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テンソルネットワーク法を用いたキタエフスピン液体の研究
東大理, 大久保毅
共同研究者：

Korea Univ. H.-Y. Lee,
Kinki Univ. R. Kaneko
ISSP N. Kawashima,

Ref:• T.O. et al, Phys. Rev. B 96, 054434 (2017).
• H.-Y. Lee, R. Kanako, T.O. and N. Kawashima, PRL 123, 087203 (2019),
PRB 101, 035140 (2020).

コンテンツ•はじめに•

量子多体状態のテンソルネットワーク表現•

ハニカム格子キタエフ模型•

ギャップレスキタエフスピン液体のテンソルネットワーク状態•

テンソルネットワークによる有限温度シミュレーション•

まとめコンテンツ•はじめに•

量子多体状態のテンソルネットワーク表現•

ハニカム格子キタエフ模型•

ギャップレスキタエフスピン液体のテンソルネットワーク状態•

テンソルネットワークによる有限温度シミュレーション•まとめ量子多体問題物質科学における多彩な現象
•

化学反応
超伝導
トポロジカル状態

•

...

•
•

多数の"粒子"が量子力学に従って"運動"量子多体問題

wikipedia"マイスナー効果", "トーラス"より

量子力学の支配方程式＝シュレディンガー方程式：ハミルトニアン：状態ベクトル
時間に依存
しない場合

エネルギー

= 固有値問題量子多体問題の困難
シュレディンガー方程式：••

ベクトル空間の次元は"粒子数"に対して指数関数的に大きい
量子多体問題〜「巨大な行列」の固有値問題

古典的な計算機でこの問題を（厳密に）解くには、膨大なメモリと計算時間が必要
例：量子ビットの厳密シミュレーションは50 qubit程度が限界制御された量子系：••

量子シミュレータ
万能量子コンピュータ

Optical Lattice, https://www.nist.gov より

量子多体系の情報をそのまま扱え、（理想的には）指数関数の困難はない

量子多体問題の例：スピン模型の基底状態
例：横磁場イジングモデル

Disorder(+x方向)

強磁性(±z方向)
1
S=
2

量子スピンの行列表示<latexit sha1_base64="9jEMPZqqdJlvTrkzUC3kT88IHy0=">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</latexit>

物性物理の研究対象の一つ•

どんな相があるか？
••

長距離秩序、トポロジカル秩序、…

それらを分ける相転移の性質は？

<latexit sha1_base64="WGPOYK4P8+EtP+RG4T2pDeTgX5o=">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</latexit><latexit sha1_base64="ueUTnz6aYO08gQxt2hihgPUxD+g=">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</latexit>

量子多体問題に対する計算科学のアプローチ
•

•

対角化法
•

そのまま固有値問題を解くため厳密•

S=1/2の量子スピン系では50スピン程度が限界

量子モンテカルロ法
•
•
•

•

変分法
•

•
•

統計誤差の範囲で厳密な答えが得られる
対角化よりもずっと大きな系が取り扱える符号問題により、適用できる系が限られる

状態ベクトル（波動関数）の形を仮定して計算量を減らす• 仮定した形によるバイアスが存在
どんな系にでも適用できる（ことが多い）
テンソルネットワーク法はこのカテゴリ

変分法
例：最低エネルギー状態

コスト関数：

Fの最小値
その時の

変分法
•
•

固有値問題の近似解を得る方法の一つ
Fの最小値を制限された空間の範囲で探すの形を仮定する＝試行関数、変分波動関数
例：平均場近似、テンソルネットワーク状態、ニューラルネットワーク, ...
•良い試行関数→高精度の最低エネルギー•

複雑な試行関数→コスト関数の計算量が増大

テンソルネットワークによる情報圧縮
指数関数的に大きな状態空間を全て扱うことは不可能
実効的な次元を減らしたい
テンソルネットワーク状態：
情報のエンタングルメントに注目することで、適切な部分空間を構成

テンソルネットワーク分解

一般の量子状態<latexit sha1_base64="XkC8uXv0uT1Qilx+79HHlBouX2k=">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</latexit>

