Spread of entanglement in a Sachdev-Ye-Kitaev chain

Journal of High Energy Physics, Sep 2017

We study the spread of Rényi entropy between two halves of a Sachdev-Ye-Kitaev (SYK) chain of Majorana fermions, prepared in a thermofield double (TFD) state. The SYK chain model is a model of chaotic many-body systems, which describes a one-dimensional lattice of Majorana fermions, with spatially local random quartic interaction. We find that for integer Rényi index n > 1, the Rényi entanglement entropy saturates at a parametrically smaller value than expected. This implies that the TFD state of the SYK chain does not rapidly thermalize, despite being maximally chaotic: instead, it rapidly approaches a prethermal state. We compare our results to the signatures of thermalization observed in other quenches in the SYK model, and to intuition from nearly-AdS2 gravity.

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Spread of entanglement in a Sachdev-Ye-Kitaev chain

HJE Spread of entanglement in a Sachdev-Ye-Kitaev chain Yingfei Gu 0 Andrew Lucas 0 Xiao-Liang Qi 0 Via Pueblo Mall 0 Stanford 0 U.S.A. 0 0 Department of Physics, Stanford University We study the spread of Renyi entropy between two halves of a Sachdev-YeKitaev (SYK) chain of Majorana fermions, prepared in a thermo eld double (TFD) state. The SYK chain model is a model of chaotic many-body systems, which describes a onedimensional lattice of Majorana fermions, with spatially local random quartic interaction. We nd that for integer Renyi index n > 1, the Renyi entanglement entropy saturates at a parametrically smaller value than expected. This implies that the TFD state of the SYK chain does not rapidly thermalize, despite being maximally chaotic: instead, it rapidly approaches a prethermal state. We compare our results to the signatures of thermalization observed in other quenches in the SYK model, and to intuition from nearly-AdS2 gravity. 1 Introduction 2 Setup 3 4 5 6 7 2.1 2.2 2.3 2.4 2.5 4.1 4.2 5.1 5.2 5.3 5.4 The SYK model Thermo eld double state and global quench Twist operators and Renyi entropy The path integral for the SYK chain Large N and replica diagonal partition function Weak link limit Deviation from linear growth: Gaussian correction E ective action of the reparameterization eld Gaussian correction Long time saturation: geometric interpretation Two site e ective action and the mapping to a geometric problem Explicit form of the action Analytic continuation Numerical and analytic results Comparison to Holography Discussion A.1 In nite chain A.2 Two-sites link limit A Details of the Gaussian approximation A.3 Comparison of the two-site result with the geometric minimization at weak B Derivation of the geometric interpretation B.1 The Schwarzian action term B.2 The twist operator term C Some details of the geometric minimization C.1 Solving the constraint C.2 Minimization of I(D) at t ! 1 limit D Holographic von Neumann entanglement velocity { 1 { Introduction Intuition from statistical mechanics suggests that generic interacting sytsems should thermalize. For an isolated quantum system, this seems counter-intuitive since a pure state always stays pure under unitary time evolution. Nevertheless, there is a deep sense in which a highly excited pure state can nevertheless \look thermal". If we study the reduced density matrix of a small subregion of the total quantum system, we expect that in a generic thermalizing quantum system, the reduced density matrix of such a region is very close to a thermal density matrix [1{3]. Such a thermal reduced density matrix has entanglement entropy proportional to the volume of the subregion, in contrast to the vanishing HJEP09(217) entanglement entropy of a pure state. Therefore thermalization of an isolated system is fundamentally related to the dynamics of the entanglement entropy between subsystems. A direct probe of thermalization is thus to start with a highly excited state with low entanglement, and to evolve it forward in time. For example, suppose that we start in the ground state j i of a quantum system for t < 0, and at t = 0 abruptly change the Hamiltonian so that j i is now highly excited. By studying the entanglement growth in a subregion of the quantum system after such a quench, we learn how the system thermalizes; as we have seen, the spread of entanglement is necessary for thermalization. In many strongly interacting quantum systems, it is observed that (for large enough regions) the rate of change of the (von Neumann) entanglement entropy of a region of surface area A is given by vE 6 vB: { 2 { dt dSE = sth vEA; where sth is the entropy density of the resulting thermal state, and vE is an \entanglement velocity" [4{6]. The entanglement velocity gives a simple measure of how rapidly the density matrix appears thermal, and | at least by dimensional analysis | de nes a velocity scale for the thermalization of an interacting quantum system. Another perspective on thermalization arises from quantum chaos. In a chaotic system, quantum information present in a small subregion at time t = 0 becomes spread out quickly. Consider an operator Ox with support near a point x at t = 0. After time evolution, the operator Ox(t) = eiHt Oxe iHt can become a `large' operator with support in a ball of radius / t [7]. The process by which these operators become delocalized, and so one must look at a large region of the system to recover a small amount of information, is coined `scrambling' [8]. The spatial dynamics of scrambling is governed by a di erent velocity scale, called the butter y velocity vB [9]. In order for a large subregion of a quantum system to thermalize, certainly operators localized inside of the subregion at t = 0 must begin to extend outside of the subregion by the thermalization time. Thus, the dynamics of entanglement cannot be entirely independent from the dynamics of scrambling: both are intimately connected with thermalization. There are plausible arguments [10, 11] that scrambling should (begin) to occur rst : (1.1) (1.2) because the growth of entanglement is impossible without the spread of information.1 However, many open questions remain. Is (1.1) a universal property of chaotic, thermalizing quantum systems? If so, is vE a physical speed in the quantum system, and if it is, does something locally well-de ned propagate at this speed? What velocity scale | if either | limits the onset of classical hydrodynamics, and could thus bound sound speeds and di usion constants [13], and why? While many of these questions become quite delicate for many-body localized systems [14] (or systems close to a many-body localization transition) [15{18], they are also not well understood for highly chaotic systems. In this paper, we will focus on the generalized Sachdev-Ye-Kitaev (SYK) models [19{ 22] as a solvable model of a chaotic system. The solvability of this model, and various generalizations of it [22{38], allows for detailed studies of quantum chaos and thermalization. Here we will initiate a study of the spatial dynamics of entanglement in the SYK chain model proposed in ref. [22].We will describe speci c details of this model later. For now, let us simply emphasize that it is a highly disordered quantum system with a single energy scale J , and a large number N of degrees of freedom per lattice site. At temperatures T J (commonly denoted J 1, with inverse temperature = 1=T ), the SYK model exhibits near-conformal invariance, becomes maximally chaotic in its early time behavior, as measured by the Lyapunov exponent, and has exponentially many low-lying excited states, leading to a non-zero entropy at zero temperature if we rst take the large N limit. Due to being maximally chaotic, we would anticipate that the SYK chain is an e ective thermalizer. More speci cally, we study the entropy growth in the generalized SYK model after a global quench from a special initial state, the thermo eld double (TFD) state [39]. To construct the TFD, we tensor product two copies of the original Hilbert space of the SYK HR. We have denoted the copies as left (L) and right (R). The TFD chain: H = HL state at time t = 0 is jTFDi / e (HL+HR)=4jIi: (1.3) The state jIi is a direct product of local EPR pairs between the two systems L and R (See gure 1). For a spatial entanglement cut, jIi has no entanglement entropy. HL and HR are suitable notions of the SYK-chain Hamiltonian acting on only the L or R degrees of freedom (see section 2.2 for precise de nitions). Upon tracing out either the L or the R degrees of freedom, the resulting density matrix is thermal. However, we may de ne a Hamiltonian for the combined LR system such that jTFDi is not an eigenstate. If we look at a suitable subregion A with support in both the L and R chains, such as the left half of the chain shown in gure 1, we can observe the spatial spread of entanglement in this doubled system. More speci cally, by using the replica trick, we compute the n-th Renyi entanglement entropy SA;n (with integer n > 1) of the reduced density matrix of jTFD(t)i in one half of the TFD chain. When the coupling between neighboring sites is the smallest energy scale, 1One can prove [12] that vE 6 vLR, the Lieb-Robinson velocity [7], though this is in general a far weaker bound [10]. { 3 { HJEP09(217) M sites e 4 HL EPR pairs e 4 HR entanglement cut AL AR e itHL e 4 HL EPR pairs e 4 HR e itHR HJEP09(217) (a) Thermo eld double state (b) Entanglement cut on jTFD(t)i e 4 HL and e 4 HR to a state jIi of the two-chain system. jIi is a direct product of local EPR pairs between the two sites in the two chains at the same spatial location. (b) The real time evolution of the TFD state by U (t) = exp[ it(HL + HR)] and our choice of entanglement cut. We study the Renyi entropies of the region A = AL [ AR, with support on both chains. we nd that the entropy increases linearly in time as in (1.1), with the growth rate dSA;n dt / T: The linear growth slows down at long time and eventually has to saturate if the length of the chain is nite. In the large N and large J limits, we study the late time behavior by two di erent approaches. The rst approach is a perturbative expansion in the coupling strength between neighboring sites, and allows us to compute the onset of deviation from linear growth at an intermediate time scale. Such an approach does not apply directly to the long time limit. In the second approach, we make a simple ansatz for correlation functions that allows us to compute the entropy at all times by a much simpler (but still nonlinear) geometric problem. Solving this problem in the long real time limit predicts a saturation of the Renyi entropy. Since we have done a restricted minimization of action, what we obtain is an upper limit of the entropy: SA;n(t = 1) Here M is the length of the chain, so that SA;n(t=1) is the entropy per site. cv is the speci c heat of the doubled SYK chain, which is a constant at low temperature. Surprisingly, for n > 1, at low temperature the Renyi entropy density upper limit is parametrically smaller than that of the thermal ensemble. This result implies that there are degrees of freedom whose thermalization time tth= diverges when N ! 