Prepared for submission to JHEP
Entanglement Entropy and its Quench Dynamics
for Pure States of the Sachdev-Ye-Kitaev model
Pengfei Zhang
a,b
a
Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA
91125, USA
b
Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA
91125, USA
E-mail:
pengfeizhang.physics@gmail.com
Abstract:
Sachdev-Ye-Kitaev (SYK) is a concrete solvable model with non-Fermi liquid
behavior and maximal chaos. In this work, we study the entanglement Rényi entropy for
the subsystems of the SYK model in the Kourkoulou-Maldacena states. We use the path-
integral approach and take the saddle point approximation in the large-
N
limit. We find
a first-order transition exist when tuning the subsystem size for the
q
= 4
case, while it
is absent for the
q
= 2
case. We further study the entanglement dynamics for such states
under the real-time evolution for noninteracting, weakly interacting and strongly interacting
SYK(-like) models.
arXiv:2004.05339v2 [hep-th] 18 Apr 2020
Contents
1 Introduction
1
2 SYK model and Kourkoulou-Maldacena states
2
3 Path-integral for pure-state entanglement entropy
4
3.1 A warm up:
〈
KM
|
KM
〉
4
3.2 Computing
S
(
n
)
A
7
4 Numerical results
8
5 Quench dynamics for pure-state entanglement entropy
10
6 Conclusion
15
1 Introduction
In recent years, the entanglement entropy and its dynamics in many-body systems have
drawn a lot of attention. As an example, the entanglement entropy has been studied both
theoretically [1–4] and experimentally [5, 6] for interacting quantum systems that satisfy
the eigenstate thermalization hypothesis (ETH). For a general energy eigenstate, it shows
volume law scaling, in contrast to the area law scaling in the many-body localization (MBL)
phase [7, 8]. The entanglement dynamics can also be related to the out-of-time-order corre-
lators [9, 10], which characterize the scrambling of quantum information [11–14]. Moreover,
the recent resolution of the information paradox [15–20] is directly from the refined under-
standing of the Ryu-Takayanagi formula [21–23] for computing the entanglement entropy
in holographic systems.
Unfortunately, the calculation of the entanglement entropy for many-body systems is
usually hard. Toy models where entanglement entropy can be computed efficiently are of
especial interest. One strategy is to construct random unitary dynamics [24–26]. Here, we
consider an alternative route by studying a specific solvable model named the Sachdev-Ye-
Kitaev model [12, 27–29], which describes
N
Majorana modes with infinite range
q
−
body
interaction. For
q
≥
4
, it is known as a non-Fermi liquid without quasiparticles which
shows maximal chaotic behavior. In the low-energy limit, the system is described in terms
of reparametrization modes, with an effective Schwarzian action [12, 28, 29]. The same
action also shows up for the Jackiw-Teitelboim gravity in 2D [29, 30].
Previously, there are studies of the SYK model from the entropy perspective. Assuming
the system satisfies ETH [1, 2, 31–33], the entanglement entropy can be related to the
thermal entropy with an effective temperature depending on the system size [34]. An
analytical approximation has also been derived based on the many-body spectrum [35].
– 1 –
Numerically, the subsystem entropy has been studied in [36] by exact diagonalization for
the ground state, and in [37, 38] by path-integral approach for thermal ensembles. There
are also studies for two copies of (coupled) SYK models prepared in the thermofield double
state, which purifies the thermal density matrix [20, 39, 40]. There are also studies for
random hopping model, which is directly related to the SYK
2
case [41, 42]. However, the
large-
N
microscopic entanglement entropy and its dynamics for pure states of a single SYK
model are still unknown
1
.
In this work, we establish the formulation for computing Rényi entanglement entropy
for the Kourkoulou-Maldacena pure states in SYK-like models [44]. The paper is organized
as follows: In section 2, we give a brief review of the SYK model and the Kourkoulou-
Maldacena states. We then derive the path-integral representation of Rényi entanglement
entropy for such pure states in section 3. The numerical results are presented in section 4.
