２次元可解量子系のエンタングルメント特性

２次元可解量量⼦子系の
エンタングルメント特性
⽥田中宗 (京⼤大基研)
Valence-‐‑‒Bond-‐‑‒Solid (VBS) 状態や、
Quantum hard-‐‑‒square モデルの基底状態について、
エンタングルメント特性を検討した。
その結果、２次元系のエンタングルメント特性と
１次元量量⼦子系の間に対応関係があることを⾒見見出した。
VBS on symmetric graphs, J. Phys. A, 43, 255303 (2010)
“VBS/CFT correspondence”, Phys. Rev. B, 84, 245128 (2011)
Quantum hard-square model, Phys. Rev. A, 86, 032326 (2012)
Nested entanglement entropy, Interdisciplinary Information Sciences, 19, 101 (2013)

共同研究者の⽅方々
桂法称 (東⼤大院理理)
Anatol N. Kirillov (RIMS, 京⼤大)
Vladimir E. Korepin (YITP, Stony Brook, USA)
川島直輝 (東⼤大物性研)
Lou Jie (Fudan Univ., 中国)
⽥田村亮亮 (NIMS)

ダイジェスト
正⽅方格⼦子VBS状態ラダー上量量⼦子格⼦子気体模型
排除体積効果
注⽬目する物理理系
VBS状態
正⽅方格⼦子 1D AF Heisenberg
蜂の巣格⼦子 1D F Heisenberg
量量⼦子格⼦子気体模型
正⽅方形ラダー 2D Ising
三⾓角形ラダー 2D 3-state Potts
1次元量量⼦子系
物理理特性 (臨臨界現象)
２次元量量⼦子系
蜂の巣格⼦子VBS状態
JPA, 43, 255303 (2010), PRB 84, 245128 (2011), PRA 86, 032326 (2012), Interdisciplinary Information Sciences, 19, 101 (2013)

導⼊入
エンタングルメント
⽬目的
数学的準備

FIG. 5. (Color online) Tensor network structure with a single
ER level and a top tensor for six coarse-grained sites. Big
solid circles denote positions of sites on the coarse-grained
lattice. The top tensor is put on the parallelogram frame.
All parallelogram frames are the same by a skew periodic
arrangement.
FIG. 6. (color o
ttes. The value
ﬁned by neares
is shown by th
respondence be
sults of the PB
Kenji Harada
Multiscale entanglement
renormalization ansatz (MERA)
University of Stuttgart
Quantum spin systems
Yazdani lab
Topological insulator
Joel E. Moore
D-Wave Systems
Quantum information processing
Karlsruhe Institute of Technology
Single molecule magnet
Computation physicsCondensed matter physics &
statistical physics
H. Bombin et al.
Toric code
EQUINOX
GRAPHICS
Cluster state
Quantum Hall systems
Strongly correlated fermionic/bosonic systems
Li-Haldane conjecture
CFT correspondence
Density matrix renormalization group (DMRG)
Tensor network
Projected entangled pair states (PEPS)
NV center in diamond
S. Benjamin and
J. Smith

Entanglement
２スピン系、ハイゼンベルク模型 H = J ˆ1 · ˆ2, (J > 0)
J
1 2
部分系 B部分系 A
Energy
縮約密度度⾏行行列列 (⽚片⽅方の部分系の⾃自由度度に関する部分和を実⾏行行)
A = TrB = 2 |s s| 2 + 2 |s s| 2 =
1
2
1 0
0 1
Entanglement entropy (e-entropy)

Entanglemententropy
全系
Schmidt 分解
規格化された
状態ベクトル
von Neumann entanglement entropy
|
部分系に分割部分系
A
部分系
B
各々の部分系における
正規直交基底
| =
=1
| [A]
| [B]
縮約密度度⾏行行列列
A = TrB | | =
=1
2
| [A] [A]
|
⾮非負実数
最⼤大エンタングル状態
S = ln
1
2
(| | + | | )例例:
S = Tr A ln A =
=1
2
ln 2
2
=
1