1 and/or J ! 1. (1.4) (1.5) { 4 { We propose that this phenomenon of prethermalization is related to the presence of a large density of almost localized states at very low energy. These states are responsible for the zero temperature entropy. They evolve slowly and do not contribute to entanglement if we rst take the large N limit. As a consequence, the entropy growth is upper bounded by cvT = sth(T ) sth(0): the change in entropy due to nite temperature T . This does not include the non-vanishing zero temperature entropy density of the SYK chain. We expect the system eventually thermalizes but that the thermalization time diverges in the large N and/or large J limit. The fact that the SYK chain only prethermalizes rapidly, but thermalizes slowly, implies that eq. (1.1) may not be a sensible de nition of an entanglement velocity. We will discuss this possibility in much more detail at the end of the paper. If one uses the de nition (1.1) for vE, then using (1.4) we nd that vE / T for the SYK chain. It is interesting to discuss the relation of our results with holographic models of strongly interacting theories. Although the holographic dual of the SYK model is not known, the SYK model shares many similar properties with holographic models containing extremal horizons [40{42]. We show that these holographic models exhibit similar `early' time von Neumann entanglement growth, with dSE=dt / T , but the von Neumann entanglement SE saturates at the thermal value, including the extremal zero temperature contribution. Although we are unable to explicitly compute the analogous holographic Renyi entropy growth, we propose that the Renyi entropy in an analogous holographic setting may behave similarly to the SYK chain. This arises due to subtleties with gravitational dynamics in AdS2. Finally, we note that two models studying thermalization in (single site) SYK models in a somewhat di erent context have recently appeared [43, 44]; both studies show evidence for rapid thermalization. Evidence for eigenstate thermalization in the SYK model has also recently appeared [45]. Our results are not inconsistent with theirs, but we defer a detailed comparison to the end of the paper. The rest of the paper is organized as follows: in section 2 we present the explicit setup of the global quench problem we are studying. We compute SA;n(t) in the limit where the coupling between di erent sites is much weaker than the on-site coupling in section 3, and in section 4 we consider the leading perturbative correction to this result. In section 5 we present a geometric interpretation of the quench problem we are studying and compute the long time saturation of the entanglement entropy. This is a regime where both previous calculations fail. In section 6 we compare our result to intuition from holography. In section 7 we discuss the broader implications of our result and compare to other recent studies on quantum quenches in the SYK model [43, 44]. 2 2.1 Setup The SYK model The Sachdev-Ye-Kitaev (SYK) model [19, 20] describes N Majorana fermions with quartic random all-to-all interactions. The Hamiltonian of this model is f j ; kg = jk; (2.1) { 5 { where fJjklmg are independent, mean-zero random couplings: 1 [20, 21, 46]. It provides a rare example of chaotic yet tractable many-body systems. Recently, many generalizations of the SYK model have been proposed. In particular, [22] studied a higher dimensional lattice generalization of the SYK model with spatial locality. For a one-dimensional chain, the Hamiltonian of the generalized SYK model is where the couplings fJjklm;xg and fJj0klm;xg are all independent Gaussian random variables with mean zero, and variances It is convenient to de ne an e ective coupling constant Jj2klm = which determines the local properties of the model. Similar to the original SYK model, this generalized model is solvable at large N and maximally chaotic at strong coupling. At leading order in N , this model has a saddle point which is equivalent to the one-site SYK model. At next-to-leading order in N , there is non-trivial spatial dynamics with dynamical critical exponent z = 1, also known as local criticality. Local criticality implies that space does not scale under renormalization group ow, and is responsible for some of the particular features of the SYK chain model. The spatial locality of the model enables us to study thermal transport and the spatial propagation of scrambling and chaos. The out-of-time-order correlation function (OTOC) takes the form 1 N 2 X i;j h ix(t) jy(0) ix(t) jy(0)i / const: + 1 N e L(t x=vB) p with Lyapunov exponent L = 2 T and butter y velocity vB = 2 T D, with D the thermal di usion constant [22]. We note that this relation between vB and thermal di usion constant D holds in holographic locally critical theories as well [47, 48]. Charge transport can also be studied in a modi ed model with charge conservation [24], though we will focus in this paper on the simpler model above. Before describing how to exactly solve this model in the limit N J 1 in section 2.4, we would like to rst de ne the TFD state described in the introduction, and discuss how to compute the Renyi entropy in this state. { 6 { We consider two copies of a single SYK chain, and consider a special initial state: the thermo eld double (TFD) state [39]. As we will see, analytic computations in such a doubled state are tractable; the qualitative features of entropy growth in the quenched time evolution of a short-range entangled initial state should not depend on details of the initial state. We rst give the general de nition of the thermo eld double state. For a system with lattice sites labeled by x, we rst choose a basis ja; xi ; a = 1; 2; : : : ; D on each site. Here D is the Hilbert space dimension of each site (D = 2N=2 for the SYK chain model). Then we consider the following state of the doubled system, which is a direct product of maximally entangled pairs on each site: explicitly, HT is de ned as Here L and R (left and right) label the two copies of the system. In state jIi, each chain is maximally entangled with the other chain, but when we consider the two sites at x together, they are unentangled with the rest of the chain. Indeed, interpreting the chain labels L and R as an \internal" label, jIi is a direct product state with no spatial entanglement. For a given Hamiltonian H of the original single chain problem, we can de ne its transpose HT by taking the matrix transpose in the basis jfaxgi x jax; xi.2 HT X faxg;fbxg hfaxgj H jfbxgi jfbxgi hfaxgj Now de ne a Hamiltonian in the doubled system HD = HL + HR; with HL = H I; HR = I HT such that H acts on the left system and HT acts on the right system. One can explicitly check that the state jIi satis es The TFD state jIi, introduced in (1.3), is then de ned as (HL HR) jIi = 0 jTFDi = Z 1=2e 4 (HL+HR) I j i with Z = tr e H the thermal partition function of the single-chain system. A key property of TFD is that the reduced density matrix of the L chain alone, or the R chain alone, is thermal, with inverse temperature . This can be directly shown by applying eq. (2.10) to obtain jTFDi = Z 1=2e 2 HL jIi and use the fact that jIi maximally 2It should be noted that transpose is a basis-dependent operation, so that it is essential to rst de ne the basis. { 7 { (2.7) More (2.8) (2.9) (2.10) (2.11) 1 2 cj;x;L = ( 2j 1;x;L + i 2j;x;L) ; cj;x;R = ( 2j;x;R i 2j 1;x;R) (cj;x;L cj;x;R) jIi = 0; cjy;x;L + cjy;x;R jIi = 0 1 2 with j = 1; 2; : : : ; N=2. In the eigenbasis of nj;x;L(R) = cjy;x;L(R)cj;x;L(R), the state is a product of j0iLj1iR + j1iLj0iR on each site. This leads to the equations for the Majorana operators ( j;x;L + i j;x;R) jIi = 0; (j = 1; 2; : : : ; N ): In this choice, one can check that the SYK chain Hamiltonian (2.3) satis es HT = H. Therefore we can take two identical chains and the TFD state is de ned as3 entangles the two chains. One can view the TFD state as a puri cation of the thermal density matrix, in which the chain R plays the role of thermal bath of chain L. Compared to a generic puri cation, the TFD state has the special property that the entanglement between the two chains is spatially local. The state jIi (which is the ! 