By varying the subsystem size, we find a first-order transition of the entanglement entropy
for the SYK
4
model, which leads to the singularity of the Page curve at half system size [45].
This is qualitatively different from the
q
= 2
case, where the entanglement entropy changes
analytically when varying the subsystem size. We further study the exact dynamics of the
entanglement entropy for the SYK
4
non-Fermi liquid, with a comparison to noninteracting
or weakly interacting SYK
2
Fermi liquids in section 5. Finally, we summarize our results
in 6.
2 SYK model and Kourkoulou-Maldacena states
The Hamiltonian of SYK
q
model [12, 28] reads:
H
=
1
q
!
∑
i
1
i
2
...i
q
i
q/
2
J
(
q
)
i
1
i
2
...i
q
χ
i
1
χ
i
2
...χ
i
q
.
(2.1)
Here
q
is an even integer and
i
= 1
,
2
...N
labels different Majorana modes. We take the
convention that
{
χ
i
,χ
j
}
=
δ
ij
.
J
(
q
)
i
1
i
2
...i
q
are independent Gaussian variables with:
|
J
(
q
)
i
1
i
2
...i
q
|
= 0
,
|
J
(
q
)
i
1
i
2
...i
q
|
2
=
(
q
−
1)!
J
2
N
q
−
1
=
2
q
−
1
(
q
−
1)!
J
2
qN
q
−
1
.
(2.2)
Here
J
is taken to be a constant in the large-
q
limit.
For a thermal ensemble, to the leading order of
1
/N
, the two-point correlator
G
th
(
τ
) =
〈T
τ
χ
i
(
τ
)
χ
i
(0)
〉
β
satisfies the self-consistent equation:
G
−
1
th
(
iω
n
) =
−
iω
n
−
Σ
th
(
iω
n
)
,
Σ
th
(
τ
) =
.
.
.
=
J
2
G
q
−
1
th
(
τ
)
,
(2.3)
where the self-energy is given by melon diagrams. By solving the Schwinger-Dyson equation,
the model is found to be a Fermi liquid with finite spectral function near
ω
∼
0
for
q
= 2
.
On contrary, for
q
≥
4
, the model has divergent spectral function
ρ
(
ω
)
∼
ω
2
/q
−
1
, which
1
There are related dicussions of entanglement entropy in [43].
– 2 –
is known as a non-Fermi liquid. Further study shows it has maximal chaos [12, 28] and
satisfies the ETH [31–33].
Considering the SYK system in some eigenstate
|
E
〉
with energy
E
=
N
, the entropy
S
A
of the a subsystem
A
containing
M
=
λN
(
λ <
1
/
2
) Majorana fermions is argued to
be [34]:
S
A
=
Ms
th
(
λ
q
−
1
2
)
,
(2.4)
for
q
≥
4
. Here
s
th
(
x
)
is the thermal entropy density in the micro-canonical ensemble with
energy density
x
. A similar statement when the total system is prepared in a thermal ensem-
ble has been tested in [37]. Approximately, we have
s
th
(
x
)
≈
(log(2)
/
2
−
arcsin(
x/
0
)
/q
2
)
with
0
being the energy density of the ground state [35]. On the other hand, for
q
= 2
,
the ground state entanglement entropy can be calculated analytically [36, 37].
In this work, we focus on a specific class of pure states of the SYK model [44]. These
states are now known as the Kourkoulou-Maldacena (KM) states. To construct them, we
first pair Majorana fermions as
c
j
=
χ
2
j
−
1
+
iχ
2
j
2
with
j
= 1
,
2
...N/
2
2
. Then (unnormalized)
KM states are given by:
|
KM
(
{
s
}
,β
)
〉
=
e
−
βH
2
|{
s
}〉
,
(2
n
j
−
1)
|{
s
}〉
=
s
j
|{
s
}〉
.