Symmetriccase
部分系
A
部分系
B
| = | [A]
| [B]
Pre-‐‑‒Schmidt 分解
規格化された
状態ベクトル
線形独⽴立立な基底
(⼀一般に⾮非直交)
部分系Aと部分系Bとを(鏡映)対称軸で
分割した場合を考える。
重なり⾏行行列列 (overlap matrix)
(M[A]
) := [A]
|
[A]
(M[B]
) := [B]
|
[B]
部分系Aと部分系Bとが対称
M[A]
= M[B]
=: M
p =
d2
d2
S = p ln p ,
e-entropy in symmetric cases
重なり⾏行行列列 M の固有値: {d }

e-entropyin1Dsystems
e-entropy の熱⼒力力学的極限での振る舞いと物理理特性の関係
state energy characterizing a phase transition already
occurs for a finite chain. Correspondingly, already for a
chain of N 20 spins it is possible to observe a distinct,
characteristic behavior of SL depending on whether the
values ; in Eq. (2) belong or not to a critical regime.
hcmcni mn i mn
any other expectatio
Wick’s theorem, in t
2
6
6
6
6
6
6
4
0
1
..
.
1 N
l
0
g l
with real coefficien
(N ! 1), by
gl
1
2
Z 2
0
d e
From Eqs. (6) and
of Eq. (4) as follows
the rows and column
chain that do not belo
10 20 30 40
NUMBER OF SITES − L −
1
1.5
2
2.5
ENTROPY−S−
FIG. 1. Noncritical entanglement is characterized by a satu-
ration of SL as a function of the block size L: noncritical Ising
chain (empty squares), HXY a 1:1; 1 ; noncritical XXZ
chain (filled squares), HXXZ 2:5; 0 . Instead, the en-
G. Vidal et al. PRL 90, 227902 (2003)
XXZ( = 2.5, = 0)
XY(a = 1.1, = 1)
L A B
XY(a = 1, = 1)
XY(a =
, = 0)
XXZ( = 1, = 0)
HXXZ =
i
x
i
x
i+1 + y
i
y
i+1 + z
i
z
i+1
z
i
HXY =
i
a
2
(1 + ) x
i
x
i+1 + (1 ) y
i
y
i+1 + z
i
S(L) =
c
3
ln L + S0
中心電荷
臨界系の場合
ギャップ系の場合
S(L) const.

Entanglementspectrum
全系
Schmidt 分解
規格化された
状態ベクトル
|
部分系に分割部分系
A
部分系
B
正規直交基底
| =
=1
| [A]
| [B]
A = TrB | | =
=1
2
| [A] [A]
|
⾮非負実数
=
=1
e | [A] [A]
|
A = e HE
熱平衡状態の密度度⾏行行列列とみなす
Entanglement Hamiltonian
(e-Hamiltonian)
Entanglement spectrum
(e-spectrum)
e-Hamiltonianの固有値
= ln 2
H. Li and M. Haldane, PRL 101, 010504 (2008)

