第4部トポロジカルなスピン状態・量子相転移
エンタングルメントエントロピー
Entanglement Entropy
Quantum Many-body System
• Novel quantum order/disorder states
• Haldane spin chains, SU(N) spin models, frustrated quantum magnets, quantum
Hall eﬀects, etc
• Quantum phase transitions between the topological quantum states and
unconventional quantum criticality
• Can not well characterized by classical local order parameters (magnetization, etc)
• Strong demand on new tools
• topological order parameters
• entanglement entropy, etc
Entanglement Entropy
• density matrix of the full system
(for T = 0)
⇢ = | g ih g |
⇢ = exp[ H]/Z (for T > 0)
• divide the system into two subsystems
A and B
• take a partial trace over subsystem B
• reduced density matrix A
⇢A = trB ⇢
B
• entanglement entropy is defined as von Neumann entropy for subsystem A
SA =
tr ⇢A log ⇢A
Entanglement Entropy
• Entanglement entropy
S=
tr ⇢A ln ⇢A
• measure of quantum entanglement (or correlations) between bipartite subsystems
• important quantity in quantum information theory and quantum statistical mechanics
• Numerical estimation of entanglement entropy of many body systems
• exact diagonalization is restricted very small systems
• DMRG works only for one-dimensional systems
• quantum Monte Carlo?
• Recent development of extended ensemble methods
• such as quantum Wang-Landau methods
• enables to calculate partition function, free energy, entropy, etc...
Rényi entanglement entropy
• n=2,3,4...
Sn =
1
1
n
ln tr(⇢nA )
• gives von Neumann entanglement entropy in n=1 limit
• Scaling property of Rényi entropy (Calabrese, Cardy 2004)
c
Sn = (1 + n
6
1
`
) ln
a
• Whole “entanglement spectrum” (encoded in set of Rényi entropies) contains much
more information than von Neumann entropy alone
(Li, Haldane 2008)
• Kn := tr⇢n
A can be evaluated by means of extended ensemble method
Path integral representation of Kn
n-sheet Riemann surface
• Kn can be expressed by a path integral
representation of n-replica system with
special imaginary-time boundary conditions
trA ⇢nA = trA [(trB ⇢)(trB ⇢) · · · (trB ⇢)]
• periodic imaginary-time boundary conditions
for subsystem A (B) for nβ (every β)
A
B
• loop (directed-loop) algorithm can be used for
updating world line configurations
△direct
evaluation by using quantum
Wang-Landau method random walk range ~ nN/Δ
cf. Melko et al (2010)
○calculating partition function ratio
Kn/(Ki Kj) with n = i + j
MC sampling on boundary conditions
• MC update “boundary conditions” in addition to loop update of world lines
K2
A
B
A
B
K1 K 1
A
B
A
B
• partition function ratio is given by appearance rate of “boundary configuration” with
#cuts = 0 and `
• Wang-Landau scheme to enhance the tunneling rate and statistics
Scaling of Rényi entropy in 1+1-dimensions
□ Heisenberg S2 (L = 64, `)
1.4
1.2
S2
■ Heisenberg S2 (L, ` = L/2)
Heisenberg
c=1
1
▲
quantum 3-state Potts
c=4/5
S2 (L, ` = L/2)
◆ quantum Ising
S2 (L, ` = L/2)
0.8
quantum Ising
c=1/2
0.6
0.4
quantum 3-state Potts
1
10
x
L
⇡`
x = sin
⇡
L
c
Sn = (1 + n
6
1
`
) ln
a
VBS状態とトポロジカル秩序
VBS state and topological order
Singlet ground state in Low-D HAF
• Haldane conjecture (1983) for Heisenberg AF chains with S=1,2,3…
• singlet GS with finite excitation gap
Δ/J = 0.41 (S=1), 0.089 (S=2), 0.010 (S=3), …
ξ = 6.02 (S=1), 49.5 (S=2), 637 (S=3), …
• valence bond solid (VBS) picture, topological order, ...