0 limit of jTFDi) has zero entanglement entropy between di erent spatial regions. jTFDi at nite temperature is obtained by a nite time imaginary time evolution of jIi. This imaginary time evolution leads to spatial entanglement. However, any resulting entanglement entropy will satisfy an area law [49], as long as HL;R are local. In the one-dimensional chain case, this means the entanglement entropy of a connected region A stays nite even if the size of A and its compliment go to in nity. The de nition of jTFDi is not unique, since it depends on a basis choice. However, di erent de nitions lead to jIi that are related by a product of local unitaries, which does not change entanglement properties of the TFD state as long as the de nition of HT is conjugated by these unitaries correspondingly. For concreteness, we give an explicit de nition of jTFDi in the SYK chain case. Denote the Majorana fermion operators by j;x;L and j;x;R, with j = 1; 2; : : : ; N . (We remind the reader that N must be even, to have a well-de ned Hilbert space at each site.) One convenient choice of the state jIi can be de ned by the following equations: A(t) := trAC jTFD(t)ihTFD(t)j: jTFDi = Z 1=2 exp 4 (H I + I H) jIi: From the perspective of time evolution, the thermo eld double state is an eigen-state of operator H I I HT but not the eigen-state of our Hamiltonian HD = H I + I HT . Therefore, we can treat HD as the initial state after a global quench, and apply the corresponding time evolution operator to obtain a time dependent state: jTFD(t)i = U (t)jTFDi; U (t) = exp i H I + I HT t Now we can look at the sub-region A = AL [ AR which is supported on two sides as shown in gure 2 (a) and consider its reduced density matrix: (2.12) (2.13) (2.14) (2.15) (2.16) AL (a) Subregion A (blue) on U (t)jTFDi (b) Reduced density matrix A (a) We picture the TFD state of the SYK chain model as a half tube. The two top edges correspond to the states in left and right Hilbert space. The subregion A = AL [AR is de ned on both sides, and colored blue. The yellow shaded region corresponds to the real time evolution U (t), and the circular portion of the tube represents the initial imaginary time evolution of (1.3). (b) The density matrix A is then found by gluing two jTFD(t)i states on Ac and leaving A (blue) `open'. This perspective is useful in computing the partition function ZA;n, where the blue lines play the role of branch cuts. Each replicated fermion shifts its replica index to + 1 when it crosses the right branch cut line (the side closer to the reader) from below and shift to 1 when it crosses the left branch cut line from the above. We can further deform the two horizontal blue branch cuts to a single vertical dashed branch cut (shown in red). By construction, each chain in the TFD state has a thermal density matrix that is invariant under time evolution. Consequently a region that is a subset of only L chain or only R chain also has a thermal time-independent density matrix. In the region we choose, AL = trAR A(t) and AR are thermal, but A is time-dependent. If region A thermalizes after a long time, A should approach the thermal density matrix AL AR . Therefore, the increase in entropy of region A during thermalization corresponds to the decrease of mutual information I(AL : AR) = S( AL )+S( AR ) S( A) between the two regions AL and AR. If the system thermalizes, the mutual information should vanish in long time (or at least becomes subleading in volume of A). Physically, the decrease of mutual information is a consequence of the scrambling of quantum information during chaotic time evolution. Correlation between operators in AL and AR evolves to more and more non-local operators that cannot be revealed in region AL and AR [16, 50]. 2.3 Twist operators and Renyi entropy It is di cult to directly calculate von Neumann entropy of the region A. Instead we use the replica trick [51] to compute the Renyi entropy for the subsystem A by a path integral: log tr nA ; tr nA = ZZAn;n ; (2.17) where the factor of Zn in the denominator arises from the de nition of the TFD state. The numerator ZA;n is the partition function evaluated on a \twisted" manifold corresponding to an n-sheeted cover of spacetime. The easiest way to understood the partition function on this twisted manifold is to consider a partition function de ned on n replicas of the original 3This discussion, and the relation HT = H, generalize to an SYK model with q-body interactions [21]. { 9 { theory. For the generalized SYK model with Majorana fermions j;x, with j = 1; 2; : : : ; N the avor index and x the lattice site coordinate, the replicated theory describes fermions j;x with = 1; : : : ; n. We then write down a path integral for all j;x, containing evolution in Euclidean time, which prepares the TFD state, followed by evolution for real time t. From the perspective of the twisted manifold, the replica index of a fermion describes which sheet of Euclidean space-time it lives on. These sheets are connected through the branch cut lines shown in blue in gure 2, such that fermions passing across a branch cut have their replica index cyclically permuted. More explicitly, the boundary condition of fermions is j;x( +) = < 8 >> j;+x1( 1 j;x ( > > : j;x( ); x 2 AL ); x 2 AR ); x 2 A (2.18) is the time location of the branch cut. ZA;n is the partition function for replicated theory with the above boundary condition. The position of the branch cut line is not important; it can be moved around by relabelling fermions in di erent replicas. The only invariant information about the twist is the end points of the branch cut.4 For our convenience, we can move the branch cut points to a \time-like" line connecting the two branch cut points (red line in gure 2 (b)). With this gauge choice, the boundary condition in the time direction remains untwisted, and the only e ect of the twist operators is to modify the spatial couplings between fermions on di erent sites. Denoting the sites separated by the boundary of A as x and x + 1, the twisted coupling is between j;x ( ) and j;+x1+1( ) when time is in the interval of the branchcut line. The coupling is diagonal in everywhere else. It is also helpful to write another equivalent expression of the Renyi entropy. If we de ne a twist operator XAn which is applied to the branch cut lines AL or AR and cyclically permutes the replica index, the Renyi entropy can also be written in a time-ordered thermal two-point function of twist operators: e (n 1)SA;n = tr XAyn ( it) n 1=2 XAn (it) n 1=2 = XAyn 2 it XAn(it) (2.19) Here h: : :i = tr n : : : is the thermal average in n copies of the single chain system. The real time evolution and imaginary time evolution can be drawn as a contour in the complex plane of time, as shown in gure 3(a). 2.4 The path integral for the SYK chain The discussion of the previous subsection was completely general. It applies to the TFD state of any system with a spatially local Hamiltonian. We will now write down the coupling more explicitly for the SYK chain model. 4We can view the local permutation of di erent replica as a gauge transformation, and the branchcut points are gauge uxes. it in complex plane z = exp(i 2 tC), where tC = + it is the complex time variable. The red part represents the hTFD(t)j in gure 2, and the black represents the jTFD(t)i. For later convenience, we name them as C1 and C2. The replicated partition function ZA;n with the boundary condition described in the previous subsection can be written as ZA;n[J ] = SJ [ x;j ] = Z X ;x Y ;j;x j j;x Jjklm;x j;x k;x l;x m;x + J 0jklm;xgx ( ) j;x k;x l;x+1 m;x+1 where C stands for a special time contour for the thermo eld double states as shown in gure 3. At the boundary of A, between sites x and x + 1, the contour is split into two parts C1 and C2 by the twist operators. The branchcut line runs along C1. The e ect of the twist is to modify the spatial coupling J 0jklm;x term by the matrix gx ( ), given by gx ( ) = ( ; +1; if 2 C1 and x = x ; elsewhere In a generic system, to compute the quenched average of the Renyi entropy SA;n over disordered couplings Jjklm;x and Jj0klm;x one should compute ZAk;n for a general integer k, and then analytically continuate to the k ! 0 limit. In the SYK model, it is known that at leading and next-to-leading order in N , the partition function is replica diagonal, such that ZAk;n ' ZA;nk [21, 22, 52]. Therefore we will directly work with ZA;n. The average over the Gaussian-distributed random couplings J and J 0 leads to the following partition ZA;n = X " Z Z X x + X j x J 2 4 Next, we rewrite this fermionic partition function as a theory of bosonic bilocal elds. De ne the Green's function G (note the replica indices): Gx ( 1; 2) := j;x( 1) j;x( 2): We impose this de nition of Gx ( 1; 2) by a Lagrangian multiplier x ( 1; 2) in the path integral. Integrating out fermions after inserting G and leads to the following e ective theory: Z ZA;n = DGD exp ( N S[G; ]) 4 + 1 2 x; X Z Z C " Up to this point, all the manipulations are exact in the large N limit. In what follows, we will treat the e ect of the twist operators by making certain approximations in the low temperature limit as well. 2.5 Large N and replica diagonal partition function Our goal is to compute the Renyi entropy: We have seen that ZA;n may be evaluated using a path integral for a generalized SYK model with n avors of replicas, with a twisted interaction on a special time contour. In this section, we aim to evaluate ZA;n with some further assumptions. In the large N limit, the partition function ZA;n can be computed using a saddle point approximation. We nd a saddle point equation for and for G. The rst equation is standard: Gx ( 1; 2) = ( 0( 1; 2) x ( 1; 2)) 1: 1 Z Z 8N 3 C !2 !2!# !4 (2.22) (2.23) (2.25) (2.26) We must explicitly consider functions of two time variables because time translation symmetry is broken due to the special time contour C, and the twisted interaction; the 1 should be read as matrix inverse in both time domain ( 1; 2) and replica indices ( ; ). The second equation depends on the location x. When x 6= 0; 1, the self energy is the same as the normal generalized SYK model (with added replica indices): )3 means an entry-wise cube of the matrix G , and not the element of G3. For x = x or x + 1, the self energy term experiences the twisted interaction: where we have omitted the time variables ( 1; 2) in G and for simplicity. Without the twist, the saddle point solution to the Schwinger-Dyson equation is diagonal in replica indices, with the form Gx ( 1; 2) = Gs( 1 2) [21]. Since the twist couples di erent replicas, it is possible that the saddle point solution becomes o -diagonal. To see whether this possibility is realized, let us start with a theory with J1 = 0. This reduces to a theory of decoupled SYK sites and the saddle point solution is diagonal. When a small J1 is gradually turned on, it is natural to consider a perturbative solution to the Schwinger-Dyson equations (2.27){(2.28). However, according to eq. (2.28) the self-energy x is always proportional to Gx . Consequently, if we start from the J1 = 0 diagonal solution and solve the equations iteratively, we nd that the solution stays diagonal to all orders of the coupling J12. Therefore we conclude that the solution either stays diagonal or that the solution is non-diagonal, but the o -diagonal contributions are non-perturbatively small as J1 ! 