e-spectrumintopologicalsystems
gure 2 shows the spectra of the system of the same size
Fig. 1, i.e., Ne 16 and Norb 30, but for the ground
of the Coulomb interaction projected into the second
au level, obtained by direct diagonalization.
estingly, the low-lying levels have the same counting
ure as the corresponding Moore-Read case. We iden-
hese low-lying levels as the ‘‘CFT’’ part of the spec-
分数量量⼦子ホール効果
F. Pollmann et al., PRB 81, 064439 (2010)
T. Ohta, S. Tanaka, I. Danshita, and K. Totsuka, in preparation.
␹=80 for the simulations. The double degeneracy of the en-
tanglement spectrum is used to identify the Haldane phase.
Example 1. We begin with the original Hamiltonian H0 in
Eq. ͑3͒. This model is translation invariant, invariant under
spatial inversion, under e−i␲Sx
and under e−i␲Sy
ϫTR. Using
the above argument, we know that inversion symmetry alone
−1 −0.5 0 0.5 1 1.5 2
0
2
4
6
8
10
12
Uzz
/J
−2log()
Z
2
z
Haldane TRI
FIG. 2. ͑Color online͒ Entanglement spectrum of Hamiltonian
H0 in Eq. ͑3͒ for Bx=0 ͑only the lower part of the spectrum is
shown͒. The dots show the multiplicity of the Schmidt values,
which is even in the entire Haldane phase.
OPOLOGICAL PHASE… PHYSICAL REVIEW B 81, 064439 ͑2010͒
J. Phys. Soc. Jpn.
0
3
6
9
0 0.5 1 1.5 2
ξi
JYY
/JXZX
JYZY
/JXZX
0
2
4
6
0.5 1 1.5
ξi
(b)
(a)
(c)
Subsystem A Subsystem BSubsystem B
Lsub dependent Hamiltonian
H(t) = −JXZX
N
i=1
σx
i σz
i+1σx
i+2 + J(t)
where the interaction parameter chang
speed as
J(t)/JXZX
= 2t/τ, 0 ≤
We call τ as the sweep time. The system
ical point at t = 0.5τ. Near the critical
of the relaxation time prevents the syste
cally, which implies the ﬁnal state after
number of defects. We study how the de
dependence of the string correlation fu
量量⼦子スピン系
トポロジカル相
⾮非Landau型相
　局所秩序変数による
　検出不不可
　⾮非局所秩序変数による記述
　ストリング秩序変数等
　 e-spectrum の基底状態の
　縮退数
e-spectrum の準位構造と
端に⽣生じる低エネルギー
励起構造の類似性

⽬目的
２次元量量⼦子系における新しいエンタングルメント特性の発⾒見見
基底状態が厳密に構成できる統計⼒力力学模型に着⽬目
２次元 Valence-Bond-Solid (VBS) 状態
量量⼦子 hard-‐‑‒square modelの基底状態
e-spectrum に続く新しい概念念の検討
e-Hamiltonian の基底状態に注⽬目し、エンタングルメントを⾒見見る

VBS状態
VBS状態の性質
エンタングルメントの性質
(e-entropy, e-spectrum, nested e-entropy)
１次元量量⼦子系との対応

Valence-Bond-Solid (VBS) 状態
Valence bond = singlet pair
1
2
(| | )
VBS 状態
Valence bond で
敷き詰められた状態
Valence bond
S = 1 への射影
１次元VBS状態
H =
i
Si · Si+1 +
1
3
Si · Si+1
2
(S = 1)
１次元VBS状態を基底状態とするHamiltonian (AKLT model)
I. Aﬄeck, T. Kennedy, E. Lieb, and H. Tasaki, PRL 59, 799 (1987).

H =
i
Si · Si+1 +
1
3
Si · Si+1
2
(S = 1)
１次元VBS状態を基底状態とするHamiltonian (AKLT model)
I. Aﬄeck, T. Kennedy, E. Lieb, and H. Tasaki, PRL 59, 799 (1987).
M. Yamashita et al., Coordination Chemistry Reviews 198, 347 (2000).
S=1 VBS状態が厳密な基底状態である。
S=1 VBS状態が唯⼀一の基底状態である。
Haldane gapの厳密な証明を与える。
反強磁性相関関数が指数関数的に減衰する。
MnCl3bpy NINO

F. Verstraete and J. I. Cirac, PRA 70, 060302 (2004)

D. Gross and J. Eisert, PRL 98, 220503 (2007)
T-C. Wei et al., PRL 106, 070501 (2011)
A. Miyake, Ann. Phys. 326, 1656 (2011)
正⽅方格⼦子蜂の巣格⼦子
VBS状態を⽤用いた測定型量量⼦子計算(MBQC)
VBS state = singlet-covering state