• 2D analogue of Haldane states - SU(N) Heisenberg model
Nc#
n#
Arovas and Auerbach 1988 Read and Sachdev 1989
Lieb-Schultz-Mattis の定理 (1961)
• 系を一様にひねり、低励起状態が作れるかどうかを見る
• 唯一の基底状態の上に有限のギャップが残るための必要条件を与える
• Lieb-Schultz-Mattis の定理
• ハミルトニアンが並進対称性をもつ
• 全スピンの z 成分が保存量 (ハミルトニアンと可換)
• ハミルトニアンが空間反転に対して不変
• 有限系の基底状態が唯一かつ Sz = 0
• このとき以下が成り立つ
• 単位胞注のスピンの大きさの和を S とすると、S が整数でないならば、熱力学
的極限で、ギャップレスの励起状態が存在するか、基底状態に縮退が存在
• 唯一の基底状態の上に有限のギャップが残るには S は整数でなければならない
Lieb-Schultz-Mattis のひねり演算子
• ひねり演算子 (ユニタリ変換)
L
U=
L
Uj = exp[i
j
z
S
j j]
j
j
2
=
j
L
• ひねり演算子による基底状態の変換
| t = U | 0
• エネルギーの増加量
E=
t |H|
0 |H|
t
0
=
1
0 |[U
HU
H]|
0
= O(L
• ひねった状態ともとの基底状態の直交性
0|
t
=
=
0 |U |
0
=
2
0 |U exp[ i
L
0 |T
1
UT|
0
Sjz ] exp[2 iS1z ]|
j
• 全 Sz = 0 かつ S が半奇数の場合には2つの状態は直交
0
1
)
Lieb-Schultz-Mattis の定理の帰結
• 反強磁性ハイゼンベルグ鎖：S が半奇数の場合にはギャップレス or 多重縮退。S が
整数の場合にはギャップが開いても良い
• ボンド交替反強磁性鎖：単位胞が 2 倍
ギャップが開いても良い
• J1-J2ジグザグスピン鎖：もとの基底状態が Marshall-Lieb-Mattis の前提条件(二部格
子)を満たさないため、Lieb-Schultz-Mattis はそのままでは適用不可
• 二次元系：ある特定の方向にひねった場合
エネルギー増加 = O(1)
• 有限磁化への拡張：Oshikawa-Yamanaka-Aﬄeck (1997)
• 数値計算の援用：Nakamura-Todo (2002)
Oshikawa-Yamanaka-Affleck の定理 (1997)
• Lieb-Schlutz-Mattis の定理の有限磁化(全 Sz = M)への拡張
M
(1 exp[ 2 i ] exp[2 iS]) 0 |
L
• S が半奇数の時：M/L が半奇数でない限り
0 | t = 0
• S が整数の時：M/L が整数でない限り
0|
t
=0
• まとめると、磁化プラトーの出現条件: S - M/L が整数
t
=0
Haldane gap of S=1,2,3 chains
∆
method
S=1 0.413(7)
MCPM (Nightingale-Blöte 1986)
0.4150(2) 6.03(2)
DMRG (White-Huse 1992)
0.41049(2) 6.2
ED (Golinelli et al 1994)
0.408(12)
QMC (Yamamoto 1995)
0.41048(6) 6.0164(2) QMC+loop (Todo-Kato 2001)
S=2 0.074(16)
QMC (Yamamoto 1995)
0.055(15)
DMRG (Nishiyama et al 1995)
0.085(5)
49(1)
DMRG (Schollwöck-Jolicœur 1995)
0.090(5)
50(1)
QMC+loop (Kim et al 1997)
0.0876(13)
DMRG (Wang et al 1999)
0.08916(5) 49.49(1) QMC+loop (Todo-Kato 2001)
S=3 0.01002(3) 637(1)
QMC+loop (Todo-Kato 2001)
ξx
Scale of simulations
• S=3 chain with L=5792 and T=0.001
• mapped into (1+1)-dimensional classical system with 34752 sites, 208512 bonds,
length in imaginary time direction 1000
• volume of phase space
• 256 processors x 100 hours = 3 CPU years in 1999
• S=4 chain (T2K Tsukuba, ISSP Kashiwa 8192 nodes)
• N ~ 222 β ~ 213
N β ~ 235 ~ 3・1010
• S=5 chain (K Computer)
• N~ 226 β ~ 217
N β ~ 243 ~ 8・1012
Haldane gap for S=4 and 5 chains
12
• Haldane conjecture (1983)
S(S + 1)e
C
10
S(S+1)
⇥
⇥
S(S+1)
• from S=3 results
⇥ 7.0
10
⇥ 4.5
10
4
,
⇥ 1.2
104
,
⇥ 2.2
10
for S=5
8
7
6
5
4
3
• expected simulation scale (CPU time, memory)
S
2
(10⇥)
10
Matsuo-Todo
5
for S=4
5
9
S
1
11
(x10 )
C S(S + 1)e
⇥
Nakano-Terai (2009)
Todo-Kato (2001)
1
2
3
S
102 ⇥ 1012 (for S=4) and 1015 (for S=5)
= 0.000799(5),
4
5
= 10400(70)
for S=4 (Matsuo-Todo)
Bond-alternating spin chains
H=
i
[1 + ( 1)i ] Si · Si+1
• Quantum phase transitions between VBS phases rearrangement of VBS pattern
S=1
S=3/2
S=1/2
• Hidden string order and string order parameter (for uniform S=1 chain) [den Nijs and Rommelse (1989)]
• Generalization of string order parameter
• generalized twist angle for S>1 [Oshikawa (1992)]
• string order parameter for spin-1 ladder [Todo et al (2001)]
!