量子エンタングルメ
ントの構造を利用

ヒルベルト空間

部分空間

ダイアグラムを用いたテンソル表記
•

ベクトル

：

•

行列

：

•

テンソル

：

テンソルの積（縮約）の表現＊n階のテンソル＝n本の足

C

＝

B

A
C

D

＝ B

A

量子多体状態のテンソルネットワーク表現
量子多体状態：基底

量子スピン・bit：
量子化学：

i = 原子軌道・分子軌道の占有数

係数はテンソル量子多体状態(a)(b)

テンソルネットワーク分解(c)PEPS, TPS (for 2d system)<latexit sha1_base64="XkC8uXv0uT1Qilx+79HHlBouX2k=">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</latexit>

量子相関の小さいテンソルに分解特徴を利用した近似~eNの独立要素~O(N)の独立要素

良いネットワークを選ぶことで、量子多体状態を効率的に表現できる
ex. TPS: 独立な要素数~ ND4指数関数→線形

D: テンソルTの各足の大きさ

良いネットワークの選び方：

エンタングルメントエントロピーの面積則
エンタングルメントエントロピー（EE）：
部分系の縮約密度行列:A

B

EE= ρA のvon Neumann エントロピー

B

一般の状態ベクトル：
EE は 部分系の体積（スピン数）に比例

基底状態ベクトル：L（c.f. ランダムベクトル）

多くの低エネルギー状態では, EE は面積に比例
J. Eisert, M. Cramer, and M. B. Plenio, Rev. Mod. Phys, 277, 82 (2010)

基底状態はヒルベルト空間の狭い部分空間で表現可能

A

テンソル積状態（TPS）：面積則を満たすTNS
TPS (Tensor Product State) (AKLT, T. Nishino, K. Okunishi, …)
PEPS (Projected Entangled-Pair State)(F. Verstraete and J. Cirac, arXiv:cond-mat/0407066)

例：2次元正方格子のTPS

4+1 階のテンソルが敷き詰められたネットワーク
局所自由度：s
Virtual自由度：i, j, k, l
各インデックスの次元＝ボンド次元（D）
変分波動関数としての精度に関係するパラメタ （D→∞で厳密に）TPSを変分波動関数とする変分法••

面積則を満たすため、有限Dでも精度の良い近似•
無限系も直接、有限のDで計算できる：iTPS
テンソルネットワークのみを仮定した、バイアスの少ない変分波動関数•
ボンド次元の増大により、系統的に精度を改善できる

TPS (PEPS)のエンタングルメントエントロピー
ボンド次元 = DA

B

領域 A と B をつなぐボンドの数（正方格子）
Nc (L) = 4L
Nc (L) = 2dLd 1 （d 次元の超立方格子)

rank ⇢A  D

SA =TPS は面積則を満たすことができる!

Nc (L)⇠D

2dLd

Tr ⇢A log ⇢A  2dL

d 1

1

log D

TPSを用いることで高次元の量子多体状態を効率的に近似できる＊これは、有限DのTPSが無限系を効率的に近似できることも示唆
infinite TPS = iTPS

may be found in Kitaev (2003).
We consider a square lattice, possibly embedded into a nontrivial surface su
a torus, and place spins on the edges, as in Fig. 3.1. The Hamiltonian is given b

iTPSの例：可解模型の基底状態
The two-dimensional toric code

HT = −Je!
s

As − Jm!

Bppwhere s runs over the vertices (stars) of the lattice and p runs over the plaqu
The star operator acts on the four spins surrounding a vertex s,

Toric code

! 303, 2 (2003).
(A. Kitaev, Ann. Phys.
x
σj

As =

(3.2)

j∈star(s)nal toric code
while the plaquette operator acts on the four spins surrounding a plaquette,
!
!
x
Bp =
σjz .
σj
(3.2)
As =
j∈star(s)