0. In the following we will assume that G and remain diagonal at nite J1. Recall that the Renyi entropy in large N limit will be determined by the saddle point with maximal contribution to the partition function. Hence, if the true minimum of the action occurs for an o -diagonal solution, then the value of SA;n that arises from a diagonal ansatz serves as an upper bound on the true value of SA;n.5 With the diagonal assumption Gx ( 1; 2) = Gx( 1; 2); x ( 1; 2) = ( 1; 2), the e ective action is simpli ed to 1 n X x x) + 5It should be noted that a subtlety may arise due to a non-trivial integration contour for the G and elds in the path integral (2.24), which is required for the path integral to be convergent [21]. Only saddle points on the integration contour contribute to the partition function. We will assume that real-valued reparameterizations (a certain ansatz for G, which we will de ne in (3.3)) lie on this integration contour. We thank Douglas Stanford for helpful discussion on this issue. J 2 1 8 Z C1 Z C2 n S0[G; ] + 4 1 Gx( 1; 2)2Gx+1( 1; 2) Z C2 Z C1 d 2 + (2.29) the original action of SYK chain, and n1 With the replica diagonal ansatz, the e ective action is proportional to replica number n, so that we divide the overall n to the left side. n1 S0 denotes the rst two lines, which is S denotes the third line which is the extra action o -diagonal term Gx; ++11 or Gx++11; . This is the origin of the extra term. cost caused by the twisted coupling. When only one of 1 and 2 is on the twisted contour C1, the J12 term in the action vanishes since it couples a replica diagonal term Gx to an 3 Weak link limit Even with the replica diagonal ansatz, the Schwinger-Dyson equation is still hard to solve, especially because of the lack of time translation invariance. However, we can start with a simple limit where N J 1; and 1 J J 2 1 In this limit, the twisted coupling term / J12 can be treated perturbatively. To do a perturbative calculation of SA;n, we begin by reviewing the untwisted SYK model at large N and strong coupling N J 1. The saddle point solution approximately follows a conformal form: Gc( 1; 2) = 1 (4 J 2)1=4 sin ( 1 2) 1=2 General uctuations around this saddle costs order N action. However, there is a special class of the uctuations that cost action NJ in the long wave-length limit [22]. These uctuations correspond to a time reparametrization fx 2 Di (S1) of the conformal solution Gc( 1; 2), which has the form f Gx( 1; 2) := f x0( 1)f x0( 2) Gc(fx( 1); fx( 2)); = 1 4 : In the limit J12=J 2 1= J , the twisted interaction is so weak that even these reparameterization modes will not be sourced. The rst order shift to the e ective action is thus obtained by evaluating the twisted term at the conformal saddle Gc. The e ect of the reparameterization modes will be considered later in section 4 and 5. It is convenient to rst work in imaginary (Euclidean) time and then analytically continue to real time by taking integral with twist operators inserted at time ! it. The imaginary time problem involves a path and 2 , as is shown in gure 4. (3.1) (3.2) (3.3) 2 = 2 1 = The calculation needs to be regularized by introducing a small separation between C1 and C2 by 1;2, both of which are of order J 1 . At rst order in J12, the Renyi entropy is simply obtained by evaluating the e ective action on the conformal solution Gc. We nd HJEP09(217) log Zn;A C1 C2 d 2Gc( 1; 2)4 = n sin sin 21 where is a small UV regulator.6 The corresponding Renyi entropy is Z n := 1 : 8 J 2 sin sin 21 (3.4) (3.5) (3.6) (3.7) . At (3.8) (3.9) ! 0 For convenience, we will de ne Analytically continuing to real time by taking 21 = 2 2it, we obtain: which includes a time-dependent piece and a constant piece coming from the cut-o large real time t , the entropy grows linearly: SA;n(t) N ' const: + n n 2 1 t = const : + n n Denote the Renyi entropy of each site in thermal equilibrium as stnh, we can de ne a Renyi entanglement velocity as in (1.1): with of order J 1. 6Physically, this cut-o is of order J 1: the arti cial divergence arises from approximating the actual saddle point two-point function by the conformal solution Gc( ) in eq. (3.2). The conformal approximation applies to the IR region J 1 [21], but in the UV limit . J 1 the conformal saddle diverges at while the true saddle Gtrue( ! 0) ! 21 . The e ect of this deviation can be described by a cuto term log dt dSA;n = 2stAh;nvE;n The factor of 2 comes from the fact that the TFD state is de ned on two chains. At low temperature, stnh approaches the nite zero temperature entropy, so that we conclude vE;n / T at low temperature. However, as we have discussed in the introduction (and in more detail in section 5), the entanglement entropy in this system actually does not saturate to the thermal value at long time. Thus, as we have noted previously, a conventional de nition of vE;n may not apply. Usually, taking n ! 1 in SA;n leads to the von Neumann entropy SA. However, this limit is singular in eq. (3.8). This is a consequence of the replica diagonal ansatz, because the resulting e ective action S / n. Physically, the divergence suggests that the n 1 1 region is described by a replica non-diagonal saddle point. But we emphasize that SA;n(t), as given in (3.5), is an upper bound for the Renyi entropy of the optimal non-diagonal solution. We will return to this point in section 5.4. 4 Deviation from linear growth: Gaussian correction For any chain with a nite length, entropy is upper bounded. The linear growth of entropy cannot last forever. To see the saturation of SA;n(t) as t becomes large, we need to go beyond the rst order approximation of the previous section. In this section we analyze the second order correction to the rst order linear growth. More explicitly, we will consider the change of the saddle point solution due to the additional twisted interaction term. Recall that this amounts to nding the minimum of the e ective action expressed in eq. (2.29): log ZA;n = fG; g min S[G; ] (4.1) As we noted before in eq. (3.3), so long as 1 1, the soft modes of the SYK model f J are reparameterizations of the untwisted saddle point solution. When the twisted coupling is also small: 1, we can ignore the induced change of G outside the manifold of reparameterization. In this approximation, the saddle point solution is determined by minimizing the e ective action log ZA;n[Gx( 1; 2)] over reparameterization f ( ). Since the minimal action in the restricted space of Gx( 1; 2) is always larger or equal to the f actual minimal action in the unrestricted space of two-point functions, the entropy we obtain in this approximation always bounds the actual entropy from above. In other words, our results are still meaningful as an upper bound, even when the e ect of nonreparameterization modes is not negligible. 4.1 E ective action of the reparameterization eld The form of the e ective action S[fx( )] log ZA;n hGfxi can be explicitly written down. For simplicity, we will consider a chain with an even number M 2 2Z of sites x = 1; M2 + 2; : : : ; M2 1; M2 .We choose open boundary conditions, and an entanglement cut in the middle, between sites x = 0 and x + 1 = 1. In this case, the system has re ection M2 + symmetry after the random average, so that the saddle point solution shall satisfy fx( ) = f x+1( ). In particular, f0( ) = f1( ). With this simpli cation one can write S = S0 + The S0 term controls the dynamics of the reparametrization eld in SYK chain model [22] without the twisted interaction;. S 0:01 is the numerical coe cient of the Schwarzian term [21], and this coe cient also determines the speci c heat: cv = N 8 2 S .7 Without the J twist term, the SYK chain model has a saddle point solution f ( ) = , which corresponds to the conformal solution Gc. Using the explicit form of G saddle Gc, we can further write f0 in terms of the conformal 1 n S = J12 Z 4 f(C1) df ( 1) Z f(C2) df ( 2)Gc(f ( 1); f ( 2))4 (4.2) (4.3) (4.4) (4.5) (4.6) This integral diverges due to the UV divergence of Gc at 1 is arti cial, since the actual saddle point two-point function should saturate to the UV 2 ! 0. This divergence value 12 for 1 2 . J 1 . To take into account of this UV regularization, we introduce UV cut-o for C1 and C2. On the imaginary time circle, this corresponds to de ning C1 and C2 as interval [ 1; 2] and [ 2 + 2 ; + 1 1], respectively, as shown in gure 4(b). The resulting integral can be evaluated explicitly, using the functional form of Gc given in eq. (3.2) 8 and the result is 1 n 2 S = log [fx ]; where [fx ] is the reparametrized cross ratio: As expected, the result is explicitly invariant under the global conformal group SL(2; R). Therefore we have computed the twisted interactions, restricted to the reparameterization modes fx( ). In general, it is still di cult to rigorously study the e ective action of reparameterization elds. In the remainder of this section we will study the e ective action and Renyi entropy using a Gaussian approximation. In the next section we will study the full nonlinear e ective action, but mostly for the simpli ed case of two-site problem. 7This speci c heat is for the TFD state, which is twice of the speci c heat for a single chain. [22] 8The evaluation essentially corresponds to the explicit integral I = This is simply the cross ratio (4.6) for four points on the imaginary time circle, for f0( ) = . 