Ostr = − lim S0z exp πi
ℓ→∞
2ℓ−1
"
k=1
#
z
Skz S2ℓ
大きな S ・ボンド交替系への拡張
• 一般化された string 秩序パラメタ
• 数値計算の結果
(m,n)
Ostr
2
= lim (iS0z )m exp
1
Skz (iS2z )n
i
k=1
S=1/2 0.6
S=1
S=1/2
0.7
0.6
0.5
O(0,1)
str
0.4
O(0,2)
str
0.5
O(1,0)
str
O(2,0)
str
O(1,1)
str
0.4
0.3
S=1
0.3
0.2
0.2
0.1
0
S=3/2
0.1
-1
-0.5
0
0.5
1
0
S=2
-1
-0.5
0
0.5
1
6
S=3/2
1.6
5
1.4
O(0,3)
str
1.2
O(0,4)
str
O(3,0)
str
O(4,0)
str
4
1
O(1,2)
str
0.8
3
O(2,1)
str
0.6
O(1,3)
str
2
O(2,2)
str
0.4
1
0.2
0
O(3,1)
str
-1
-0.5
0
0.5
1
0
-1
-0.5
0
0.5
1
Spin-1 ladder
• Plaquette singlet solid (PSS) state in S=1 ladder
J
K
J
Haldane state
uniform susceptibility
0.3
0.25
PSS state
~
0.2
dimer state
0.15
0.3
0.25
0.2
0.15
0.1
0.05
0
0.01
0.1
0.05
0
0
0.5
1
~
T
0.1
1
1.5
2
String order parameter for spin-1 ladder
• Single- vs double-chain string order parameter
(⌅)
O2 =
z
S1,⌅
exp
(1)
(2)
⇥
L/2
⇤
z
z
Sk,⌅
SL/2+1,⌅
(⌅ = 1, 2)
i
k=2
z
O2 O2 = S1,1
S1,2 exp
O4
J
K
J
R=
J
J +K
⇥
L/2
i
⇤
z
z
z
(Sk,1
+ Sk,2
) SL/2+1,1
SL/2+1,2
TODO, MATSUMOTO, YASUDA, AND TAKAYAMA
k=2
ひねり演算子の期待値の計算
• ひねった状態と基底状態の重なり積分は、基底状態におけるひねり演算子の期待値と
して計算できる。Nakamura-Todo (2002)
zL =
0|
t
=
0 |U |
0
• スピン S ボンド交替鎖への応用 (量子モンテカルロによる数値計算)
1
H=
S= 1
0.5
!
i
[1 − (−1)i δ] Si · Si+1
zL
(1)
S= 2
0
S=1/2
-0.5
-1
3
S= 2
-1
1
S= 2
-0.5
0
δ
0.5
1
S=1
S=3/2
Sign of twist order parameter
• contribution from a dimer
i
π
cos[ |i − j|]
L
j
• S=1 dimer phase: (2,0) VBS state
.....
1
2
3
4
L-1
L
• Haldane phase: (1,1) VBS state
zL → (−1)
b
.....