Bp

(3.3)

j∈∂p

As

この模型の基底状態は
Z2 aスピン液体
operator acts
ons the four spins surrounding
plaquette,p!
As Bp = Bp A*量子コンピュータの誤り訂正にも使われる(3.4)
sFig. 3.1 A piece of the toric code. The spins live on the edges of the square lattice
z
Bp =
σj .
(3.3)
spins
adjacent to a star operator A and a plaquette operator B are shown.
Clearly, the A all commute with one another, as do the B . Slightly less trivially,

because any given star and plaquette share an even number of edges (either none orj∈∂p
このスピン液体状態は
D=2TPSで厳密に表現可能two) and
therefore the minus signs arisingfromthe commutationof σ x and σ z on
those withedges onecancel.
Since allof HT commute,we expect to be able toommuteanother,
as of
dothe
theterms
Bp . Slightly
less trivially,(F. Verstraete, et al, Phys. Rev. Lett. 96, 220601 (2006).
solve it term by term.
As BweBpsolve
As HT working in the σz basis. Define(3.4)
In particular,
willclassical variablesp =sj = ±1
label the σ z ofbasis
each classicalspin{s}, we can
0,1:toeigenstate
σz states. For(Non-zeroelements
of configuration
tensor)tar define
and plaquetteshare
an even number of edges (either none or
the plaquette
flux
0
0
1
1
1 !
1
0 x 0
z
the minus signs
arisingfromthecommutation
of
σ
and
σ
on
w
(s)
=
s
.
(3.5)
p
j
0
0
1
1
s

0
1 to be able toSince all of 0the terms of1 HT commute,weexpectj∈∂p
1
0
1
m. If wp = −1, we0 say that there
is a vortex on plaquette p.
will solve HT0 working
in1the 1σ z basis.
classical
00
1 Define01variables
statesσ z 3.1
basis Groundstates.
For
spin
configuration{s},
0 each classical0
1
1 we can01
10"
fluxTo find the ground0
1 of HT , we 0
1statesneed to minimize
the energy, which means! |ΨpiTPS を用いた変分計算（の概略）
最適化：虚時間発展法
Approximatin

Costinformation Accuracy

Simple update

O(D5)

local

bad

Full update

O(D10)

globalbetterUpdateはだいたい 103~ 105 ステップの計算

物理量評価: 全体のネットワークの（近似的）縮約
角転送行列繰り込み群

Cost ~ O(D10)
典型的には数十step(R. J. Baxter (1968), T. Nishino, et al (1998), R. Orus et al (2009) ...)
Environment

C

e

e

e
C

C

e

C

テンソルネットワーク法の適用例
例：（QMCのできない）フラストレート磁性体

H. YAMAGUCHI et al.

フラストレート正方格子模型

M/Msat

1

5/9

0.4

3/9
Spin

1

0

0

1

site 10.1
3B/J

g. 1 Calculated magnetization process for the spin-1/2 KAFM with theearest-neighborinteractionD.
J. Nakamura,
The tensor network
methodwith theR. Okuma,
T. Okubo
et al,
rojected entangledNat.
pair Commun.
state (PEPS)10,
is used.
vertical and horizontal
1229The(2019).
xes represent magnetization M divided by saturated magnetization Ms and0site 1

Hsite 2

1.0
0.5

1.5

(c)

2.0

site 1

0.3 site 2

0.2

2

(a)

0.4

0.3

2

Magnon

1/9
0

(b)

<Sz >

M /MS

7/9

1.0
J6 J1
0.9
site 1
0.8
J2 J5
0.7
J4 J3 site 2
0.6
0.5
0.4
0.3
H=0
0.2
0.1
0
0
0.5
0.5

Intensity

カゴメ格子ハイゼンベルグ模型の磁化曲線(< Sx>2+<Sy>2+<Sz>2 )1/2

RE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-09063-7

0.2

site 2
0

0.5

1.0

1.5

0.1

H/|J1|

0

0.5

1.0

1.5

FIG. 4. The ground states under magnetic fields obtained from
H. Yamaguchi,
Y. Sasaki, T.
Okubo,
D = 6 iTPS calculation.
(a) Normalized
magnetization
curve, (b)
Phys.
Rev.
B
98,
094402(2018).
average local magnetization, (c) average local moment at T = 0
calculated using the tensor network method assuming the ratios of

var
P(1
den

コメント：テンソル繰り込み群
Tensor renormalization group (TRG) cf. M. Levin and C. P. Nave, Phys. Rev. Lett. 99, 120601 (2007)
ネットワークの粗視化によるテンソルネットワークの近似的な縮約法(L×L)/2

