2 2 2 2 2 2 2 2 4 2 2 2 2 4 2 12 (2 )3 3 c(5c+22) 3 720 +O( 3= 9) 5c+22 2 (5c+43) ! 2880 6 c(5c+22) 10 3 (2 )3 3 c(5c + 22) c(5c + 22)(7c + 74) 4320 + O( 3= 9) c(5c + 22) 2880 6 c(5c + 22) 10 (2 )3 c(5c + 22)(7c + 74) 5c + 22 2 (5c + 43) ! 3 where we have suppressed the 3 = 9 corrections. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [1] N. Lashkari, A. Dymarsky and H. Liu, Eigenstate Thermalization Hypothesis in Conformal Field Theory, J. Stat. Mech. Theor. Exp. 2018 (2018) 033101 [arXiv:1610.00302] [INSPIRE]. [2] M. Srednicki, Chaos and quantum thermalization, Phys. Rev. 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Nima Lashkari, Anatoly Dymarsky, Hong Liu. Universality of quantum information in chaotic CFTs, Journal of High Energy Physics, 2018, 70, DOI: 10.1007/JHEP03(2018)070