Supplementary Materials for “ Efficient Simulation of Low Temperature Physics in
One-Dimensional Gapless Systems ”
Yuya Kusuki,
1, 2
Kotaro Tamaoka,
3
Zixia Wei,
4, 2, 5
and Yasushi Yoneta
6, 7
1
Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA
2
Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), RIKEN, Wako, Saitama 351-0198, Japan
3
Department of Physics, College of Humanities and Sciences, Nihon University, Tokyo 156-8550, Japan
4
Center for the Fundamental Laws of Nature & Society of Fellows,
Harvard University, Cambridge, MA 02138, USA
5
Yukawa Institute for Theoretical Physics, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan
6
Center for Quantum Computing, RIKEN, Wako, Saitama 351-0198, Japan
7
Department of Basic Science, The University of Tokyo, Meguro, Tokyo 153-8902, Japan
PROOF OF THEOREM 1
Here we present a detailed proof for Theorem 1. appearing in the main text. As explained in the main text, the
proof utilizes concavity of R ́enyi entropy for R ́enyi index 0
< q <
1 and that of von Neumann entropy. The proof
presented here is parallel to the proof of theorem 7 in [1], which is a related statement for von Neumann entropy.
Proof.
Take sets of product states
{|
P
i
⟩
A
}
i
and
{|
P
j
⟩
B
}
j
of subsystems
A
and
B
, respectively, so that
{|
P
i
⟩}
i
=
{|
P
i
⟩
A
⊗|
P
j
⟩
B
}
i,j
. Consider the projection
K
j
=
I
A
L
⊗
I
B
L
⊗
I
A
R
⊗|
P
j
⟩⟨
P
j
|
B
R
.
(S1)
Then, from the completeness of
K
j
, we get
Tr
B
L
B
R
[
|
TFD
⟩⟨
TFD
|
]
=
X
j
Tr
B
L
B
R
h
K
j
|
TFD
⟩⟨
TFD
|
K
j
†
i
.
(S2)
Therefore, we have
S
(
q
)
A
L
A
R
(
|
TFD
⟩
) =
S
(
q
)
X
j
s
j
ρ
K
j
|
TFD
⟩
A
L
A
R
,
(S3)
where
s
j
=
⟨
TFD
|
K
j
|
TFD
⟩
⟨
TFD
|
TFD
⟩
.
(S4)
Further, using the matrix concavity of the R ́enyi entropy
S
(
q
)
with 0
< q <
1 [2] and with
q
= 1 (i.e., the concavity
of the von Neumann entropy), we obtain
S
(
q
)
A
L
A
R
(
|
TFD
⟩
)
≥
X
j
s
j
S
(
q
)
A
L
A
R
(
K
j
|
TFD
⟩
)
.
(S5)
In the same way, we also obtain
S
(
q
)
A
L
A
R
(
K
j
|
TFD
⟩
) =
S
(
q
)
B
L
B
R
(
K
j
|
TFD
⟩
)
≥
X
j
r
ij
S
(
q
)
B
L
B
R
(
J
i
K
j
|
TFD
⟩
)
=
X
j
r
ij
S
(
q
)
A
L
A
R
(
J
i
K
j
|
TFD
⟩
)
,
(S6)
2
where
J
i
=
I
A
L
⊗
I
B
L
⊗|
P
i
⟩⟨
P
i
|
A
R
⊗
I
B
R
,
(S7)
r
ij
=
⟨
TFD
|
J
i
K
j
|
TFD
⟩
⟨
TFD
|
K
j
|
TFD
⟩
.
(S8)
Thus, we have
S
(
q
)
A
L
A
R
(
|
TFD
⟩
)
≥
X
i,j
r
ij
s
j
S
(
q
)
A
L
A
R
(
J
i
K
j
|
TFD
⟩
)
.
(S9)
Using the decomposition of
|
TFD
⟩
shown in (6),
J
i
K
j
|
TFD
⟩
=
e
−
βH/
2
|
P
ij
⟩
L
⊗|
P
ij
⟩
R
,
(S10)
r
ij
s
j
=
⟨
P
ij
|
e
−
βH
|
P
ij
⟩
P
k
P
l
⟨
P
kl
|
e
−
βH
|
P
kl
⟩
≡
p
ij
,
(S11)
where
|
P
ij
⟩≡|
P
i
⟩
A
⊗|
P
j
⟩
B
. Then, from the additivity and non-negativity of the R ́enyi entropy, we have
S
(
q
)
A
L
A
R
(
|
TFD
⟩
)
≥
X
i,j
p
ij
S
(
q
)
A
L
A
R
(
|
μ
ij
⟩
L
⊗|
P
ij
⟩
R
)
=
X
i,j
p
ij
n
S
(
q
)
A
(
|
μ
ij
⟩
) +
S
(
q
)
A
(
|
P
ij
⟩
)
o
≥
X
i,j
p
ij
S
(
q
)
A
(
|
μ
ij
⟩
)
.
(S12)
Hence (7) is shown for 0
< q
≤
1.
FINITE-SIZE EFFECTS ON THE AVERAGE ENTANGLEMENT ENTROPY OF METTS
0
0
.
2
0
.
4
0
.
6
0
.
8
1
.
0
10
100
1000
S
(
q
)
A
N
q
= 0
.
5
q
→
1
q
= 2
(a) Ising model (
ν
=
x
)
0
0
.
2
0
.
4
0
.
6
0
.
8
1
.
0
10
100
1000
S
(
q
)
A
N
q
= 0
.
5
q
→
1
q
= 2
(b) Ising model (
ν
=
z
)
0
0
.
2
0
.
4
0
.
6
0
.
8
1
.
0
10
100
1000
S
(
q
)
A
N
q
= 0
.
5
q
→
1
q
= 2
(c) Heisenberg model
FIG. S1.
N
dependence of the average entanglement entropy of METTS for the half-chain
A
=
{
1
,
2
,
···
,N/
2
}
in the
critical transverse-field Ising model (a, b) and in the critical Heisenberg model with the next-nearest-neighbor interaction (c).
Imaginary-time evolution is carried out using the second-order Trotter decomposition with time step
δτ
= 0
.
04 for the Ising
model and
δτ
= 0
.
01 for the Heisenberg model. The data is average over 10000 samples for the Ising model and 5000 samples
for the Heisenberg model.
In the main text, we perform numerical calculations on sufficiently large systems to neglect finite-size effects when
investigating the inverse temperature dependence of the average entanglement entropy of METTS. Here, we examine
the finite-size effects on the average entanglement entropy of METTS and justify the parameters adopted in our
numerical calculations.
In the main text, numerical calculations are conducted for METTS at inverse temperatures ranging from 0
≤
β
≤
β
max
= 128 for the critical transverse-field Ising model and 0
≤
β
≤
β
max
= 16 for the critical Heisenberg model with
3
next-nearest-neighbor interaction. Then, in Fig. S1, we plot the system size dependence of the average entanglement
entropy of METTS at the lowest temperature
β
=
β
max
, where finite-size effects are most prominent within that range,
for each model. We can confirm that setting
N
= 640 for the critical transverse-field Ising model and
N
= 1280 for
the critical Heisenberg model with next-nearest-neighbor interaction is sufficient to ensure that finite-size effects can
be adequately neglected.
[1] Z. Wei and Y. Yoneta, Counting atypical black hole microstates from entanglement wedges, JHEP
05
, 251, arXiv:2211.11787
[hep-th].
[2] A. E. Rastegin, Some General Properties of Unified Entropies, J. Stat. Phys.
143
, 1120 (2011), arXiv:1012.5356.