4.2 In the spirit of our previous perturbative calculation as ! 0, we now compute the second order correction to SA;n in . De ning fx( ) = + x( ), we must expand the action up to the quadratic order in ( ), compute the rst order correction ( ) caused by the twisted interaction term in the action, and then evaluate the action to order 2, accounting for the non-zero ( ). The e ective action for small ( ) is given by 1 n The rst line is the quadratic expansion of S0, which has been obtained in [22]. The second line corresponds to n1 S. 0 = [f ( ) = ] is the cross ratio for the trivial reparameterization, which corresponds to the rst order contribution to the entropy we obtained earlier in eq. (3.5). The twist term contributes a linear in term, which directly sources the rst order correction ( ) . We have ignored the quadratic term in S; it will modify SA;n at higher order in . Minimizing S[ ] is straightforward: where h i denotes expectation values with respect to the quadratic action for ( ) given above. In appendix A.1, we explicitly perform this expectation value, and we nd that min S[ ] = log 0 2 1 * 2 2+ ; min S[ ] ' 2 t 1 p 2 The negative correction / t3=2 indicates that the entropy growth starts to become slower than linear around the characteristic time t discussed here, this time scale t J S . In the weak link limit is much longer than the thermal time . We can further estimate the amount of entropy growth by the time t : SA;n = SA;n(t ) SA;n(0) N n n 1 2 t n n 1 N S J which is of order cvT , the speci c heat's contribution to the thermal entropy. As we will see later in section 5, the entropy actually saturates to a nal value that is comparable with our estimation here. 5 Long time saturation: geometric interpretation In this section, we will evaluate the partition function ZA;n for a simple case, when the chain has only two coupled SYK sites (M = 2). The entanglement cut is between the two sites. For the two-site problem, the full non-linear saddle point problem can be solved (4.8) (4.9) 1 J (4.10) using a geometric interpretation of the action. Although this special case does not directly determine the entropy growth in a longer chain, the results will help us to understand qualitative features of this system, especially the long time saturation of the (Renyi) entanglement entropy.9 As we will discuss in section 5.4, the two-site calculation can be generalized to longer chains, and by doing so, we provide an upper bound of the entropy growth and saturation in that case. Two site e ective action and the mapping to a geometric problem As we discussed in the previous section, the two-site problem with an entanglement cut between the two sites has a re ection symmetry, so that the saddle point solution should be given by identical reparameterization elds on the two sites: f1( ) = f2( ) = f ( ). The e ective action is thus a functional of a single f ( ), and the Renyi entropy has the following form: n n 1 N min f J d Sch tan f ( ); + 2 log f + 2 S The rst term is the Schwarzian action for the reparametrizations f on two sites (hence the factor of 2), and the second term arises from the twisted interaction between the two sites. The last term, coming from the log Z term, is the constant piece of the Schwarzian, which cancels the value of the rst term when f ( ) = . Therefore, our goal here is to nd the minimal value of SA;n by varying all reparametrizations f 2 Di (S1): I(t) = min f J d Sch tan f ( ); + 2 log f The time dependence comes from the second term log f , where f is the cross ratio of the reparametrization of four time coordinates: ( 1; 2; 2 + 2 ; + 1 1) (cf. (4.6)) and the end of the calculation. In the limit 1;2= ( J ) 1 1, f is simpli ed to 1;2 are cut-o s of order J 1 . 1 and 2 will be analytic continued to it and 2 it towards (5.1) (5.2) (5.3) (5.4) (5.5) HJEP09(217) growth will stop at late time. sin (f ( 2) f ( 1))2 f ' ( )2 1 2f 0( 1)f 0( 2) L = J 1 ds2 = d 2 + sinh2 d 2 It is manifest that both terms in the two-site action (5.2) are SL(2; R) invariant. The Schwarzian action term has a geometric interpretation [41, 53], which corresponds to the area enclosed by a curve in hyperbolic space with xed length More explicitly, one can consider a hyperbolic disk with coordinates ( ; ) and metric: We can then use the Gauss-Bonnet theorem to compute the area of triangles. As all extrinsic curvature of the triangular regions is located at the corners, we simply sum the inner angles. Two of the inner angles are known. The nontrivial inner angle is \CXO := in gure 5(b) which can be computed by considering the angle between radial line OX=CX and geodesic X1X2: cot 21 cosh 1 cot 22 cosh 2 1 = arctan ; 2 = arctan ; = 1 2 (5.12) Therefore, the total area of the two triangles is AM = part with hyperbolic geometry, we obtain Using eqs. (5.9), (5.10), and (5.13), we can express the action I(D) as a function of D. I(t) will be the minimal value of I(D) when varying D. Analytic continuation The imaginary time minimization problem can be solved numerically, which leads to I( ) as a function of imaginary time . However, only knowing I( ) numerically makes the analytic continuation di cult. Instead, we address the real time problem directly, and analytically continue the equations (5.9), (5.10) and (5.13). The analytic continuation is de ned by 2 ! i2t , or x = 12 i 2t in eqs. (5.9) and (5.10). This leads to complex 1;2 and 1;2. In general, this would lead to a complex action (and thus complex entropy), which would be unphysical. However, we notice that 1 sinh 1 = L( 12 i 2t ) and 2 sinh 2 = L( 12 + i 2t ) are complex conjugates. Both eqs. (5.9) and (5.10) have a Z2 symmetry: Thus we can look for saddle point solution that is invariant under this Z2 symmetry, which satis es 2 = 1 ; 2 = 1 ; D 2 R. In this case eqs. (5.9) and (5.10) require 1 $ 2 ; 1 $ 2; D $ D sinh = D 2 L 1 2 1 i 2t sin 1 2 2 R; and the area term becomes A = 2Re 66 L2 2 6s 6 4 1 2 i 2t 2 + 12 + 2 arctan BBB s 0 B C7 C7 C7 C7 + 1 A5 The complex angle 1 is determined by D (which remains real) in eq. (5.15). The resulting action I(D) is the sum of (5.16) and 2 log cosh D, and both terms are manifestly real. We can now minimize I(D) with respect to D directly for real time t. (5.14) (5.15) 2 (5.16) n 1 is measured in unit 2N Sn (the coe cient of the area term in the expression of entropy (5.1)). Left panel shows the entropy growth for xed L = 20 and di erent (higher corresponds to curves with higher entropy). Right panel shows the entropy growth for di erent L 2 [10; 20] for xed 2 S = 0:1. Numerical and analytic results With the analytically continuated e ective action de ned above, we can minimize the action with respect to D numerically, and obtain the Renyi entropy SA;n(t). The numerical result is shown in gure 6. We see that the entropy grows quadratically at very early time, and subsequently crosses over to a linear growth. The linear growth rate agrees with the perturbative calculation for small . At longer times, the growth slows down and eventually SA;n saturates to a nite value as t ! 1. Therefore the result appears to be qualitatively similar to the expectation for a thermalizing system. However, as we will show, this system has not actually thermalized. To gain better understanding of the long time saturation behavior, we have observed numerically that at long time the saddle point value of D corresponds to Re 1 ! 2 . Therefore we can expand around this point and study the long time behavior. In this limit, eq. (5.15) requires 1 to take the following form (for details, see appendix C.1): 1 = 2 2 2 + 4t + i with 2 R. In the limit which corresponds to the entropy 1; L 1; L 1, the saddle point value is = where the saturation value is approximately given by n nN 1 SA;n(t ! +1) ' t 2 n nN 1 SA;n(1) ' 4 2 S + J (5.17) 16 LSt , (5.18) (5.19) and the coe cient for the saturation term is given by c ' L S 3 L 128 2 S + 2 + 2 : (5.20) Details of this calculation are presented in appendix C.2. It should be noticed that the rst term is -independent, which clearly shows that the result is non-perturbative in (as we have rst taken t ! 1), although we still consider a small The entropy at t = 0 can be computed perturbatively, leading to L 1. 1 SA;n(0) ' 2 log L2 J 1 2 cvT In the limit where t ! 1, and then ! 0, the entropy growth is thus given by with cv the speci c heat of the doubled SYK model we are studying. The entropy growth is independent from the UV cut-o . The relation to speci c heat is interesting, since cvT = Sth(T ) Sth(0) is the thermal entropy minus the zero temperature entropy. Therefore, even after a long time, the TFD state has not thermalized, and the entropy is smaller than the thermal entropy by an amount that is determined by the zero temperature extremal entropy. The result of our geometric minimization procedure can also be veri ed by applying the Gaussian approximation method in section 4 to the two-site problem. We nd that the early time linear growth will be corrected by a quadratic term around time scale t in the weak link limit L 1: SA;n n 1 n 2 t N 1 8 J t S In appendix A.2, we present the detail of the Gaussian approximation calculation, and compare the result with the geometric formula in appendix A.3. This early time result suggests that the t2 term becomes signi cant when the entropy approaches / is consistent with the late time saturation value we get above. N JS , which Thus, we conclude that the Renyi entanglement entropy in the time-evolved TFD state saturates to a sub-thermal value, and in the low temperature and weak inter-site coupling limit, the entropy is proportional to the \near-extremal entropy" SAth;n(T ) SAth;n(T = 0). This result indicates that the system does not completely thermalize, but instead reaches a \pre-thermalized" state. Roughly speaking, pre-thermalization occurs because the degrees of freedom in this system are separated into fast modes (the reparameterization quasiGoldstone modes) and slow modes (which are responsible for the zero temperature entropy). The latter are almost localized, and so it is natural that they require a long time to thermalize. The lack of rapid thermalization for these slow modes seems consistent with the fact that the non-reparameterization modes in the coupled SYK chain have exponentially decaying correlation functions, with correlation length at the order of lattice constant [22]. (5.21) HJEP09(217) (5.22) S L (5.23) Renyi entropy. The entropies are measured by the zero temperature entropy, and the curves are plotted for cSv0T = 0:1. The thermalization time tth, de ned when the Renyi entropy approaches its thermal value, must therefore grow faster than . In particular, it should diverge when either N ! 