1
2
3
4
π L
[cos ] → +1
L
L-1
L
π L−1
π
[cos ]
[cos (L − 1)] → −1
L
L
Quantum phase transition in 2D HAF
• Quantum phase transition from spin-gapped state to AF-LRO phase
H=
!
i,j
S2i,j · S2i+1,j + α
!
i,j
S2i+1,j · S2i+2,j + J ′
!
i,j
S=1/2
Si,j · Si,j+1
S=1
2.5
J’
H-II
2
10
AF
1.5
J’
1
1
1
J’
Gapped
0.5
AF
0.1
0.08
AF
0.06
0
0
0.2
0.4
0.6
0.8
1
H-II
0.04
0.02
0
H-I
0
0.2
0.4
0.6
0.8
1
Matsumoto, Todo, et al (2002)
3D Heisenberg (classical O(3)) universality
局所Z2ベリー位相
Local Z2 Berry phase
Hatsugai 2006
Local Z2 Berry Phase
• Introduce a local perturbation parameterized by θ
• ex) quantum Spin model: local twist on a particular bond
SI+ SJ + SI SJ+
!
e
i
SI+ SJ + ei SI SJ+
• Consider a close path in the parameter space (e.g. θ: 0 → 2π)
• Assumptions:
• unique ground state and finite gap during the operation
• the Hamiltonian has time-reversal symmetry
• The Berry connection and the Berry phase can be defined for this operation
=
i
Z
d
⇥gs(⇥)|
|gs(⇥)⇤ d⇥ = 0
d⇥
• The Berry phase takes only quantized values: 0 or π (mod 2π)
or
⇤
mod 2⇤
Local Berry Phase for a Singlet Dimer
• Two-state system
HD ( ) =
0
e
ei
0
i
• The ground state and the Berry connection
1
| ( ) =
2
d
( )|d ( ) =
2
1
i
e
1
e
·
i
0
ie
i
=
id
2
• The Berry phase: γ = π
• Similarly a spin-1/2 Heisenberg dimer has the Berry phase π
JS1 · S2
JS1z S2z
J
+
e
2
i
S1+ S2 + ei S1 S2+
• The local Berry phase (0 or π) can detect the dimer pattern of the ground state
Local Z2 Berry Phase
• Can detect the pattern of singlet dimers of the ground state
• Takes the quantized values → clearly distinguishes two ground states with diﬀerent
topological order
• Can be used to classify the topological order states and identify the phase transition
point between them
• Expected to work not only in one dimension but also in higher dimensions
• Calculation of local Berry phase
• standard method: exact diagonalization
• the system size is very limited (especially in higher dimensions)
• quantum Monte Carlo
• how to calculate the Berry connection (i.e. inner product of two wave functions)
in QMC?
• complex weight problem?
SI+ SJ + SI SJ+
!
e
i
SI+ SJ + ei SI SJ+
QMC Calculation of Berry Phase
• Discretization by using KSV-formula (King-Smith and Vanderbilt 1993)
= lim
M !1
M
X
arg
gs(✓i )|gs(✓i+1 )⇥
i=1
• Evaluation of inner product of wave functions by projector method
⇥ ⇥gs( i )|gs(
i+1 )⇤⇥ = ⇥⇥| e
⇥⇥| e
2
H( i )
2
H( i )
⇥⇥| e
2
e
e
H(0) e
2
H(
i+1 )
|⇥⇤
2
H(
i+1 )
|⇥⇤
2
H(0)
|⇥⇤
• Fixed wave function at imaginary time boundary τ=0 and β
• Replace imaginary time propagation using two diﬀerent Hamiltonians by the
Hamiltonian at θ=0 (note: the denominator is always real)
Projector Monte Carlo
⌧=
• World line QMC using the Hamiltonian at θ=0
• Fixed configuration at imaginary-time boundary
(e.g. classical Néel state for bipartite lattice)
• Evaluate the mean value of
exp i
!
(N
j+1
bottom
(
e
!
