#### 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.
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] J.M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43 2046
[2] M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50 (1994) 888
[3] M. Rigol, V. Dunjko and M. Olshanii, Thermalization and its mechanism for generic isolated
quantum systems, Nature 452 (2008) 854 [arXiv:0708.1324].
interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].
[4] T. Hartman and J. Maldacena, Time evolution of entanglement entropy from black hole
[5] H. Liu and S.J. Suh, Entanglement Tsunami: universal scaling in holographic thermalization,
Phys. Rev. Lett. 112 (2014) 011601 [arXiv:1305.7244] [INSPIRE].
[6] H. Liu and S.J. Suh, Entanglement growth during thermalization in holographic systems,
Phys. Rev. D 89 (2014) 066012 [arXiv:1311.1200] [INSPIRE].
[7] E.H. Lieb and D. Robinson, The nite group velocity of quantum spin systems, Commun.
Math. Phys. 28 (1972) 251.
[8] Y. Sekino and L. Susskind, Fast scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096]
[9] S.H. Shenker and D. Stanford, Black holes and the butter y e ect, JHEP 03 (2014) 067
[10] D.A. Roberts and B. Swingle, Lieb-Robinson bound and the butter y e ect in quantum eld
theories, Phys. Rev. Lett. 117 (2016) 091602 [arXiv:1603.09298] [INSPIRE].
[11] M. Mezei and D. Stanford, On entanglement spreading in chaotic systems, JHEP 05 (2017)
065 [arXiv:1608.05101] [INSPIRE].
[12] T. Hartman and N. Afkhami-Jeddi, Speed Limits for Entanglement, arXiv:1512.02695
arXiv:1706.00019 [INSPIRE].
[13] T. Hartman, S.A. Hartnoll and R. Mahajan, An upper bound on transport,
[14] R. Nandkishore and D.A. Huse, Many body localization and thermalization in quantum
statistical mechanics, Ann. Rev. Condensed Matter Phys. 6 (2015) 15 [arXiv:1404.0686]
[15] Y. Huang, Y. Zhang and X. Chen, Out-of-time-ordered correlators in many-body localized
systems, Annalen Phys. 529 (2017) 1600318 [arXiv:1608.01091] [INSPIRE].
[17] Y. Chen, Quantum logarithmic butter y in many body localization, arXiv:1608.02765
95 (2017) 060201 [arXiv:1608.03280] [INSPIRE].
[18] B. Swingle and D. Chowdhury, Slow scrambling in disordered quantum systems, Phys. Rev. B
[19] S. Sachdev and J. Ye, Gapless spin
uid ground state in a random, quantum Heisenberg
magnet, Phys. Rev. Lett. 70 (1993) 3339 [cond-mat/9212030] [INSPIRE].
[20] A. Kitaev, A simple model of quantum holography, talks given at the KITP Program:
Entanglement in Strongly-Correlated Quantum Matter, April 7 and May 27, Santa Barbara,
U.S.A. (2015).
(2016) 106002 [arXiv:1604.07818] [INSPIRE].
[21] J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94
[22] Y. Gu, X.-L. Qi and D. Stanford, Local criticality, di usion and chaos in generalized
Sachdev-Ye-Kitaev models, JHEP 05 (2017) 125 [arXiv:1609.07832] [INSPIRE].
[23] S. Sachdev, Bekenstein-Hawking entropy and strange metals, Phys. Rev. X 5 (2015) 041025
[arXiv:1506.05111] [INSPIRE].
[INSPIRE].
[INSPIRE].
[24] R.A. Davison, W. Fu, A. Georges, Y. Gu, K. Jensen and S. Sachdev, Thermoelectric
transport in disordered metals without quasiparticles: The Sachdev-Ye-Kitaev models and
holography, Phys. Rev. B 95 (2017) 155131 [arXiv:1612.00849] [INSPIRE].
[25] D.J. Gross and V. Rosenhaus, A generalization of Sachdev-Ye-Kitaev, JHEP 02 (2017) 093
[arXiv:1610.01569] [INSPIRE].
[26] S. Banerjee and E. Altman, Solvable model for a dynamical quantum phase transition from
fast to slow scrambling, Phys. Rev. B 95 (2017) 134302 [arXiv:1610.04619] [INSPIRE].
[27] Z. Bi, C.-M. Jian, Y.-Z. You, K.A. Pawlak and C. Xu, Instability of the non-Fermi liquid
state of the Sachdev-Ye-Kitaev Model, Phys. Rev. B 95 (2017) 205105 [arXiv:1701.07081]
[28] Y. Gu, A. Lucas and X.-L. Qi, Energy di usion and the butter y e ect in inhomogeneous
Sachdev-Ye-Kitaev chains, SciPost Phys. 2 (2017) 018 [arXiv:1702.08462] [INSPIRE].
[29] X.-Y. Song, C.-M. Jian and L. Balents, A strongly correlated metal built from
Sachdev-Ye-Kitaev models, arXiv:1705.00117 [INSPIRE].
[30] Y. Chen, H. Zhai and P. Zhang, Tunable quantum chaos in the Sachdev-Ye-Kitaev model
coupled to a thermal bath, JHEP 07 (2017) 150 [arXiv:1705.09818] [INSPIRE].