Schwinger boson representation
(スピンと２種類のボゾンの関係)
| = a†
|vac , | = b†
|vac
n
(b)
k = b†
kbk
n
(a)
k = a†
kak
0 1 2 3 4
0
1
2
3
4
S=0 1/2 1 3/2 2
a†
kak + b†
kbk = 2Sk
a ボゾンの個数
b ボゾンの個数
a,b ボゾンの個数の和に関する
拘束条件
|VBS =
k,l
a†
kb†
l b†
ka†
l |vac
VBS state
a†
kak + b†
kbk = 2Sk

部分系
A
部分系
B
鏡映対称軸で分割

|VBS =
k,l
a†
kb†
l b†
ka†
l |vac
=
{ }
| [A]
| [B]
局所ゲージ変換
鏡映対称性
{ } = { 1, · · · , | A|}
補助空間のスピン： i = ±1/2
端ボンドの本数
重なり⾏行行列列 : ⾏行行列列M{ },{ } 2| A|
2| A|
重なり⾏行行列列の各要素：モンテカルロ法で求める。
SU(N)の場合にも容易易に適⽤用可能な⽅方法を構築
J. Lou, S. Tanaka, H. Katsura, and N. Kawashima, PRB, 84, 245128 (2011)

H. Katsura, J. Stat. Mech. P01006 (2015)

エンタングルメント
e-entropy
e-spectrum
nested e-entropy

２次元 VBS 状態
Lx
Ly
開放端
周期境界条件

S=1
S=3
S=4
S=6
S=8
S=2
１次元 VBS 状態の e-entropy
H. Katsura, T. Hirano, and Y. Hatsugai, PRB 76, 012401 (2007).
|VBS =
N
i=0
a†
i b†
i+1 b†
i a†
i+1
S
|vac
S = ln (# edge states)
熱⼒力力学的極限
e-entropy
S = Tr A ln A =
=1
2
ln 2

２次元 VBS 状態の e-entropy
S
| A|
= ln 2
端ボンドの本数
0, 1D = 0, square > hexagonal
square > hexagonal
H. Katsura, T. Hirano, and
Y. Hatsugai, PRB 76, 012401 (2007).
Lx
Ly
開放端
周期境界条件
H. Katsura, N. Kawashima, A.N.Kirillov, V. E.Korepin, S. Tanaka, JPA, 43, 255303 (2010)

ENTANGLEMENT SPECTRA OF THE TWO-DIMENSIONAL . . .
0.58
0.6
0.62
0.64
0.66
S/Ly
(a) (b)Lx=1
Lx=2
Lx=3
Lx=4
Lx=5
0.684
0.685
0.686
0.687
2 4 6 8 10 12 14 16
S/Ly
(c)
2 4 6 8 10 12 14 16
(d)Lx=1
Lx=2
Lx=3
2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16
#Bonds #Bonds
S/#Bonds
S
Ly
= +
C1
Ly
+
C2
Ly ln Ly
S
Ly
exp( Ly/ )
. PHYSICAL REVIEW B 84, 245128 (2011)
TABLE I. Obtained ﬁtting parameters for entanglement entropy
by the method of least squares.
Square
OBC PBC
σ C1 C2 σ ξ
Lx = 1 0.61277(3) 0.0865(3) 0.0009(2) 0.6129(1) 1.54(5)
Lx = 2 0.59601(4) 0.1127(4) 0.0034(3) 0.5966(2) 2.4(1)
Lx = 3 0.59322(5) 0.1196(5) 0.0049(3) 0.5942(3) 2.9(2)
Lx = 4 0.59259(5) 0.1223(5) 0.006(3) 0.5936(4) 3.1(2)
Lx = 5 0.59246(4) 0.1227(5) 0.006(3) 0.5934(4) 3.1(2)
(ln 2 = 0.693147 · · · )
EE is less than ln 2.
S = Tr A ln A
正⽅方格⼦子 (PBC)正⽅方格⼦子 (OBC)