(2.5)
Here
n
j
=
c
†
j
c
j
and
s
j
∈ {±
1
}
. We could always redefine
χ
2
j
→
s
j
χ
2
j
to set
s
j
= 1
for all
j
. As result, all single states are equivalent after averaging over the ensemble of random
interaction. We will focus on
{
s
}
=
{
1
}
for most parts of the manuscript. Moreover, for
simplicity, from now on we keep the
β
= 1
/T
dependence of
|
KM
〉
implicit.
The hallmark of these states is that to the leading order of
1
/N
, the correlation func-
tions of
χ
i
can be related to thermal correlators [44], under the assumption of the disorder
replica diagonal [29, 46–48]. As an example, two-point functions
G
ij
(
τ,τ
′
) =
1
Z
KM
〈{
1
}|T
τ
e
−
∫
β
0
dτH
χ
i
(
τ
)
χ
i
(
τ
′
)
|{
1
}〉
,
with
Z
KM
≡〈
KM
|
KM
〉≡
e
−
I
KM
, can be expressed in terms of the thermal Green’s function
G
th
(
τ
)
. Explicitly, all non-zero components are
G
ii
(
τ,τ
′
) =
G
th
(
τ
−
τ
′
)
, G
2
j
−
1
,
2
j
(
τ,τ
′
) =
−
i
2
G
th
(
τ
)
G
th
(
τ
′
)
.
(2.6)
Here we have
τ,τ
′
∈
[0
,β
]
. This shows that the diagonal component
G
ii
take the same form
as a thermal Green’s function, while the off-diagonal part
G
2
j
−
1
,
2
j
(
β/
2
,β/
2)
characterize
the deviation from a thermalized state (at the two-point function level). For
βJ
→∞
,
|
KM
〉
selects one state from the ground state sector of the SYK model, and
G
2
j
−
1
,
2
j
(
β/
2
,β/
2)
→
0
.
We are mainly interested in the Rényi entanglement entropy of such pure states. For
such states, we define subsystem A consisting of
M/
2
complex fermions
3
. The reduced den-
sity matrix
ρ
A
=
1
Z
KM
tr
B
|
KM
〉〈
KM
| ≡
̃
ρ
A
/Z
KM
is given by tracing out its complimentary
2
Since there is a permutation symmetry for different modes
i
, this choice is general.
3
Note that although the KM pure states are expected to be dual to an AdS
2
geometry with a brane,
there is no index
i
degree of freedom, and consequently no direct analogy of this entanglement entropy in
the gravity picture.
– 3 –
B
. The
n
−
th Rényi entropy is then given by
S
(
n
)
A
=
1
1
−
n
log(
tr
A
ρ
n
A
)
,
(2.7)
with
S
A
≡ S
(1)
A
being the Von Neumann entropy. Since the full system is in a pure state,
we expect
S
(
n
)
A
being symmetric under a reflection along
λ
= 1
/
2
. For
β
= 0
,
|
KM
〉
is a
product state, and we have
S
(
n
)
A
= 0
. On the contrary, if we take
βJ
→∞
, and we expect
S
(
n
)
A
follow a Page curve [45] with energy density depending on subsystem size (2.4).
3 Path-integral for pure-state entanglement entropy
In this section, we derive the path-integral representation of
S
(
n
)
A
for KM pure states. We
begin with a warm up by computing the normalization factor
Z
KM
in section 3.1, which
is also needed then computing the entanglement entropy. The path integral formula for
computing
S
(
n
)
A
is then derived in 3.2.
3.1 A warm up:
〈
KM
|
KM
〉
Let us first consider the path-integral representation of
Z
KM
=
e
−
I
KM
=
〈
KM
|
KM
〉
. Note
that this is in fact not essential, since
Z
KM
can be directly related to the the thermal
partition function
Z
=
tr
e
−
βH
[37]. However, the trick developed in this subsection is
useful when computing the entanglement entropy.