Nbottom ) + i j (Ntop
i
j
Ntop
⇥ = /2
)
• Nbottom(top) : number of right (left) going kinks
on the twisted bond in bottom half 0 < < /2
(top half /2 < < )
• example:
Nbottom = Ntop = 1, Nbottom = Ntop = 0
ei(
j+1
ei
j+1
j)
⌧ =0
twisted bond
Spin-1/2 Bond Alternating Chain
[1 + ( 1) ] Si · Si+1
i
π
1+ bond
1- bond
0
3
2.5
2
1.5
1
0.5
0
0
-0.5
-1
-1.5
-1
-0.5
)
J(1 + )
3.5
Berry phase
H=
J(1
i
0
bond alternating
0.5
1
quantum critical point
N = 32
T = 1/100
Level Crossing at the Critical Point
• θ-dependence of the energy gap
• For δ≠0, the energy gap remains finite during the operation
• the Berry phase is well defined and takes 0 or π
• At δ=0, the gap closes at θ=π (level crossing)
• the Berry phase can not be defined
0.6
• At θ=π, the energy gap is quite
sensitive to the value of δ
= 0.1
= 0.01
=0
0.5
energy gap
0.4
0.3
0.2
H=
i
[1 + ( 1)i ] Si · Si+1
L=8 exact diagonalization
0.1
0
0
0.2
0.4
0.6
0.8
1
/
1.2
1.4
1.6
1.8
2
Spin-1/2 Bond Alternating Spin Ladder
J(1
)
Jrung
J(1 + )
π
1
0.8
1.42
1.4
0.6
γ/ π
1.38
1.36
1.34
J'c/J
0.4
1.32
1.3
0.2
1.28
1.26
0
0
1.1
1.15
1.2
1.25
J'/J
1.3
1.35
1.4
1.24
1.22
0
0.02
0.04
0.06
1/L
0.08
0.1
0.12
0.14
Ground state of 2D SU(N) Heisenberg model
• 2D analogue of Haldane states - SU(N) Heisenberg model
Nc#
n#
1.9
L = 16
L = 32
L = 64
L= ∞
1.8
1.7
Arovas and Auerbach 1988 Read and Sachdev 1989
Nγ/2π
1.6
1.5
1.4
1.3
1.2
1.1
1
0.9
2
3
4
5
N
6
7
8
脱閉じ込め臨界現象
Deconfined Critical Phenomena
脱閉じ込め転移
Senthil, et al. (Science, 2004)
Paramagnetic
T
Anti-Ferromagnetic
(AF)
Spinon
Deconfined
Singlet
Valence Bond Solid
(VBS)
Non-Landau Ginzburg Wilson!
0
gc
g
core of such a vortex there is a site with an unpaired spin—
i.e., a spin that is not part of any valence bond. It is easy to
see that this is a general property of any such vortex pattern
of the VBS order parameter. Furthermore, translating the entire valence bond pattern by one lattice spacing reverses the
direction of the winding—thus the Z4 vortices are associated
with one sublattice, say the A sublattice, and the Z4 antivortices with the B sublattice.
Thus in this particular quantum problem, the Z4 vortices
(and antivortices) carry an uncompensated spin-1 / 2 moment.
They may therefore be identified with “spinons.” In the
VBS-ordered phase, the energy required to separate a vortex
from an antivortex increases linearly with distance, since
such a pair is necessarily accompanied by four domain walls
connecting the two defects (see Fig. 3). This means that the
spinons are confined and do not exist as free excitations in
this phase.
正方格子上のVBS秩序
state with a Z4 order parameter. Such an ordinary state obtains for instance in a simple lattice quantum O!2" rotor
model with a fourfold anisotropy. In this case the Z4 vortices
in the ordered state have featureless cores. The disordering
transition in this simple model may be described by the usual
three-dimensional classical Z4 model and is hence in the 3D
XY universality class (since the clock anisotropy is irrelevant). In contrast, disordering transitions out of the VBS
phase must necessarily take into account the presence of the
spin-1 / 2 moment in the cores of the Z4 vortices. Any mapping to a classical 3D Z4 model is then complicated by the
need to incorporate this vortex structure.
Consider moving out of the VBS phase by proliferating
and condensing the Z4 vortices. Clearly once the vortices
proliferate, long-ranged Z4 order cannot be sustained. Furthermore, as these vortices carry spin, the resulting state will
break spin symmetry, and as argued below may be identified
as the Néel state.
These simple considerations, therefore, provide a mechanism for a direct second-order transition between the VBS
and Néel phases. As for the usual Z4 model, it is reasonable
to expect that the clock anisotropy will be irrelevant at this
transition as well. Indeed, as we will argue later, this is
strongly supported by the evidence from Refs. 1 and 2. For
the present, let us explore the consequences of the expected
irrelevance of the clock anisotropy.