1, or perhaps as J ! 1. For example, one possibility is that the corrections to the Schwarzian action at higher orders in 1= J lead to a qualitative change to the optimal reparameterization f which we have computed. We expect that at nite N and/or nite J , the thermalization time for the SYK chain is nite, as it seems unlikely that the slow modes of the SYK chain have many-body localized. Our result is based on several assumptions. We have taken a replica diagonal ansatz of the two-point function, and then further restricted ourselves to the reparameterization of the unperturbed saddle point. The entropy we obtain is determined by minimization of the e ective action in this restricted space. As we have previously noted, the actual entropy obtained by unrestricted minimization will be smaller or equal to what we obtained. Thus, the pre-thermalization feature we have uncovered is independent of the validity of our approximations, and remains true even for the \actual" saddle point solution Gx ( 1; 2). We have computed the Renyi entanglement entropy of the SYK chain in a TFD state, whereas most previous studies of similar systems have studied the von Neumann entropy. As such, we now discuss the analytic continuation of the Renyi entropy to n ! 1. We can compare the long time entropy (5.22) with the Renyi entropy of the thermal ensemble for the same system A, which is SAth;n = S0 + 1 + 1 n 1 2 cvT (5.24) The comparison of SAth;n and SA;n(t ! +1) is illustrated in gure 7. Since the entropy should always be smaller or equal to the thermal value, we conclude that our approximation must fail near n ! 1 where SA;n > SAth;n. The actual entropy is upper bounded by both our result SA;n and by the thermal entropy SAth;n, so that it is below both curves in gure 7. M cv2T bound sublinear growth slower linear growth Gaussian correction S J M S J t is the bound derived using the geometric interpretation with global/collective reparametrizations. It grows linearly until a time scale proportional to M S . The black line denotes the early time weak link result, which contains a linear growth with the same rate, and receives a correction at an M -independent value of order JS . After that, we have no de nitive predictions using the tools we have developed in this paper. It is unclear whether the entropy continue to grow linearly (but presumably with a slower growth rate) until it reaches the late time bound we have derived. The growth towards this bound may be sublinear, and it is even possible that the entropy saturates at J a value sublinear in M as t ! 1. The crossing of the two curves occur at n ' 1 + cSv0T at low temperature, which serves as an estimation of where our approximations fail. For chains with more than two sites, it is di cult to solve the non-linear equation determining the saddle point of fx( ). However, we can use the same argument above and obtain an upper limit of the Renyi entropy. If we consider a uniform ansatz fx( ) = f ( ), the e ective action reduces to the same form as the two-site case, with the parameter 2 S rescaled to M S when there are M sites. Therefore the minimization problem can be mapped to the same geometric problem, with the e ective coupling parameters S; replaced by M S=2; . In the large M limit, so long as we take the uniform ansatz above, we are in fact closer to the perturbative limit since =M S ! 0 as M ! 1. The saddle point gives a long time saturation entropy that is simply M=2 times the two site value: n 1 n M cvT 4 + O(1); SA;n(0) = O(1) (5.25) Therefore the entropy grows from area law to volume law, but the nal entropy density is still lower than the thermal value. Since this is an upper bound of the actual entropy, we conclude that the chain with generic size M also reaches a pre-thermalized state at long time. Furthermore, as we noted in section 4.2, there is likely a signi cant correction to the growth rate of entanglement at the time scale t t = JS , independent from the length of the chain. This deviation indicates that the upper bound we found for the long chain is not tight. There are two possibilities about the fate of entropy growth in a long chain. The nal entropy in large M may either be proportional to M or grow slower than M . In the latter case, we would consider (at least part of the system) to be many-body localized. An illustration is given in gure 8. Physically we expect that a volume law entropy is more likely, although the growth rate may be quite slow. 6 Comparison to Holography In this section we brie y compare our results to intuition from gauge-gravity duality. Many features of the SYK model are known to be shared with models of (nearly) AdS2 gravity: in particular, a common e ective action [21, 40, 41], and similar thermoelectric transport properties [24]. Hence it is natural to ask whether the spread of entanglement shares any qualitative features. Unlike in the SYK model, in holography it is easier to study von Neumann entropy rather than Renyi entropy. There is a well-known formula for vE in holography in the TFD quench [4{6, 54]. Applying this formula to geometries with nearly AdS2 Rd infrared geometries (d 1 is required for the interpretation that entanglement is owing across a spatial surface), we nd that vE / T: Details of this calculation are found in appendix D. We emphasize that this de nition assumes that vE has been de ned as in (1.1). This result agrees with our intuition based on dimensional analysis, presented in the introduction, and our early time result from the SYK chain. However, there are important di erences between the late time behavior of the holographic von Neumann entropy SE(t) and that of the SYK Renyi entropy. For a region of large but nite width R (analogous to the nite length chains studied above), the holographic saturation entanglement is given by (6.1) (6.2) (6.3) (6.4) (6.5) HJEP09(217) SE(1) 2sthR; tsat R 2vE / T 1 : where we calculate the length of a cosmic brane of tension / n 1, stretching between the boundaries of (two-sided) AdS subject to suitable boundary conditions. Because this and the saturation time will be approximately This behavior is quite di erent from the SYK chain. Strictly speaking, the calculation was performed in a di erent limit. In the SYK chain model, we required that n 1 1 was an integer, and the holographic calculation of the previous paragraph was performed exactly at n = 1. While it is di cult to reliably perform a calculation in both models for the same value of n, we can make some preliminary comments about the behavior of the holographic calculation for n > 1. If we de ne then [55] has shown that n 1 n SA;n ; SeA;n / Area(brane of tension / n 1); brane has nite tension, we must account for its gravitational backreaction. In AdS2, the lack of gravitational dynamics means that this backreaction is expected to be very severe. A preliminary hint for the outcome comes from the following argument: in nearly AdS2 geometries, the gravitational dynamics are associated with the movement of the boundary of the near-horizon region [41, 42, 44]. Stretching a brane of nite tension from one side of AdS2 to the other will thus warp the geometry to bring the two sides closer together. This is likely analogous to the geometric calculation that we performed in the previous section, although the physical interpretation of the tensionful brane is somewhat di erent. In the geometric calculation we did, we observed that for any nite string tension , the backreaction of a tensionful brane stretching between two points on the boundary is so strong that as real time t ! 1, the length of the brane remains nite: t!1 lim cosh D(t) = (6.6) Hence, we expect a qualitative change in the Renyi entanglement entropy vs. the von Neumann entanglement entropy, and that the Renyi entropy for n > 1 may saturate at a parametrically smaller value than the von Neumann entropy at n = 1. There are two subtleties to note, which make a precise comparison between holography and the SYK chain di cult. Firstly, SeA;n vanishes when SA;n / n=(n 1), as we found in our replica diagonal SYK calculation. To the extent that SYK can recover the holographic results described above as n ! 1, it is crucial that one nds a replica non-diagonal saddle point of the action (2.24). Secondly, the calculation of holographic Renyi entropy requires calculating the backreaction of tensionful branes in AdS2 Rd, not in purely AdS2. Perhaps additional spatial dimensions modify the intuition about gravitational dynamics in AdS2 that we presented in the previous paragraph. 