The state
|
KM
〉
is given by an imaginary-time evolution of a initial state
|{
1
}〉
. The
graphic representation is:
|
KM
〉
=
e
−
βH
2
|{
1
}〉
=
|{
s
}〉
0
β
2
0
β
2
χ
2
χ
1
(3.1)
Here we have explicitly separated out fermions with odd/even indices:
χ
1
/
2
represents
Majorana fermions
χ
2
j
−
1
/χ
2
j
with odd/even indices. The solid lines denote the imaginary-
time evolution and the dotted lines represent interactions between fermions. Two points
connected by the dotted line are at the same imaginary time. The black dots represent the
boundary condition
c
j
|{
1
}〉
= 0
, or in terms of Majorana fermions
(
χ
2
j
−
1
+
iχ
2
j
)
|{
1
}〉
= 0
.
Similarly, the normalization
Z
KM
is given by
Z
KM
=
e
−
I
KM
=
〈{
1
}|
e
−
βH
|{
1
}〉
=
|{
s
}〉
〈{
s
}|
0
β
0
β
χ
2
χ
1
(3.2)
– 4 –
Here we have another boundary condition
〈{
1
}|
(
χ
2
j
−
1
−
iχ
2
j
) = 0
at imaginary time
β
.
The path-integral representation of
Z
KM
is then
e
−
I
KM
=
∫
b.c.
D
χ
i
(
τ
) exp(
−
S
KM
[
χ
i
])
,
S
KM
=
∫
β
0
dτ
1
2
∑
i
χ
i
∂
τ
χ
i
+
1
q
!
∑
i
1
i
2
...i
q
i
q/
2
J
(
q
)
i
1
i
2
...i
q
χ
i
1
χ
i
2
...χ
i
q
.
(3.3)
Here b.c. indicates the the boundary condition at
τ
= 0
and
τ
=
β
:
χ
2
j
−
1
(0) =
−
iχ
2
j
(0)
, χ
2
j
−
1
(
β
) =
iχ
2
j
(
β
)
.
(3.4)
We further take the disorder average of random interaction
J
(
q
)
i
1
i
2
...i
q
. As for the thermal
ensemble, we expect the replica diagonal assumption works well in the large-
N
limit and we
could neglect the difference between
exp(
−
I
KM
)
and
exp(
−
I
KM
)
. Consequently, we keep
the disorder average implicitly from now on.
To proceed, we use the standard trick by introducing bilocal fields
G
and
Σ
[28]. Since
fields with even or odd indices are in-equivalent, we should define two sets of fields
G
11
/
22
and
Σ
11
/
22
. The definition of
G
11
and
G
22
is
G
11
(
τ,τ
′
) =
2
N
∑
j
χ
2
j
−
1
(
τ
)
χ
2
j
−
1
(
τ
′
)
, G
22
(
τ,τ
′
) =
2
N
∑
j
χ
2
j
(
τ
)
χ
2
j
(
τ
′
)
.
(3.5)
Σ
11
and
Σ
22
are introduced in order to impose the relation between
G
and
χχ
:
δ
(
G
11
−
2
N
∑
i
∈
odd
χ
i
χ
i
)
=
∫
D
Σ
22
e
1
2
∫
dτdτ
′
Σ
11
(
τ,τ
′
)
(
∑
odd
χ
i
(
τ
)
χ
i
(
τ
′
)
−
N
2
G
11
(
τ,τ
′
)
)
,
δ
(
G
22
−
2
N
∑
i
∈
even
χ
i
χ
i
)
=
∫
D
Σ
22
e
1
2
∫
dτdτ
′
Σ
22
(
τ,τ
′
)
(
∑
even
χ
i
(
τ
)
χ
i
(
τ
′
)
−
N
2
G
22
(
τ,τ
′
)
)
.
(3.6)
Then by integrating out the Majorana fields, we find
e
−
I
KM
=
∫
D
G
11
D
G
22
D
Σ
11
D
Σ
22
exp(
−
S
eff
KM
[
G,
Σ])
.
(3.7)
Here the effective
G
-
Σ
action
S
KM
is given by
4
S
eff
KM
N
=
−
1
4
log det
b.c.