L×L

指数関数的な縮約コストがサイト数の多項式時間に低減
長いスケールへの粗視化（繰り込み）Aaa1: D×D×D×D: D×D×D×D

T

(a2)

A

•
•
•

TRG型の方法は、古典模型の分配関数計算によく用いられる
経路積分表示を通じて量子多体問題に適用可能
TRGはテンソルネットワーク状態と密接に関係
•
•

•

B

短距離相関（short-range entanglement)の除去
MERAとの関連

S2

F

B C
S1
S
D A 1

D

E

D A

E

b

F

F

C

C

S2

E

S2

<latexit sha1_base64="O7Oz7Fgh
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S1

E

S2

S2

F

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S2

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S1
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T
S2

(b3)

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S1

我々も新しい手法開発をしている。
• D. Adachi, T.O. and S. Todo, PRB 102 054432 (2020); arXiv:2011.01679.

S1

S1

S1
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T

S1<latexit sha1_base64="RIqFXjV+ZuGcOnyDo4zZSzPYh5w=">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</latexit><latexit sha1_base64="yUI40PNIgJog+wedqvtgVkB8HS8=">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</latexit>Tコンテンツ•はじめに•

量子多体状態のテンソルネットワーク表現•

ハニカム格子キタエフ模型•

ギャップレスキタエフスピン液体のテンソルネットワーク状態•

テンソルネットワークによる有限温度シミュレーション•まとめ（量子）スピン模型と典型的な（基底状態）相図
例：横磁場イジングモデル

Disorder(+x方向)

強磁性(±z方向)
1
S=
2

量子スピンの行列表示<latexit sha1_base64="WGPOYK4P8+EtP+RG4T2pDeTgX5o=">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</latexit><latexit sha1_base64="ueUTnz6aYO08gQxt2hihgPUxD+g=">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</latexit><latexit sha1_base64="9jEMPZqqdJlvTrkzUC3kT88IHy0=">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</latexit>

通常、量子スピン系の基底状態は（磁気的な）長距離秩序を持つ:
・自発的対称性の破れ
・外部磁場の効果

量子スピン液体

theory of interacting metals: they propose that the ground state has acertain structure and a set of elementary excitations that are consistent
with this structure. In contrast to the Fermi liquid case, however, theelementary excitations consist of spinons and other exotic particles,
which are coupled by gauge fields. A theory of this type — that is, proposing a ‘spinon Fermi surface’ coupled to a U(1) gauge field — has
had some success in explaining data from experiments on κ-(BEDTTTF)2Cu2(CN)3 (refs 65, 66). Related theories have been proposed for
ZnCu3(OH)6Cl2 (ref. 67) and Na4Ir3O8 (ref. 68). However, comparisons（量子）スピン系の相互作用にフラストレーションがあると:
a

u
c
o

1
c
t
s
l

場合によっては、基底状態が長距離秩序を持たない

量子スピン液体b

平均場描像に基づく、スピン液体状態が数多く存在••+

Gapped spin liquid•
Z2 spin liquid•
Chiral spin liquid•...
Gapless spin liquid•U(1) spin liquids(L. Balents, Nature (2010))c

Spin liquid (RVB)+…

•

稀な例：可解模型

Figure 3 | Valence-bond states of frustrated antiferromagnets. In a VBS
state (a), a specific pattern of entangled pairs of spins — the valence bonds— is formed. Entangled pairs are indicated by ovals that cover two points
on the triangular lattice. By contrast, in a RVB state, the wavefunction is a