#Bonds #Bonds
S/#Bonds
0.58
0.6
0.684
0.685
0.686
0.687
2 4 6 8 10 12 14 16
S/Ly
Ly
(c)
2 4 6 8 10 12 14 16
Ly
(d)Lx=1
Lx=2
Lx=3
FIG. 2. (Color online) Entanglement entropy S per valence bond
across the boundary as a function of Ly for (a) square lattices with
rectangular geometry (OBC), (b) square lattices with cylindrical
geometry (PBC), (c) hexagonal lattices with OBC, and (d) hexagonal
lattices with PBC. A further increase of Lx does not affect the results
. . PHYSICAL REVIEW B 84, 245128 (2011)
6
d
h
l
l
s
n
TABLE I. Obtained ﬁtting parameters for entanglement entropy
by the method of least squares.
Square
OBC PBC
σ C1 C2 σ ξ
Lx = 1 0.61277(3) 0.0865(3) 0.0009(2) 0.6129(1) 1.54(5)
Lx = 2 0.59601(4) 0.1127(4) 0.0034(3) 0.5966(2) 2.4(1)
Lx = 3 0.59322(5) 0.1196(5) 0.0049(3) 0.5942(3) 2.9(2)
Lx = 4 0.59259(5) 0.1223(5) 0.006(3) 0.5936(4) 3.1(2)
Lx = 5 0.59246(4) 0.1227(5) 0.006(3) 0.5934(4) 3.1(2)
Hexagonal
OBC PBC
σ C1 C2 σ ξ
Lx = 1 0.68502(2) 0.0092(6) 0.0011(7) 0.68508(6) 0.4(1)
Lx = 2 0.68469(3) 0.0098(6) 0.0012(8) 0.684757(4) 0.6(1)
Lx = 3 0.68468(3) 0.0098(6) 0.0013(7) 0.684754(5) 0.7(1)
S
Ly
= +
C1
Ly
+
C2
Ly ln Ly
S
Ly
exp( Ly/ )
(ln 2 = 0.693147 · · · )
S = Tr A ln A
EE is less than ln 2.
蜂の巣格⼦子 (PBC)蜂の巣格⼦子 (OBC)

２次元 VBS 状態の e-spectrum
e-spectrum
e-Hamiltonian の固有値 = ln 2
LOU, TANAKA, KATSURA, AND KAWASHIMA
the FM spe
left panel c
spectrum in
the excitatio
spin-wave s
to the trans
theorem app
are exact eig
In order
the hologra
Heisenberg
nested enta
properties. O
0
(Lx=5, Ly=16)
(Lx=5, Ly=32)
J. I. Cirac, et al., PRB 83, 245134 (2011)

e-spectrum
the FM spe
left panel c
spectrum in
the excitatio
spin-wave s
to the trans
theorem app
are exact eig
In order
the hologra
Heisenberg
nested enta
properties. O
0
(Lx=5, Ly=16)
(Lx=5, Ly=32)
J. I. Cirac, et al., PRB 83, 245134 (2011)
des Cloizeaux-Pearson mode
1D AF Heisenberg model

e-spectrum
the FM spe
left panel c
spectrum in
the excitatio
spin-wave s
to the trans
theorem app
are exact eig
In order
the hologra
Heisenberg
nested enta
properties. O
0
(Lx=5, Ly=16)
(Lx=5, Ly=32)
J. I. Cirac, et al., PRB 83, 245134 (2011)
des Cloizeaux-Pearson mode
1D AF Heisenberg model
Spin wave
1D F Heisenberg model