(
∂
τ
−
Σ
11
0
0
∂
τ
−
Σ
22
)
−
J
2
2
q
∫
dτdτ
′
(
G
11
(
τ,τ
′
) +
G
22
(
τ,τ
′
)
2
)
q
+
1
4
∫
dτdτ
′
G
11
(
τ,τ
′
)Σ
11
(
τ,τ
′
) +
1
4
∫
dτdτ
′
G
22
(
τ,τ
′
)Σ
22
(
τ,τ
′
)
.
(3.8)
Here b.c. denotes the boundary condition 3.4. Note that although the self-energy is blocked
diagonal, the boundary condition would mix modes with even/odd indices. In the large-
N
4
There could be additional boundary terms. Nevertheless, they cancel out when computing the entan-
glement entropy.
– 5 –
limit, we take the saddle point of (3.15). The saddle point equation reads
(
G
11
G
12
G
21
G
22
)
=
(
∂
τ
−
Σ
11
0
0
∂
τ
−
Σ
22
)
−
1
b.c.
,
Σ
11
(
τ,τ
′
) = Σ
22
(
τ,τ
′
) =
J
2
(
G
11
(
τ,τ
′
) +
G
22
(
τ,τ
′
)
2
)
q
−
1
.
(3.9)
Here we have also introduced
G
12
(
τ,τ
′
) =
2
N
∑
j
〈
T
τ
χ
2
j
−
1
(
τ
)
χ
2
j
(
τ
′
)
〉
, G
21
(
τ,τ
′
) =
2
N
∑
j
〈
T
τ
χ
2
j
(
τ
)
χ
2
j
−
1
(
τ
′
)
〉
,
for completeness. In terms of bilocal fields
G
ab
with
a,b
∈ {
1
,
2
}
, the boundary condition
(3.4) becomes
G
1
a
(0
,τ
) =
−
iG
2
a
(0
,τ
)
, G
1
a
(
β,τ
) =
iG
2
a
(
β,τ
)
,
G
a
1
(
τ,
0) =
−
iG
a
2
(
τ,
0)
, G
a
1
(
τ,β
) =
iG
a
2
(
τ,β
)
.
(3.10)
Solving the equation (3.9) with boundary condition (3.10), and substituting the solution
into (3.15) already gives the on-shell action
I
KM
and thus
Z
KM
. However, it is more
convenient to introduce a different parametrization of the contour. The key observation is
that if we define a Majorana field
χ
j
(
s
)
with parameter
s
∈
[0
,
2
β
)
:
0
2
β
β
χ
(
s
)
:
χ
j
(
s
) =
{
χ
2
j
(
s
)
for
s
∈
[0
,β
)
−
iχ
2
j
−
1
(2
β
−
s
)
for
s
∈
[
β,
2
β
)
,
(3.11)
here
j
= 1
,
2
...N/
2
, the boundary condition (3.4) becomes the traditional continuous and
anti-periodic boundary condition
χ
j
(2
β
−
) =
−
χ
j
(0
+
)
, as for a thermal ensemble. The
Green’s function
G
(
s,s
′
) =
〈T
C
χ
j
(
s
)
χ
j
(
s
′
)
〉
is then given by
G
(
s,s
′
) =
(
G
22
(
s,s
′
)
−
iG
21
(
s,
2
β
−
s
′
)
−
iG
12
(2
β
−
s,s
′
)
−
G
11
(2
β
−
s,
2
β
−
s
′
)
)
.
(3.12)
The self-consistent equation for
G
(
s,s
′
)
is then
G
(
s,s
′
) = (
∂
s
−
Σ)
−
1
(
s,s
′
)
≡
(
∂
s
−
Σ
22
(
s,s
′
)
0
0
∂
s
+ Σ
11
(2
β
−
s,
2
β
−
s
′
)
)
−
1
.