基底状態がスピン液体であることが分かっている最も有名な例
204（ハニカム格子）キタエフ模型
199-208 Insight - Balents NS.indd 204

s
b
(
m

© 2010 Macmillan Publishers Lハニカム格子キタエフ模型

A. Kitaev, Annals of Physics 321, 2 (2006)

ハニカム格子キタエフ模型x-bond：相互作用の方向

y-bond相互作用の方向に応じて、異なるスピン成分がイジング型に相互作用

この模型は「マヨラナフェルミ粒子」
を用いて、自由粒子の問題に変換可能
Four Majorana fermionsSpin<latexit sha1_base64="ndACOmdHyEFw6+5WGcFenobMeBk=">AAACjHichVFNS8NAEH2NX7VqW/UieCkWxZNMpKAoQkEQj7a1VVApSdzW0DQJSVrU4h8QvOrBk4IH8R949eIf8NCfIB4rePHgJA2IijrL7r59O2/2LaPahu56RO2I1NPb1z8QHYwNDY/EE8nRsZJrNRxNFDXLsJxtVXGFoZui6OmeIbZtRyh11RBbam3Vv99qCsfVLXPTO7LFXl2pmnpF1xSPqYJaPiwn0zRHQaR+AjkEaYSxYSXvsYt9WNDQQB0CJjzGBhS4PHYgg2Azt4cWcw4jPbgXOEGMtQ3OEpyhMFvjtcqnnZA1+ezXdAO1xq8YPB1WpjBNT3RLHXqkO3qm919rtYIavpcj3tWuVtjlxOlE4e1fVZ13Dwefqj89e6hgMfCqs3c7YPxfaF198/iiU1jKT7dm6Jpe2P8VtemBf2A2X7WbnMhf/uFHZS9BFR9xm+TvTfkJSvNzMuNcJp1dDBsWxSSmMMtdWUAW69hAkV+p4gznuJDiUkZalla6qVIk1IzjS0hrH1AZkeU=</latexit><latexit sha1_base64="3gETdZIAl8JUGeODBmxZoUDFJFk=">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</latexit><latexit sha1_base64="lLrAZ/tp5/wPcyeKhOqE996ETYw=">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</latexit><latexit sha1_base64="TrXX2ogSIADGydGmb9qDh09yNSE=">AAACs3ichVG7SgNBFD2u73fURrBZDEoqmYiiCELAxjIxRgNJDLPrGAf3xe4mEIM/YGFrYaVgIbZWgjY2/oCFnyCWCjYW3t2siAb1Ljtz7pl77pzhao4hPZ+xxzalvaOzq7unt69/YHBoODYyuuHZVVcXOd02bDevcU8Y0hI5X/qGyDuu4KZmiE1tbyU436wJ15O2te7XHVEyecWSO1LnPlHlWGI2u1WscNPk6rJa9GTF5F+5VLXPRC/H4myGhaG2gmQE4ogibceuUcQ2bOiowoSABZ+wAQ6PvgKSYHCIK6FBnEtIhucCB+gjbZWqBFVwYvdorVBWiFiL8qCnF6p1usWg3yWliin2wC7YC7tnl+yJvf/aqxH2CLzUadeaWuGUhw/Hs2//qkzafex+qf707GMHi6FXSd6dkAleoTf1tf3jl+zS2lRjmp2xZ/J/yh7ZHb3Aqr3q5xmxdvKHH428hF0CRGNK/hxKK9iYnUkSzszFU4vRwHowgUkkaCoLSGEVaeToliNc4Qa3yrxSUDRlu1mqtEWaMXwLxfwA3yqf4A==</latexit><latexit sha1_base64="LtKosN5CZYigK8F2h3ebzQz5vl8=">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</latexit>

Majorana fermions:<latexit sha1_base64="kqIgGyyyBrWrhMRGzjoxS6YeDVI=">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</latexit><latexit sha1_base64="K+3tkgcYpuPv39HNdEfKJhjnLbY=">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</latexit>

z-bondキタエフ模型の保存量：フラックス
フラックス演算子<latexit sha1_base64="W682k5HZHC55Sa9dltWKkHJWjdg=">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</latexit>

1

の性質：

<latexit sha1_base64="W682k5HZHC55Sa9dltWKkHJWjdg=">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</latexit>