Nestede-entropy
Entanglement ground state = e-Hamiltonianの基底状態
HE| 0 = Egs| 0
(e-ground state)
A = TrB | | =
=1
2
| [A] [A]
|
=
=1
e | [A] [A]
|
A = e HE
熱平衡状態の密度度⾏行行列列とみなす
Entanglement Hamiltonian
(e-Hamiltonian)
( ) := Tr +1,··· ,L [| 0 0|]
Nested reduced density matrix
Nested e-entropy
S( , L) = Tr1,··· , [ ( ) ln ( )]
１次元量量⼦子臨臨界系(周期境界条件)
P. Calabrese and J. Cardy, J. Stat. Mech. (2004) P06002.

Nestede-entropyENTANGLEMENT SPECTRA OF THE TWO-DIMENS
0.6
0.8
1
1.2
1.4
0 2 4 6 8 10 12 14 16
S(l,16)
l
(a)
s1=0.77(4)
c=1.01(7) Lx=5, Ly=16
fitting
0.5
0.6
0.7
0.8
0 2 4 6 8 1
S(l,16)
l
(b)
a=0.393(1)
c1/v=0.093(3)
Lx=5, Ly
fit
FIG. 4. (Color online) Nested entanglement entro
as a function of the subchain length ℓ for Lx = 5 an
(a) and (b) show results obtained for square VBS states wi
正⽅方格⼦子 (PBC)
Square ladd
(OBC)
A B
中⼼心電荷 c=1 1D AF Heisenberg model
VBS/CFT correspondence

基底状態
エンタングルメントの性質
(e-entropy, e-spectrum, nested e-entropy)
２次元古典系との対応
Quantum hard-square model, Phys. Rev. A, 86, 032326 (2012)

Rydberg Atom
Max Planck Institute
Ground state
Rydberg atom
(excited state)
Interaction
H =
i
x
i +
i
ni + V
i,j
ninj
|ri rj|

ハードコア格⼦子気体模型
ハードコア：１つのサイトに１粒粒⼦子まで⼊入ることが可能
排除体積効果
排除体積効果のあるハードコア格⼦子気体模型
コントロールパラメータ
化学ポテンシャル
排除体積効果の距離離
D. S. Gaunt and M. E. Fisher, J. Chem. Phys. 43, 2840 (1965)
R. J. Baxter et al., J. Stat. Phys. 22, 465 (1980)
S. Todo and M. Suzuki, Int. J. Mod. Phys. C7, 811 (1996)
R. J. Baxter, J. Phys. A, 13, L61 (1980)
R. J. Baxter, Exactly Solved Models in Statistical Mechanics (1982)

ハードコア格⼦子気体模型
ハードコア：１つのサイトに１粒粒⼦子まで⼊入ることが可能
排除体積効果
排除体積効果のあるハードコア格⼦子気体模型
コントロールパラメータ
排除体積効果の距離離
D. S. Gaunt and M. E. Fisher, J. Chem. Phys. 43, 2840 (1965)
R. J. Baxter et al., J. Stat. Phys. 22, 465 (1980)
S. Todo and M. Suzuki, Int. J. Mod. Phys. C7, 811 (1996)
R. J. Baxter, J. Phys. A, 13, L61 (1980)
R. J. Baxter, Exactly Solved Models in Statistical Mechanics (1982)
正⽅方形の敷き詰め問題
Hard-square model

量量⼦子ハードコア格⼦子気体模型
基底状態が厳密に得られる模型を構築
排除体積効果が強い場合に妥当な理理論論模型
P i :=
j Gi
(1 nj)
冷冷却原⼦子系
ground state 0 ↓
excited state
(Rydberg atom) 1 ↑
ni i
S. Ji et al. PRL 107, 060406 (2011)
I. Lesanovsky, PRL 108, 105301 (2012)
Hsol = z
i
( +
i + i )P i +
i
[(1 z)ni + z]P i
Hsol =
i
h†
i (z)hi(z), hi(z) := [ i z(1 ni)]P i
Rydberg原⼦子の⽣生成・消滅 Rydberg原⼦子間相互作⽤用(排除体積効果)