7 We have initiated a study of the spatial spread of entanglement in the SYK chain model. Although we were unable to compute the von Neumann entanglement directly, we were able to upper bound the spread of Renyi entanglement entropy. We found that this Renyi entropy did not saturate at the expected thermal value, but instead at a parametrically smaller value: SA;n / cvT: (7.1) This implies that only the light degrees of freedom in the SYK chain (the reparameterization modes) were able to quickly thermalize. The bulk of the degrees of freedom, which are responsible for the nite zero temperature entropy, appear to be `localized' on every site, and unable to quickly thermalize. Our result casts doubt upon whether the conventional interpretation of vE as a physical velocity scale at which entanglement propagates is sensible, at least for Renyi entropy. De ned as in (1.1), we have shown that vE / T for the SYK chain. vE / T can be justi ed on dimensional grounds, using the local criticality of the SYK chain: as length does not scale under renormalization group ows, we expect that v has the dimensions of energy. However, for Renyi entropies, the small saturation value of SA;n suggests that the speed at which entanglement can spread spatially is actually better thought of as e vE = vE sth SA;n / T 0: (7.2) e This may seem surprising, as vE vB, in contrast with the conjecture of [10]. Of course, there are a few caveats to the direct interpretation of vE as the correct de nition of entanglement velocity. (i ) We only know that the early time entropy growth rate dSA;n e dt / T in the weakly coupled limit. In the low temperature limit > 1J , our result is only an upper bound of the true entropy, and so it is possible that the entropy growth rate is much slower, leading to a smaller vE. (ii ) It is also possible that indeed veE > vB, which does e not directly violate the inequality vE 6 vB since the latter is based on the assumption of thermalization [10, 11]. Thus, it is not straightforward in this model what the \correct" entanglement velocity is, or even how it scales with temperature. We contrast this with two dimensional conformal eld theories,where the n = 1 and n > 1 entropies behave qualitatively similarly in such a vB / p quench [4]. In fact, there is already a well-known holographic model where vB is not the fastest \infrared" velocity scale. The theory holographically dual to the planar extremal AdS-Reissner-Nordstrom black hole has a speed of sound vsound / T 0 [56], even while T [48]. Sound waves are classical hydrodynamic excitations that only exist after the onset of local thermalization (at the very least in a sector containing the energymomentum tensor). We do expect that both vsound and veE are smaller than the LiebRobinson velocity [7]. While there have been some proposals for bounds and relations between the velocities of scrambling, entanglement, and sound [13], it is clear that they must be made more precise. Two complementary recent studies of quantum quenches in a single site SYK model have recently appeared [43, 44], and both show evidence for rapid thermalization. In particular, in an extension of the SYK model involving a q interaction instead of 4 interaction in (2.1), [43] was able to solve for the non-equilibrium h i two-point function exactly at q = 1, in a simple quantum quench involving a change in the Hamiltonian at time t = 0. In this limit it was observed that (i ) this two-point function was instantaneously thermal after the quench,11 and (ii ) h i appears to come entirely from the light reparameterization modes in the SYK model. Generalizing our analysis of the two-site SYK chain to a model with nite q reveals that (i ) the saturation entropy is approximately q-independent. Hence, the entropy saturation time is / q 2, which means the light degrees of freedom does thermalize instantaneously in the large q limit. We expect that correlation functions which are dominated by these light degrees of freedom do not detect the slow thermalization of the `heavy' modes that we have found by studying SA;n. It will also be interesting to compare our results with non-perturbative approaches of computing correlation functions in the Schwarzian action [58, 59]. As we discussed in eq. (2.19), the Renyi entropy calculation can be considered as a twist-operator thermal SA;n / q 2 while (ii ) dSA;n=dt 11Such instantaneous thermalization can occur in the context of holographic Vaidya geometries as well [57], for an instantaneous quench. two-point function. In the perturbative limit, the Renyi entropy we obtained in eq. (3.5) corresponds to a two-point function DXAy;n( 1)XA;n( 2) E sin ( 1 2) nN (7.3) from which we see that XA;n behaves like a dimension nN 2 eld in the 0 + 1-dimensional conformal quantum mechanics. Beyond the perturbative limit, it will be interesting to apply the techniques in refs. [58, 59] to this heavy operator two-point function problem, as a comparison to our results. In summary, it appears that the SYK chain is both maximally chaotic at short times and takes a long time to completely thermalize, at least in the special state that we have prepared. These statements are not inconsistent: the rapid scrambling of the SYK model comes entirely from the Schwarzian action for the reparameterization modes. It would be interesting if there are other notable consequences of the remaining, slowly thermalizing, heavy modes. We have also proposed that a similar prethermalization phenomenon may arise in the maximally chaotic holographic models with AdS2 infrared geometries. It would also be worth studying this more closely in the future. We have left open the possibility that the von Neumann entanglement entropy saturates at the thermal value in our quench setup, even as we have shown that all higher Renyi entropies saturate at a parametrically smaller value. It would be interesting to resolve this question in future work. Even were this to occur, the reduced density matrix of one half of the chain should not appear thermal. Interestingly, it is known that in holographic theories there are examples in which two density matrices have identical von Neumann entropy but distinct Renyi entropies to the leading order of N in the large N limit. For two disjoint regions A and B with their minimal surface also disjointed, the density matrix AB and A B have identical von Neumann entropy to the leading order of N (thus vanishing mutual information I(A : B)), but di erent Renyi entropies [55]. We do not know whether such a qualitative discrepancy between n = 1 and n > 1 Renyi entropies signals something more profound about the dynamics of the model. In conformal eld theories in two, three and four spacetime dimensions, vE, vB, vsound and the speed of light c di er by, at most, about a factor of 2. The SYK chain is one example of a class of theories where these quantities can be parametrically di erent. It will serve as an excellent model for sharpening and making precise a deep yet mysterious relationship between chaos, thermalization, entanglement and hydrodynamics. Acknowledgments We would like to thank Dmitry Bagrets, Mike Blake, Bowen Chen, Alexei Kitaev, Subir Sachdev, Douglas Stanford and Herman Verlinde for helpful discussions. YG and XLQ are supported by the David and Lucile Packard Foundation. AL is supported by the Gordon and Betty Moore Foundation's EPiQS Initiative through Grant GBMF4302. AL also thanks the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611, for hospitality while this work was in progress. 1 2 4 branch cut term in the integrand (A.4) should also be deformed to the left half plane Re x 6 0, such that the integrand is analytic on the right half Re x > 0 except the poles at discrete integer points x = 1; 2; 3; : : :. A Details of the Gaussian approximation In this appendix, we present the details of the calculation of the second order correction to SA;n(t) in . A.1 In nite chain The propagator of the reparametrization eld x;t is determined by the quadratic action: 2D(1 cos p))(n2 where we replace p2 by its lattice regularized form 2(1 cos p) as we are going to integrate over the whole Brillouin zone. For simplicity, we will neglect the large N factor in the e ective action in this appendix, since it is an overall prefactor and only rescales the nal answer. The two-point function of that arises from (A.1) is S J X jnj(jnj + 2 J 1 S M (2 )4 2 2 2 J J X 2 cos 2 n Now we need to compute the in nite summation over integer n > 2. Let us denote X X (A.2) (A.3) with a > 0 and 0 6 Matsubara trick: < 2 . The summation over integers n can be done through the dx e ix 1 (x2 1)x3=2p(x + a) Z C1 dx 1 1 e2 ix (x2 eix 1)x3=2p(x + a) where C1 is an integration contour that winds integer points 2; 3; : : : clockwise, see gure 9 for an illustration. The integrand is analytic in the right half plane Re x > 0 except the integer points x = 1; 2; : : :. Therefore we can deform the contour to C2 : i1 + ! i1 + as shown in gure 9 with the cost of a double pole at x = 1: dx Z C2 + 2 i Res 1 e ix 1 (x2 1 1)x3=2p(x + a) e ix Z C2 1 dx 1 1 eix (A.4) ; x = 1 (A.5) (A.6) (A.8) (A.9) The residue can be computed explicitly: 2 i Res = (4a + 5) cos 2(a + 1)( ) sin 2(a + 1)3=2 while the rst two integrals need further treatment. We notice the integrands diverge near x = 0 and exponentially decay when going to large imaginary x in the contour C2. One can show that at large real time t, corresponding to large Im( ), it is safe to approximate the integrands by their form near x = 0: dx Z C2 1 a 1 a 1 e 2 ix Z 1 1 Z 1 0 1 dx dx 1 (x2 i e x 2 x (ix)3=2 + e ix Z 1 1 i ex( 2 ) + e x 2 x 3 2 dx 2 x (ix)3=2 Z 0 dx 1 = 2 x 4 3 2 a i ex + ex(2 ) ! (ix)3=2 (2 (2 )3=2 + 3=2 )3=2 + 3=2 (A.7) Thus, the propagator has an approximate form: (2 )4 2 S J h x; x;0i ' 4 3 2 a (2 )3=2 + 3=2 + (4a + 5) cos 2(a + 1)( ) sin 2(a + 1)3=2 where = 2 and a = 2 D is small. We can further simplify the second term to: 1)x3=2p(x + a) Z C2 dx 1 1 e2 ix (x2 eix 1)x3=2p(x + a) 5 cos ) sin This step amounts to replacing px(x + a) by x when a is small. We can now evaluate the leading growing term in the Gaussian correction for the e ective action at large real time = 2 + i2t, or * : tan 2 2 0x; + 0x;0 !2+ J q 2 D t 3=2 Notice the di usion constant D here can be expressed in terms of the parameter and S using (following [22]): J 4J12 3p2J 2 K J J 2 1 1 = J 12 S (A.10) J2 = 8 J2 1 (A.11) (A.12) (A.13) (A.14) (A.15) (A.16) Therefore, we can express the nal minimum of the action in following form: 1 * 2 S 2+ 2 1 8 (2 )2 J S t 3=2 = 2 t 3 2 r J t S We factor the 2 t out for easy comparison with the linear t growth term. The formula indicates the linear t growth will receive a correction at time scale: 3 64 q J 12 S t S J 2 as claimed