The critical theory will then be that of a (quantum) XY
model in D = 2 + 1 but with vortices that carry spin-1 / 2 (See
Fig. 5). The spinon nature of these vortices will change the
universality class from D = 3 XY to something different.
Clearly to expose this difference and to obtain a description
of the resulting new universality class, it will be most convenient to go to a dual basis in terms of the vortices and their
interactions (analogous to the familiar Coulomb gas description of classical 2D XY models).
四重縮退カラムナー
VBS(Valence Bond Solid) 状態
Z4 回転対称性の破れ
FIG. 3. (Color online) Macroscopic picture of a Z4 vortex as a
point where four oriented elementary domain walls meet and end.
VBS 場（複素場） Ψ
220403-2
Read & Sachdev, PRB 42, 4568 (1990).
Levin & Senthil, PRB 70, 220403(R) (2004).
⌘ ( 1)
x+y
⇥
（ベクトル秩序パラメータ）
P(x,y),(x+1,y) + i P(x,y),(x,y+1)
⇤
together and terminate at a point. It is clear that such termination points may be associated with Z4 vortices—the clock
angle winds by 2! upon encircling such a termination point
(see Fig. 3). Z4 antivortices may be similarly defined.
What do such Z4 vortices correspond to in terms of the
underlying VBS configurations? An example is illustrated in
Fig. 4. A remarkable property of this cartoon is that at the
core of such a vortex there is a site with an unpaired spin—
i.e., a spin that is not part of any valence bond. It is easy to
see that this is a general property of any such vortex pattern
of the VBS order parameter. Furthermore, translating the entire valence bond pattern by one lattice spacing reverses the
direction of the winding—thus the Z4 vortices are associated
with one sublattice, say the A sublattice, and the Z4 antivortices with the B sublattice.
Thus in this particular quantum problem, the Z4 vortices
(and antivortices) carry an uncompensated spin-1 / 2 moment.
They may therefore be identified with “spinons.” In the
VBS-ordered phase, the energy required to separate a vortex
from an antivortex increases linearly with distance, since
such a pair is necessarily accompanied by four domain walls
connecting the two defects (see Fig. 3). This means that the
spinons are confined and do not exist as free excitations in
this phase.
VBS相のスピノン描像
•Levin & Senthil (2004)
FIG. 3. (Color online) Macroscopic picture of a Z4 vortex as a
point where four oriented elementary domain walls meet and end.
FIG. 4. (Color online) The Z4 vortex in the columnar VBS state.
The blue lines represent the four elementary domain walls. At the
core of the vortex there is an unpaired site with a free spin-1 / 2
moment.
It is the nontrivial structure of the Z4 vortex in this problem that distinguishes the VBS state from a more ordinary
state with a Z4 order parameter. Such an ordinary state obtains for instance in a simple lattice quantum O!2" rotor
model with a fourfold anisotropy. In this case the Z4 vortices
in the ordered state have featureless cores. The disordering
transition in this simple model may be described by the usual
three-dimensional classical Z4 model and is hence in the 3D
XY universality class (since the clock anisotropy is irrelevant). In contrast, disordering transitions out of the VBS
phase must necessarily take into account the presence of the
spin-1 / 2 moment in the cores of the Z4 vortices. Any mapping to a classical 3D Z4 model is then complicated by the
need to incorporate this vortex structure.
Consider moving out of the VBS phase by proliferating
and condensing the Z4 vortices. Clearly once the vortices
proliferate, long-ranged Z4 order cannot be sustained. Furthermore, as these vortices carry spin, the resulting state will
break spin symmetry, and as argued below may be identified
as the Néel state.
These simple considerations, therefore, provide a mechanism for a direct second-order transition between the VBS
and Néel phases. As for the usual Z4 model, it is reasonable
to expect that the clock anisotropy will be irrelevant at this
transition as well. Indeed, as we will argue later, this is
strongly supported by the evidence from Refs. 1 and 2. For
the present, let us explore the consequences of the expected
irrelevance of the clock anisotropy.
The critical theory will then be that of a (quantum) XY
model in D = 2 + 1 but with vortices that carry spin-1 / 2 (See
Fig. 5). The spinon nature of these vortices will change the
universality class from D = 3 XY to something different.