•
•
•

2

6

固有値は±1

ハミルトニアンと可換＝保存量
異なる場所のフラックスも可換

354

基底状態は全ての六角形でフラックスが1のセクターの中<latexit sha1_base64="s3koN5z1pYei9Ww0KRpq0huHBkU=">AAACnnichVHLSsNAFD2N7/po1I3gJlgUFyITERRBEARxIz5qbUFrSeK0BicPkrRQiz/gD7hwIQqCxT9w68YfcOEniEsFNy68SQOiRb3DzJw5c8+dM1zdFaYfMPaUkNraOzq7unuSvX39Ayl5cGjHdyqewbOGIxwvr2s+F6bNs4EZCJ53Pa5ZuuA5/Wg5vM9Vueebjr0d1FxesLSybZZMQwuIKsrD+3slx9OEUNwpJVd0lUVFLcppNs2iUFqBGoM04thw5Dvs4QAODFRggcNGQFhAg09jFyoYXOIKqBPnETKje44TJElboSxOGRqxR7SW6bQbszadw5p+pDboFUHTI6WCcfbIGuyVPbBb9sw+fq1Vj2qEXmq0600td4up05HM+78qi/YAh1+qPz0HKGE+8mqSdzdiwl8YTX31+Ow1s7A1Xp9gV+yF/F+yJ3ZPP7Crb8b1Jt86/8OPTl6iKiGiNqk/m9IKdmamVcKbs+ml+bhh3RjFGCapK3NYwio2kKVXarjADRqSIq1Ia9J6M1VKxJphfAsp/wmF45dn</latexit>(Vortex free condition)<latexit sha1_base64="E0T+XqpTjGXqn5suN9e7QeU/0Lw=">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</latexit>キタエフ模型の基底状態
キタエフ模型

A. Kitaev, Annals of Physics 321, 2 (2006)
x-bondγ：ボンドの方向

y-bond

２つのスピン液体
非等方領域 (A) : gapped spin liquid•z-bond基底状態と励起状態に有限のエネルギーギャップ•

量子誤り訂正に使われるtoric code状態と"同じ"状態•

Toric codeにはコンパクトなTN表現が存在

基底状態相図Quasiparticle statistics in the gapped phase
J! = (0, 0, 1)

Az

等方領域(B) : gapless spin liquid•••

gappedB

励起エネルギーにエネルギーギャップがない
磁場の印加でトポロジカル相が実現

コンパクトなTN表現が未知（だった）gapless

Ax

gappedJ! = (1, 0, 0)

AygappedJ! = (0, 1, 0)

Fig. 5.3 Phase diagram of honeycomb model. This is a slice through the positive octan
J! coupling space along the Jx + Jy + Jz = 1 plane. The other octants are analogous.

キタエフ物質とテンソルネットワークでの計算例
T. Okubo, K. Shinjo, Y. Yamaji et al, Phys. Rev. B 96, 054434 (2017).
強いスピン軌道相互作用

実際の物質でキタエフ相互作用が実現
G.Jackeli, et al., PRL 102, 017205 (2009)

Na2IrO3の第一原理スピンハミルトニアン

この物質の基底状態をiTPS(Y. Yamaji et al. Phys. Rev. Lett. 113, 107201(2014))を使って明らかに

Kitaev + Heisenberg + Off-diagonal interactions+
2nd and 3rd nearest neighbor interactions••

キタエフ相互作用以外の相互作用の影響で、基底状態はスピン液体
ではなく磁気秩序状態
その場合、iTPSでの計算は、第一原理ハミルトニアンの基底状態を正しく実現できる-5

三方晶歪みを変えた場合の相図

Zigzag-5.4 (X,Y) (Z)16-sites

-5.2

0.04

120°

IC

-5.6
-5.8

iPEPS

-6.2

zigzag phase isconsistent withthe experiments

-6.4

iPEPS

-6

-6.6
-60

-40

-20

Na2IrO3

0.05

0

20

40

0.03
0.02
0.01

60

0

スピン液体計算の困難
PHYSICAL REVIEW B 90, 195102 (2014)

(3)

with
ave
ally
ergy
used
low
23].
vide
nrom
all
uent
at it

0.390
0.391

OSORIO IREGUI, CORBOZ, AND TROYER

(a)

FM
AFM
Exact

0.54

(a)

XX x bond
YY y bond
ZZ z bond

iTPSを用いた変分法で、虚時間発展のような通常の最適化をすると
0.53
0.392
Esite

初期状態に強く依存したバイアスのかかった結果が得られる
0.52
0.393

J.O. Iregui, P. Corboz, and M. Troyer, PRB 90, 195102 (2014)