(3.13)
Moreover, both the action (3.15) and boundary condition (3.4) are invariant under
χ
2
j
−
1
→
χ
2
j
and
χ
2
j
→ −
χ
2
j
−
1
. As a result, we have
G
11
(
τ,τ
′
) =
G
22
(
τ,τ
′
)
. Instead of (3.9), we
can then use
Σ(
s,s
′
) =
J
2
G
q
−
1
(
s,s
′
)
P
(
s,s
′
)
,
P
(
s,s
′
) =
[
θ
(
β
−
s
)
θ
(
β
−
s
′
) +
θ
(
s
−
β
)
θ
(
s
′
−
β
)
]
.
(3.14)
Here
P
(
s,s
′
)
is a projector. We have
P
(
s,s
′
) = 1
if both
χ
(
s
)
and
χ
(
s
′
)
represents the same
field (
χ
1
or
χ
2
) and otherwise zero. This definition for
P
(
s,s
′
)
is more general if we choose
– 6 –
a different
s
= 0
point on the contour (3.11). Note that comparing to a thermofield double
state with inverse temperature
2
β
and
N/
2
fermions, the main difference is the presence of
P
(
s,s
′
)
, which breaks the time translational invariance.
We can then choose to solve the equation (3.14) and
G
= (
∂
s
−
Σ)
−
1
self-consistently.
Moreover, we could also express the on-shell action in terms of
G
(
s,s
′
)
and
Σ(
s,s
′
)
:
I
KM
N
=
1
4
log det
G
+
q
−
1
4
q
∫
dsds
′
G
(
s,s
′
)Σ(
s,s
′
)
.
(3.15)
This gives an alternative route to compute
Z
KM
.
3.2 Computing
S
(
n
)
A
Having illustrated the trick of parameterizing the contour by
s
, we consider the path-integral
representation of
S
(
n
)
A
in this subsection.
To compute
S
(2)
A
, we first consider the path-integral representation of
|
KM
〉〈
KM
|
. Sep-
arating out the modes in system
A
and
B
, a graphic representation is
|
KM
〉〈
KM
|
=
A
B
χ
A
1
χ
A
2
χ
B
2
χ
B
1
|
KM
〉
〈
KM
|
0
β
2
,
(3.16)
here the red/blue solid line represents the contour for subsystem
A/B
.
χ
S
1
/
2
represents
Majorana fermions in subsystem
S
with odd/even indices, with
S
∈{
A,B
}
.
The unnormalized density matrix
̃
ρ
A
=
tr
B
|
KM
〉〈
KM
|
is then given by tracing out the
B
subsystem or, graphically, by connecting the (blue) contours of
B
. tr
A
̃
ρ
n
A
can then be
computed by sewing
n
copies of
̃
ρ
A
. To be concrete, in this work we focus on the
n
= 2
case. The corresponding contour is then given by
tr
A
̃
ρ
2
A
=
tr
A
(
tr
B
|
KM
〉〈
KM
|
)
2
=
A
B
A
B
̃
ρ
A
̃
ρ
A
0
2
β
β
2
β
4
β
3
β
χ
A
(
s
)
χ
B
(
s
)
,
(3.17)
where the symmetry of interchanging
A
and
B
becomes obvious. Here we have parametrized
the contour by
s
∈
[0
,
4
β
]
anticlockwise. We define
χ
S
(
s
)
∝
χ
S
2
(
s
)
for
s
∈
[
β/
2
,
3
β/
2]
∪
[5
β/
2
,
7
β/
2]
and
χ
S
(
s
)
∝
χ
S
1
(
s
)
otherwise. Similar to the previous section, the boundary
condition again becomes the traditional continuous and anti-periodic boundary condition
for
χ
S
:
χ
A
(0
+
) =
−
χ
A
(2
β
−
)
, χ
A
(2
β
+
) =
−
χ
A
(4
β
−
)
,
χ
B
(0
+
) =
−
χ
B
(4
β
−
)
, χ
B
(
β
−
) =
χ
B
(3
β
+
)
, χ
B
(
β
+
) =
−
χ
B
(3
β
−
)
.