Clearly to expose this difference and to obtain a description
of the resulting new universality class, it will be most convenient to go to a dual basis in terms of the vortices and their
interactions (analogous to the familiar Coulomb gas description of classical 2D XY models).
220403-2
3D XY + Z4
蜂の巣格子上のVBS状態
三重縮退カラムナーVBS秩序
Z3 回転対称性の破れ
VBS場 Ψ =
Read & Sachdev, PRB 42, 4568 (1990).
Pujari, et al., PRL 111, 087203 (2013).
Block, et al., PRL 111, 137202 (2013).
繰込み時のフロー
1
Senthil, et al., PRB 69, 224416 (2004).
Q
J
VBS
格子上のベリー位相から生ずる摂動項
q(
q
+(
† q
) )
q = 3 （蜂の巣格子）
q
q = 4 （正方格子）
dangerous irrelevant
Néel
DQC
g
U(1)
1
蜂の巣格子と正方格子で、臨界現象の
ユニバーサリティクラスを共有？
数値的に検証！
SU(2) JQモデル
• JQ model (Sandvik, 2007）
H=
J
X
Pij
<ij>
Q
X
[Pij Pkl + Pik Pjl ]
<ijkl>
Pij ＝スピン１重項への射影
Pij Pkl
i
j
i
j
Pij
Pik Pjl
k
l
J項は磁化，Q項はVBS秩序を引き起こす
• SU(N)一般化：SU(2)➤SU(3)➤SU(4)➤SU(N>4)
• 格子一般化：蜂の巣格子
磁気秩序の有限サイズスケーリング解析
Square L = 32
L = 48
L = 64
L = 96
Honeycomb L = 36
L = 48
L = 72
-2
L = 96
10
2 2x
B 〈m 〉 L m
0.8
0.6
-3
10
-1.4
R
q = 0.3337
q = 0.3344
q = 0.3351
0.4
SU(3) JQ models
Cm(R)
1
10-4
0.2
1
10
100
R
0
-120
-100
-80
-60
-40
1/ν
-20
0
20
A (q-qc) L
(1) 同一の臨界指数
1/⌫ = y = 1.87, ⌘m = 2xm
(2) スケーリング関数もほぼ同じ
1 = 0.40
VBS秩序の有限サイズスケーリング解析
1.2
Square L = 32
L = 48
L = 64
L = 96
1.1
1
Honeycomb L = 36
L = 48
L = 72
L = 96
SU(3) JQ models
10-2
R-1.47
q = 0.3337
q = 0.3344
q = 0.3351
0.6
0.5
0.4
1
10-4
100
10
R
0.3
0.2
-60
10-3
CΨ(R)
2xΨ
0.7
B 〈|Ψ| 〉 L
0.8
2
0.9
-50
-40
-30
-20
-10
1/ν
0
10
20
A (q-qc) L
(1) 同一の臨界指数
1/⌫ = y = 1.72, ⌘ = 2x
(2) スケーリング関数もほぼ同じ
1 = 0.47
システムサイズ依存性
1/ν = d+z = 3
2.5
2
1.5
1
SU(3)
2.5
2
1.5
1
0.5
0
SU(4)
2
1.5
1
2
1.5
1
2
1.5
Square: 1/ν
2xm
2xΨ
1
Honeycomb: 1/ν
2xm
2xΨ
0.5
0
16
32
64
Lmax
128
256
1/ν
１次転移の方にシフト
2x
SU(2)とSU(3)のどちらも
SU(2)
1/ν
2x
もし、１次転移だったら
2.5
2
1.5
1
1/ν
例： L=32, 48, 64, and 96
2x
四つ組データ (Lmax/3 から Lmax)
主要な結果
蜂の巣格子と正方格子のSU(N) JQ モデル
▶ 両格子でユニバーサリティクラスを共有 ◎
▶ ユニバーサリティクラスを特徴づける臨界指数に ?
システムサイズ依存性（１次転移へのシフト）
K. Harada, T. Suzuki, T. Okubo, H. Matsuo, J. Lou, H. Watanabe, S. Todo, N.
Kawashima, Physical Review B 88, 220408(R) (2014).
๏ 大規模並列化した量子モンテカルロ計算(ALPS/LOOPER)
๏ 有限サイズスケーリング解 ベイズ流スケーリング解析 : Harada, Phys. Rev. E 84, 056704 (2011)