in the main text. A.2 Two-sites The calculations for two sites are much simpler. Due to the symmetry between the two sites, we can neglect the p2 terms in (A.1), and compute the two-point function of as J S 1 2 (2 )4 n>2 X n2(n2 1) The extra 12 con rms that the global reparametrization is further suppressed by the system size. Now the evaluation of the summation is much simpler because there is no longer a branch cut of the summand. We can complete the contour nicely as shown in gure 10. In short, we can compute the in nite sum I( ) := 1) X n6=0; 1 e in n2(n2 1) by a contour deformation to a simple contour C that only includes three poles (two double poles at x = 1 and a triple pole at x = 0): X n>2 e ix I( ) = Z C1+C2 e 2 ix 1 (x2 1 1)x2 = Z e ix 1 C e 2 ix 1 (x2 1)x2 1 (x2 ) sin 6 ) sin 6 (A.17) (A.18) (A.19) HJEP09(217) The correlator has the form [21] h 0i = 2 S (2 )4 + i4 t integer points x 2 Z. Therefore we can deform the contour C1 [ C2 that encloses jxj > 2 integer points to C which only encloses three points 0; 1. Therefore we can easily compute the integral by the residue theorem: X 2 i Res( e ix 3 2 J 15 cos + 2: 2 Now we again evaluate the leading growing term in the Gaussian correction by setting 1 * 2 S 2+ 2 1 8 (2 )2 2 S J 4 t 2 = S t 2 weak link limit Again, we see that the linear growth receives a correction at time scale JS . A.3 Comparison of the two-site result with the geometric minimization at We can do one further self-consistency check for the two-site problem in the weak link limit. We now use the geometric interpretation to reproduce the above two-site result. The strategy is to start with the circle solution, which is a saddle point for the area term, and then expand around the circle solution and nd the minimal value when we include the twisted interaction. In the geometric picture, we can treat 1;2 as function of y = cosh D, and expand the area function around the saddle point: 1 = 2 x; 2 = 2 (1 y = 1 + 2 (A.20) L 2 2 sin2 x The saddle point represents a circle geometrically, which is expected to be the shape with maximal area under constrain. So we must expand the area term to quadratic order in the deviation y y : A = A(y ) 1 4 6 2 L5 Q(x)(y y )2: The constant term has simple expression A := A(y ) = L + 2L2 2 and the linear term vanishes since we are expanding around a saddle point for the area. The most important piece is the quadratic term. We have de ne a function Q(x) as follows: Q(x) := 1 (sin x)4(1 + (1 x) cot x)(1 x cot x) and it is clear that Q(x) determines the cost of uctuations near saddle point y . Note that Q(x) has a minimum at x = 12 of y, which we may also expand to quadratic order: . Now the twisted term 2 log y is a logarithmic function log y = log y + y ) 1 y 1 1 2 (y )2 (y y )2: (A.21) (A.22) (A.23) L1 , or (A.24) (A.25) x) cot x)(1 x cot x) ' L 2 to simplify the result. This J2 1 J2 y while the area term has L 5. So expanding around y is useful in the limit Notice that y is of order L2; therefore the quadratic term from 2 log y is of order L 4, 1J . In this limit, we get a correction for the saddle point free energy: I = min 2 S y 1 4 6 2 L5 Q(x)(y y )2 + 1 2 y y ) = L5 2 64 6 S(y )2Q(x) Notice y = 1+2 2L 2 sin2 x and we focus on x away from 0 and 1, so we can approximate 2 2L 2 sin2 x, therefore: I ' L 2 16 S 2 (1 + (1 where we have taken x = 12 expression precisely agrees with (A.19). i 2t and large real time t B Derivation of the geometric interpretation In this appendix we provide the derivation of eq. (5.8). B.1 The Schwarzian action term The relation between the Schwarzian action and the area enclosed by a closed curve in hyperbolic space has been discussed in [41, 53]. To make our discussion self-contained, we include a derivation here. We embed our curve in a global Euclidean AdS2 Poincare disk, so that our mapping is a little di erent from that in [41]. Consider the Poincare disk with metric: ds2 = d 2 + sinh2 d 2 = 4 dr2 + r2d 2 (1 r2)2 r = tanh 2 (B.1) Put a curve parametrized by : (r( ); ( )), total length L = J = , where J = 1 2 [0; ) in the hyperbolic disk, with a large 1. The physical time is required to be proportional to the arc length parameter of the curve. Thus, we x the induced metric along the curve g = 12 , i.e. 4(r02 + r2 02) (1 r2)2 : r ' 1 0 + ( 0)2 2 + O( 3) A = drd (1 4r r2)2 = Z d 2 r( )2 2 ; We always consider the case when the curve is close to the boundary. Thus, r 1 and 1. The metric constraint then implies that Using this formula, we can rewrite the area enclosed by the curve as an integral: Therefore, we have proven the geometric interpretation of the Schwarzian action [41, 53] (B.2) (B.3) (B.4) (B.5) (B.6) (B.7) 1 1 r 0 00 2# 1 2 Z d 2 002 2 02 1 + O( 2) + O( 2) 1 02 2 1 002 2 02 + 00 0 0 r02 2r2 02 + O( 4) Replacing 1 2r2 using the constraint (B.2), we obtain A + 4 = r02 + r2 02 1=2 = = L + 2 + d 1 + Z 0 + This integral is the same as the Schwarzian action: Z Z d Sch tan ; 2 1 02 2 Z d 2 d Sch tan ; = A L + 2 Here is the renormalized reparametrization of time := 2 f ( ). Each reparametrization f ( ) determines a angular coordinate ( ) and further determines the curve in the hyperbolic disk, by using the constraint (B.2). The Schwarzian action corresponds to the area enclosed by the curve with a xed length L = J . B.2 The twist operator term Next we need to show the twist operator term / also has a simple interpretation: log f = log cosh D(X1; X2); (B.8) where D(X1; X2) is the distance between two marked points (determined by 1;2 = 2 f ( 1;2)) on the curve. For this, it is helpful to introduce the embedding coordinates X1; X2; X3 = (sinh cos ; sinh sin ; cosh ) ; (B.9) which live on the hyperboloid X X = 1 in a 3 dimensional space with metric (1; 1; 1). The distance betwen X1 and X2 is related to the inner product: cosh D(X1; X2) = X1 X2 where we use cosh ' sinh ' 2 ' 1 r ' 2 2 0 = 2J0 in the 1 1;2 are UV cut-o s of order J 1, therefore 1 2J2 is a constant of order 1 whose accurate 2 determination is unimportant for our purposes. We have assumed that 1 2J2 = 1 in the 1 limit. Here main text for simplicity. Thus we arrive at log f ' log cosh D(X1; X2): A di erent choice of cut-o 1;2 leads to an additional O(1) constant, proportional to and independent from time t. This does not have any important e ect on our results. C Some details of the geometric minimization e 1+ 2 4 J 2 hsin (f ( 1) 2 sin 1 2 2 2 f ( 2)) ' 2 J 2 = f 2 e 2 f 0( 1)f 0( 2) sin 1 2 2 1 2J 2 2 2 = cosh 1 cosh 2 sinh 1 sinh 2 (cos 1 cos 2 + sin 1 sin 2) (B.10) (B.11) (C.1) +i : (C.2) (C.3) Here we provide some details for the geometric optimization problem at large real time t ! 1. In particular we derive the asymptotic form of the two terms in the action. C.1 Solving the constraint As t ! 1, our numerics suggests that the angle 1 ! 2 . We can use this knowledge to derive a constraint between the real and imaginary parts of 1. The starting point is the reality condition This can be rewritten as an equation relating the real and imaginary part of 1 = 2 Im 1 2 i 2t 1 sin 1 2 1 + tanh 2 ) tan 2 tanh 2 + tan 2 2 = 0 = 0 ) 4t 1 Im 4t 2 2 After Taylor expanding the tan = tanh functions for small and we can solve the equation in leading order: 1 2 i 2t 1 sin 1 2 ! = 0: 2 2 = 0 ) 4t In the limit t ! 1, we can Taylor expand A(D) and keep the leading order terms: Using the constraint = 2 2 A L + 2 A ' L 1 0 2 2 1i 2t + 4 1 1 2 i 2t tan 21 1 A + L 4t for 1 = 2 4 3t + i , we have: + 2 2 2 2 t 2 sub-l{ezading 2 2 t3 + : : :CC C A Since is small, we only need to keep 16t term to minimize the action. After is determined variationally, the subleading terms will determine the subleading corrections to the nal entropy at late times. We next analyze the log term: log y 2 log 8t2 + : : : sub-l{ezading min + log Inserting this value back into the action, we obtain 1 L 2 log yi Lt p I(t) = h ' t 2 with c given in eq. (5.20). We have checked that our numerical results agree well with the analytic result in both the long-time saturation value and the t12 term, as is shown in (C.4) (C.5) (C.6) (C.7) (C.8) (D.1) (D.2) Now all the terms are explicit function of and we can proceed to nd the minimum at leading order: ds2 = L2 g(r) f (r) dr2 f (r)g(r)dt2 + dx2 vE = d pj(f g)(r ) gure 11. D Holographic von Neumann entanglement velocity In this appendix we calculate vE for a holographic model with an AdS2 the IR. The formula for vE in a generic planar geometry with metric Rd geometry in ior (5.19) (black lines) for L = 10; 2 S 2 [0:02; 0:4]. The curves with higher entropy has higher . where r+ is the location of the (outer) event horizon of (D.1), and r is the solution to 2d (f g)0(r ) (f g)(r ) : which should occur behind the horizon. Note that vE is best understood as arising from a calculation of entanglement in a TFD state [4], analogous to the case we studied in the main text. This calculation may also be done for spatial quenches [5, 6], but in this case it is important that the initial state of the quench has vanishing entropy density; otherwise the formula above is generally modi ed. If the matter which sources (D.1) gives rise to an extremal black hole, then at zero temperature r+ ! re < 1. (Note that re d temperature T , we expect that near the horizon, / s > 0 [56].) At a very small but nonzero f (r)g(r) c1T c2T 2 + and that r+ > re for this new geometry if the speci c heat is positive. The coe cients c1, c2 and a are not independent of T but we will only need the fact that, to leading order in T , they are constants. (D.3) implies that At small T this equation can only be solved if where 2d a(re 2a(re c1T c1T r )2 r ) c2T 2 : re c1T r = bT 2; re ab : (D.3) (D.5) (D.6) (D.7) Using (D.2) we conclude that to leading order in T vE pc2T 2 c1T r )2 = p c2T + O(T 2): This con rms the scaling that we claimed in the main text. Open Access. 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Yingfei Gu, Andrew Lucas, Xiao-Liang Qi. Spread of entanglement in a Sachdev-Ye-Kitaev chain, Journal of High Energy Physics, 2017, 120, DOI: 10.1007/JHEP09(2017)120