[31] X. Chen, R. Fan, Y. Chen, H. Zhai and P. Zhang, Competition between chaotic and
non-chaotic phases in a quadratically coupled Sachdev-Ye-Kitaev model, arXiv:1705.03406
[32] S.-K. Jian and H. Yao, Solvable SYK models in higher dimensions: a new type of many-body
localization transition, arXiv:1703.02051 [INSPIRE].
[33] W. Fu, D. Gaiotto, J. Maldacena and S. Sachdev, Supersymmetric Sachdev-Ye-Kitaev
models, Phys. Rev. D 95 (2017) 026009 [arXiv:1610.08917] [INSPIRE].
[34] M. Berkooz, P. Narayan, M. Rozali and J. Simon, Higher dimensional generalizations of the
SYK model, JHEP 01 (2017) 138 [arXiv:1610.02422] [INSPIRE].
[35] G. Turiaci and H. Verlinde, Towards a 2d QFT analog of the SYK model, arXiv:1701.00528
[36] E. Witten, An SYK-like model without disorder, arXiv:1610.09758 [INSPIRE].
[37] I.R. Klebanov and G. Tarnopolsky, Uncolored random tensors, melon diagrams and the
[38] J. Murugan, D. Stanford and E. Witten, More on supersymmetric and 2D analogs of the
SYK model, JHEP 08 (2017) 146 [arXiv:1706.05362] [INSPIRE].
[41] J. Maldacena, D. Stanford and Z. Yang, Conformal symmetry and its breaking in two
dimensional nearly Anti-de-Sitter space, PTEP 2016 (2016) 12C104 [arXiv:1606.01857]
[42] J. Maldacena, D. Stanford and Z. Yang, Diving into traversable wormholes, Fortsch. Phys.
65 (2017) 1700034 [arXiv:1704.05333] [INSPIRE].
Sachdev-Ye-Kitaev model, arXiv:1706.07803 [INSPIRE].
[43] A. Eberlein, V. Kasper, S. Sachdev and J. Steinberg, Quantum quench of the
[44] I. Kourkoulou and J. Maldacena, Pure states in the SYK model and nearly-AdS2 gravity,
arXiv:1707.02325 [INSPIRE].
arXiv:1707.08013 [INSPIRE].
[arXiv:1503.01409] [INSPIRE].
[arXiv:1604.01754] [INSPIRE].
[arXiv:1611.09380] [INSPIRE].
[45] J. Sonner and M. Vielma, Eigenstate thermalization in the Sachdev-Ye-Kitaev model,
[46] J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106
[47] M. Blake, Universal di usion in incoherent black holes, Phys. Rev. D 94 (2016) 086014
[48] M. Blake and A. Donos, Di usion and chaos from near AdS2 horizons, JHEP 02 (2017) 013
[49] M.M. Wolf, F. Verstraete, M.B. Hastings and J.I. Cirac, Area laws in quantum systems:
mutual information and correlations, Phys. Rev. Lett. 100 (2008) 070502 [arXiv:0704.3906].
[50] P. Hosur, X.-L. Qi, D.A. Roberts and B. Yoshida, Chaos in quantum channels, JHEP 02
(2016) 004 [arXiv:1511.04021] [INSPIRE].
0406 (2004) P06002 [hep-th/0405152] [INSPIRE].
[51] P. Calabrese and J.L. Cardy, Entanglement entropy and quantum eld theory, J. Stat. Mech.
[52] J.S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S.H. Shenker et al., Black Holes
and Random Matrices, JHEP 05 (2017) 118 [arXiv:1611.04650] [INSPIRE].
[53] A. Kitaev, New Results in the SYK Model, talk given at IAS Workshop on the Chaos, the
SYK Model and AdS2, October 17{19, Princeton, U.S.A. (2016).
[54] M. Mezei, On entanglement spreading from holography, JHEP 05 (2017) 064
[arXiv:1612.00082] [INSPIRE].
[arXiv:1601.06788] [INSPIRE].
[55] X. Dong, The gravity dual of Renyi entropy, Nature Commun. 7 (2016) 12472
bootstrap, JHEP 08 (2017) 136 [arXiv:1705.08408] [INSPIRE].
localization, Sci. Bull . 62 ( 2017 ) 707 [arXiv: 1608 . 01914 ] [INSPIRE].
Sachdev-Ye-Kitaev models , Phys. Rev. D 95 ( 2017 ) 046004 [arXiv: 1611 .08915] [INSPIRE]. [39] W. Israel , Thermo- eld dynamics of black holes , Phys. Lett. A 57 ( 1976 ) 107 . [40] K. Jensen , Chaos in AdS2 holography , Phys. Rev. Lett . 117 ( 2016 ) 111601 [56] S.A. Hartnoll , A. Lucas and S. Sachdev , Holographic quantum matter, arXiv:1612 .07324 [57] V. Balasubramanian et al., Holographic thermalization , Phys. Rev. D 84 ( 2011 ) 026010 [58] D. Bagrets , A. Altland and A. Kamenev , Power-law out of time order correlation functions in the SYK model , Nucl. Phys. B 921 ( 2017 ) 727 [arXiv: 1702 .08902] [INSPIRE]. [59] T.G. Mertens , G.J. Turiaci and H.L. Verlinde , Solving the Schwarzian via the conformal