(3.18)
– 7 –
Similar to the previous subsection, we consider the Green’s function for
χ
S
:
G
A/B
(
s,s
′
) =
〈
T
C
χ
A/B
(
s
)
χ
A/B
(
s
′
)
〉
.
The Schwinger-Dyson equation then reads
G
A
(
s,s
′
) = (
∂
τ
−
Σ
A
)
−
1
A
(
s,s
′
)
, G
B
(
s,s
′
) = (
∂
τ
−
Σ
B
)
−
1
B
(
s,s
′
)
,
Σ
A
(
s,s
′
) = Σ
B
(
s,s
′
) =
J
2
(
λG
A
(
s,s
′
) + (1
−
λ
)
G
B
(
s,s
′
))
q
−
1
P
(2)
(
s,s
′
)
.
(3.19)
Here the
A/B
labels different boundary conditions for
A/B
subsystem. This self-energy
can be directly understood by the melon diagram (Here
q
= 4
for example):
Σ
A
(
s,s
′
) = Σ
B
(
s,s
′
) =
s
s
′
A/B
A/B
A/B
.
(3.20)
with
λ
or
1
−
λ
being the probability of having a mode in subsystem
A
or
B
. We have
P
(2)
(
s,s
′
) = 1
if both
χ
S
(
s
)
and
χ
S
(
s
′
)
represents the same field (
χ
S
1
or
χ
S
2
) and otherwise
zero. Note that the main difference between this expression of self-energy and that for
computing the subsystem Rényi entropy of thermal ensembles [37] is the presence of
P
(2)
.
These set of equations for
G
A/B
and
Σ
A/B
can also be directly derived by writing out
the
G
−
Σ
action and taking the saddle point approximation. Consequently, after solving
(3.19), we have tr
A
̃
ρ
2
A
=
e
−
I
(2)
with:
I
(2)
N
=
λ
4
log det
G
A
(
s,s
′
) +
λ
(
q
−
1)
4
q
∫
dsds
′
G
A
(
s,s
′
)Σ
A
(
s,s
′
)+
1
−
λ
4
log det
G
B
(
s,s
′
) +
(1
−
λ
)(
q
−
1)
4
q
∫
dsds
′
G
B
(
s,s
′
)Σ
B
(
s,s
′
)
.
(3.21)
We have
I
(2)
(
λ
= 0) =
I
(2)
(
λ
= 1) = 2
I
KM
. The second Rényi entanglement entropy is
then given by
S
(2)
A
=
−
log(
tr
A
ρ
2
A
) =
−
log
(
tr
A
̃
ρ
2
A
Z
2
KM
)
=
I
(2)
−
2
I
KM
.
(3.22)
The generalization of above discussions to
n
-th Rényi entropy is straightforward.
4 Numerical results
Because of the lack of translational invariance, the analytical study of (3.19) is difficult. In
this section, we present numerical results for the entanglement entropy with different
q
and
T/J
.
We perform the numerical iteration of (3.19) similar to that in [37, 40]. We discretize
the time
s
into
L
points, with
ds
= 4
β/L
. For
β
= 50
, we typically take
L
∼
400
∼
600
.
The equation (3.19) then becomes a matrix equation:
(
G
A
)
ij
=
(
(
G
A
0
)
−
1
−
Σ
A
)
−
1
ij
,
(
G
B
)
ij
=
(
(
G
B
0
)
−
1
−
Σ
B
)
−
1
ij
,
(
Σ
A
)
ij
=
(
Σ
B
)
ij
=
J
2
ds
2
(
λ
(
G
A
)
ij
+ (1
−
λ
)
(
G
B
)
ij
)
q
−
1
P
(2)
ij
.