0.394
0.395
0.04
0.390

Esite
M

0.391
0.03

PHYSICAL REVIEW B 90, 195102 (2014)

磁化(b)(a)

FM
AFM
Exact

0.392
0.02

OSORIO IREGUI, CORBOZ, AND TROYER
0.50
0.54

最近接スピン相関(a)(b)

XX x bond
YY y bond
ZZ z bond

0.51
0.52

0.393

0.00
0.395
0.040.00

0.51

0.50
0.53

0.52
0.51

0.01
0.394

0.53
0.50

0.05

0.10

0.15
1
D

0.20

0.25

0.30
(b)

0.54
0.00
0.50

0.03
FIG. 2. (Color online) (a) Energy per site and (b) magnetization
as a function of inverse bond dimension. Errors in the energy for the
0.02
largestvalue studied (D = 7) are• of the order of 10−4 . Magnetization
values are normalized to 1.

M

bra
vely
lled
sors
tion
. 1.
y the
d by
han
ions
tails
rbra
to
vely
lled
sors
tion
wave
. 1.
ergy
the
used
d by
23].
han
rom
ons
uent
ails
r to

0.05

0.10

0.15
1
D

0.20

0.25

(b)
0.30

0.51
キタエフ模型近傍のスピン液体研究にはFIG. 3. (Color online) Nearest-neighbor correlators for corresponding bond types in the ferromagnetically coupled case (a) andantiferromagneticallycoupled case (b). All remaining correlators are
キタエフスピン液体に近い初期状態
0.52
at most of the order of 10 for the largest value of D and exhibit• よい最適化方法の開発 similar improvement with increasing D.
0.01
0.53
From both Kitaev’sが必要
seminal paper [14] as well as later

work0.00[42] it can be gathered that the energy per site for

−3

0.54

FIG. 4. (Color
found using iPEP
phases are foundferromagnetic (daright). Magnetica
parameters in (4)phases. Phase bou
the spin liquid reg
D is increased.

スピン液体のコンパクトなテンソルネットワーク表現

Quasiparticle statistics in the gappe
GS phase diagram

キタエフスピン液体：•

J! = (0, 0, 1)

Gapped spin liquid は断熱的にToric codeに接続される•

Az

gappedToric codeはD=2 のiTPSで厳密に書けるB

gapless

Ax

gappedAygappedJ! = (1, 0, 0)
PHILIPP SCHMOLL AND ROMÁN ORÚS

J! = (0, 1, 0)

Fig. 5.3 Phase diagram of honeycomb
model. Thisis a slice
throughthe(2017
posit
PHYSICALREVIEW
B 95,
045112
J! coupling space along the Jx + Jy + Jz = 1 plane. The other octants are analo

GS for 8 sites honeycomb lattice†(ii) [âi ,(−1)n̂j ] = 0 for i "= j . Therefore, using this we canwrite the state as!
|ψ# =Cn1 n2 ··· (−1)n1 n2 (−1)(n1 +n2 )n3 · · · (Ŝ1 )n2 (Ŝ2 )n3 · · ·n1 n2 ···

•

!n2

!n1

!q1

−!q∗

!q∗

!q2

Fig. 5.4 Direct and reciprocal lattices of the honeycomb. The points ±!
q∗ are tpoints(49)of the gapless phase B.

Gapless spin liquid は簡単なテンソルネットワーク表現を持たない
where the phases come from the commutation and anticom5.4 Quasiparticle statistics in the gapped phase††× (â1 )n1 (â2 )n2 · · · |"#,•

mutation relations mentioned above. Finally, the actionof
It appears
that there are two particle types: fermions and vortices (hexthe string operators is easily computed on the Fockwpstate= −1). The vortices are associated with a Z2 gauge field, where ujk pl††(â1 )n1 (â2 )n2 · · · |"#, and the result is the quantum state of vector potential. Taking a fermion around a vortex results in the multithe state by −1 (compared to the no-vortex case). However, the details!
Cn1 n2 ··· (−1)n1 n2 (−1)(n1 +n2 )n3 · · ·|ψ# =fusion rules are not obvious.