量量⼦子ハードコア格⼦子気体模型
H =
L
i=1
P z x
i + (1 3z)ni + zni 1ni+1 + z P1次元鎖の場合
拘束条件のある横磁場イジング模型
ゼロ・エネルギー状態 Positive semi-definite Hamiltonian: 固有値は⾮非負
真空状態
Perron-Frobeniusの定理より非縮退
|z =
1
(z) i
exp z +
i P i | · · ·
基底状態が厳密に得られる模型を構築
排除体積効果が強い場合に妥当な理理論論模型
Hsol = z
i
( +
i + i )P i +
i
[(1 z)ni + z]P i
Hsol =
i
h†
i (z)hi(z), hi(z) := [ i z(1 ni)]P i
Rydberg原⼦子の⽣生成・消滅 Rydberg原⼦子間相互作⽤用(排除体積効果)

基底状態
1 2 L
1 2 L
1 2 L
1 2 L
部分系 A
部分系 B
部分系 A
部分系 B
正⽅方形はしご格⼦子三角形はしご格⼦子
| (z) = [T(z)] , | |
[T(z)] , :=
L
i=1
w ( i, i+1, i+1, i)
a b
cd
= w(a, b, c, d)
局所 Boltzmann 重み
1 z1/4
z1/2
z1/4
z1/4
z1/4
z1/2 1 z1/4
z1/2
z1/4
z1/4
z1/4
重なり⾏行行列列 M =
1
(z)
[T(z)]
T
T(z)
S. Tanaka, R. Tamura, and H. Katsura, PRA, 86, 032326 (2012)

基底状態の e-entropy
e-entropy
5
0
1
2
3
4
5
0 10 20 30
S
z
(a)
1
2
3
4
5
S
α=0.2272(3)
S0=-0.036(6)
(c)
0
2
4
6
8
10
0 50 100 150
S
z
(b)
2
4
6
8
10
S
α=0.4001(3)
S0=0.020(5)
(d)
e-entropyは極⼤大値をもつ。
z=0でゼロになる(基底状態は真空)
z が⼤大きい極限で⼀一定値に収束(基底状態の縮退数)
S = Tr A ln A =
=1
2
ln 2

臨臨界点 zc の決定
(z) :=
1
ln[p(1)(z)/p(2)(z)]
p(1)
(z)
p(2)
(z)
:重なり⾏行行列列　の最⼤大固有値M
0
1
2
3
4
5
0 5 10 15 20 25
S
L
α=0.2272(3)
S0=-0.036(6)
(c)
0
1
2
0 2 4 6
ξ(z)/L
z
(e)
0
2
4
6
8
10
0 5 10 15 20 25
S
L
α=0.4001(3)
S0=0.020(5)
(d)
0
1
2
0 5 10 15
ξ(z)/L
z
(f)
:重なり⾏行行列列　の第⼆二固有値M
相関⻑⾧長
転移点
転移点
zc ' 3.796 zc ' 11.090

有限サイズスケーリング
0
0.5
1
1.5
2
2.5
-10 0 10
ξ(z)/L
(z-zc)L1/ν
(a)
L= 6
L= 8
L=10
L=12
L=14
L=16
L=18
L=20
L=22
0
0.5
1
1.5
2
-50 0 50
ξ(z)/L
(z-zc)L1/ν
(b)
L= 6
L= 9
L=12
L=15
L=18
L=21
2D Ising
= 1 = 5/6
2D 3-state Potts