(4.1)
– 8 –
(
a
)
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(
b
)
<latexit sha1_base64="b/cMj/3MakpfvQl5kysPOUFaeE8=">AAACxnicjVHLSsNAFD2Nr1pfVZdugkWom5KooMuimy4r2gfUIsl0WoemSUgmSimCP+BWP038A/0L74xTUIvohCRnzr3nzNx7/TgQqXSc15w1N7+wuJRfLqysrq1vFDe3mmmUJYw3WBRESdv3Uh6IkDekkAFvxwn3Rn7AW/7wTMVbtzxJRRReynHMuyNvEIq+YJ4k6qLs718XS07F0cueBa4BJZhVj4ovuEIPERgyjMARQhIO4CGlpwMXDmLiupgQlxASOs5xjwJpM8rilOERO6TvgHYdw4a0V56pVjM6JaA3IaWNPdJElJcQVqfZOp5pZ8X+5j3RnupuY/r7xmtErMQNsX/pppn/1alaJPo40TUIqinWjKqOGZdMd0Xd3P5SlSSHmDiFexRPCDOtnPbZ1ppU16566+n4m85UrNozk5vhXd2SBuz+HOcsaB5U3MPKwflRqXpqRp3HDnZRpnkeo4oa6miQ9wCPeMKzVbNCK7PuPlOtnNFs49uyHj4ARM2P0A==</latexit>
=
=
=
=
=
(
)
/
=
=
=
=
(
)
/
Figure 1
. (a). The entanglement entropy
S
(2)
A
/N
of KM states at different temperature
T/J
with
q
= 4
. The black dashed line is the analytical approximation for (2.4):
S
(2)
A
(
λ
)
/N
=
x
(log(2)
/
2
−
arcsin(
x
3
/
2
)
/
16)
with
x
=
min
{
λ,
1
−
λ
}
. The red dashed line is the subsystem entropy for a
thermal ensemble with
βJ
= 50
[37]. (b). The entanglement entropy
S
(2)
A
/N
of KM states at
different temperature
T/J
with
q
= 2
. The black dashed line is the analytical formula for the SYK
2
ground state [37].
Here
G
A
0
and
G
B
0
are the Green’s functions without interaction
J
on the contour (3.17).
Their elements are either
±
1
/
2
or
0
depending on the time ordering and the connectivity
of contours. Explicitly:
(
G
A
0
)
ij
=
1
2
sign
(
i
−
j
)
for
{
i,j
}⊂
[1
,L/
2]
or
[
L/
2 + 1
,L
]
,
(
G
B
0
)
ij
=
1
2
sign
(
i
−
j
)
for
{
i,j
}⊂
[1
,L/
4]
∪
[3
L/
4 + 1
,L
]
or
[
L/
4 + 1
,
3
L/
4]
.
(4.2)
The on-shell action is then
I
(2)
N
=
λ
4
log det
[
G
A
(
G
A
0
)
−
1
]
+
λ
(
q
−
1)
4
q
tr
[
G
A
(Σ
A
)
T
]
−
1
2
log 2
+
1
−
λ
4
log det
[
G
B
(
G
B
0
)
−
1
]
+
(1
−
λ
)(
q
−
1)
4
q
tr
[
G
B
(Σ
B
)
T
]
.
(4.3)
Here we have used
log det(
G
A
0
)
−
1
= log det(
G
B
0
)
−
1
= 2 log 2
to enable the convergence. The
extrapolation towards
1
/L
→
0
is performed finally.
Now we present numerical results for the entanglement entropy of KM pure states. We
first focus on the SYK
q
model with
q
= 4
or
q
= 2
as an example of strongly interacting
systems or non-interacting systems.
The result of
S
(2)
A
for different
βJ
with
q
= 4
is shown in Figure 1 (a). We have
also plotted the analytical approximation [34] as the black dashed line and the subsystem
entropy for a thermal ensemble with
βJ
= 50
[37] as the red dashed line. For small
βJ
,
the entanglement builds up quickly as
βJ
increases, and
S
(2)
A
/N
is an analytical function
of
λ
. On the other side, for large
βJ
&
30
near
λ
∼
1
/
2
, there exist two different saddle
point solutions, and the coexistence region becomes larger as
βJ
increases. The true curve
for
S
(2)
A
is determined by the comparing actions of different saddles, leading a first-order
transition. As we will see in the next section, these two saddle points smoothly connected to
– 9 –