e-spectrum@zc
e-spectrum
0
2
4
6
0 0.25 0.5 0.75 1
λ-λ0
k/2π
0
1
2
0 0.25 0.5 0.75 1
λ-λ0
k/2π
0.6
0.8
1
1.2
1.4
0 0.5 1 1.5 2
s(l,L)
ln[g(l)]
(a)
c=1/2
0.
0.
1.
1.
s(l,L)
FIG. 7: (color online) NEE for the s
for the triangular ladder (b). Dotte
eye and indicate the slope of c/3, w
are used for the square and triang
The solid circle indicates NEE for L
are the same as in Fig. 5.
0
2
4
6
0 0.25 0.5 0.75 1
λ-λ0
k/2π
0
1
2
0 0.25 0.5 0.75 1
λ-λ0
k/2π
0.6
0.8
1
1.2
1.4
0 0.5 1 1.5 2
s(l,L)
ln[g(l)]
(a)
c=1/2
0.
0.
1.
1.
s(l,L)
FIG. 7: (color online) NEE for the s
for the triangular ladder (b). Dotte
eye and indicate the slope of c/3, w
are used for the square and triang
The solid circle indicates NEE for L
are the same as in Fig. 5.
00
M. Henkel “Conformal invariance and critical phenomena” (Springer)
正⽅方形はしご格⼦子
三角形はしご格⼦子
Primary field
Descendant field
0 =
2 v
L
(hL, + hR, )
v
hL,
hR,
: Velocity
: Holomorphic
conformal weight
: Antiholomorphic
conformal weight
hL, + hR,
Scaling dimension
2D Ising, c=1/2
2D 3-state Potts, c=4/5

Nestede-entropy@zc
Entanglement ground state = e-Hamiltonianの基底状態
HE| 0 = Egs| 0
(e-ground state)
( ) := Tr +1,··· ,L [| 0 0|]
Nested reduced density matrix
Nested e-entropy
S( , L) = Tr1,··· , [ ( ) ln ( )]
１次元量量⼦子臨臨界系(周期境界条件)
P. Calabrese and J. Cardy, J. Stat. Mech. (2004) P06002.
PHYSICAL REVIEW A 86, 032326 (2012)
e
-
,
)
d
α
e
d
l
-
g
r
h
r
r
e
e
0.6
0.8
1
1.2
1.4
0 0.5 1 1.5 2
s(l,L)
ln[g(l)]
(a)
c=1/2
0.6
0.8
1
1.2
1.4
0 0.5 1 1.5 2
s(l,L)
ln[g(l)]
(b)
c=4/5
FIG. 7. (Color online) NEE for the square ladder (a) and that
for the triangular ladder (b). Dotted lines are a guide to the eye and
indicate the slope of c/3, where c = 1/2 and c = 4/5 are used for the
square and triangular ladders, respectively. The solid circles indicate
NEE for L = 24. The other symbols are the same as in Fig. 5.
Let us now give the deﬁnition of the NEE. We divide
the system on which the holographic model lives into two
subsystems: a block of ℓ consecutive sites and the rest of the
chain. The nested reduced density matrix is then deﬁned as
SICAL REVIEW A 86, 032326 (2012)
2
0.6
0.8
1
1.2
1.4
0 0.5 1 1.5 2
s(l,L)
ln[g(l)]
(b)
c=4/5
正⽅方形はしご格⼦子
L=6-‐‑‒24
三⾓角形はしご格⼦子
L=6-‐‑‒24
2D Ising
2D 3-state
Potts

結論論
正⽅方格⼦子VBS状態ラダー上量量⼦子格⼦子気体模型
排除体積効果
VBS状態
正⽅方格⼦子 1D AF Heisenberg
蜂の巣格⼦子 1D F Heisenberg
正⽅方形ラダー 2D Ising
三⾓角形ラダー 2D 3-state Potts
1次元量量⼦子系
物理理特性 (臨臨界現象)
２次元量量⼦子系
蜂の巣格⼦子VBS状態
JPA, 43, 255303 (2010), PRB 84, 245128 (2011), PRA 86, 032326 (2012), Interdisciplinary Information Sciences, 19, 101 (2013)

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