マヨラナフェルミ粒子を用いた表現は、複雑なテンソルネットワーク

us look at the model from a different perspective. If J = J =P. Schmoll and R. Orús, Phys. Rev. B, 95 045112 (2017).the Let
system is just a set of dimers (see Fig. 5.5). Each dimer can be in tw
n1 n2 ···

x

Gapless のキタエフスピン液体を(n1 +n2 )n3y††· · · (â1 )n1 (â2 )n2 · · · |"#and ↓↓. The other two states have 2Jz higher energy. Thus, the ground sta

× (−1) (−1)!
††=
Cn1 n2 ··· (â1 )n1 (â2 )n2 · · · |"#,
n1 n2

degenerate.
(50)
If Jx , Jy $ Jz , we can use perturbation theory relative to the nonintera
n1 n2 ···
point. Let us characterize each dimer by an effective spin:!
!
!
!
where the last equation follows from the fact that the phases| ⇑ = | ↑↑ ;| ⇓ = | ↓↓ .

exactly cancel each other. The final state is nothing but
4th order of perturbation theory, we get:a fermionic state in second quantization with theAtsame
coefficients Cn1 n2 ··· as the original state for spin-1/2.
Jx2 Jy2 "(4)
Qp
Heff = const −The result is then clear: the Jordan-Wigner transformation
16Jz3 pdoes not change the overall coefficient of the state, andtherefore does not add anything quantitatively new to the
TN. In our construction, this transformation is thus taken into
account by simply saying that, from this point on, the wires in
the TN represent spins instead of fermions, so that we do notneed to worry anymore about fermionic SWAP gates for the
crossings. Mathematically, this is a site-by-site mapping fromthe Fock space of a spinless Dirac fermion to the Hilbert spaceof a spin-1/2.

「コンパクト」なテンソルネットワークでかけるか？

コンテンツ•はじめに•

量子多体状態のテンソルネットワーク表現•

ハニカム格子キタエフ模型•

ギャップレスキタエフスピン液体のテンソルネットワーク状態•

テンソルネットワークによる有限温度シミュレーション•

まとめProjector onto vortex free sector
The Kitaev spin liquid is in the vortex free sector.

1
6

Let us consider the projector onto this sector.

3
5

Exercise:
Projector on to Wi = 1:

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It can be represented by D=2 tensor network.
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2

4

Projector onto vortex free sector

H.-Y. Lee, R. Kanako, T.O. and N. Kawashima, PRL 123, 087203 (2019)

What is a tensor network representation for the vortex free projector?

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It is given by "loop gas" operator.

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Non zero elements:<latexit sha1_base64="5pGKh0Kh2GqTRcauZLwn9JCYp/Y=">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</latexit><latexit 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Loop structure of the operator

H.-Y. Lee, R. Kanako, T.O. and N. 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+

+ ...

Sum over the all closed loops!

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Notice:<latexit sha1_base64="WWTnu8rNRSnaLMIya3g6yc5ApgY=">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</latexit><latexit sha1_base64="19TEmi79boCGRayAtqTXLh65Te0=">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</latexit><latexit sha1_base64="EAIqA1OORC8xnA2FKadDmgVuZ/4=">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</latexit>:# of graphs<latexit sha1_base64="cmp8aJWgpNl0PzpCQzzPBlQsHYw=">AAACnXichVHLSsNAFD3GV33XuhFcGCwVRZCpCIqgFERQEFFrW8FKSOKowTQZkrRYiz/gD7gQBAVR8Q/cuvEHXPgJ4rKCGxfepAFRsd5hZs6cuefOGa4mTMP1GHtukBqbmltaI23tHZ1d3T3R3ljWtYuOzjO6bdrOpqa63DQsnvEMz+SbwuFqQTN5TjuY9+9zJe64hm1teGXBtwvqnmXsGrrqEaVEY7NyXjj2jiLkkSV5LKeIUSUaZ+MsCPk3SIYgjjBW7eg98tiBDR1FFMBhwSNsQoVLYwtJMAjitlEhziFkBPccx2gnbZGyOGWoxB7QukenrZC16OzXdAO1Tq+YNB1SykiwJ3bLquyR3bEX9vFnrUpQw/dSpl2rablQek760+//qgq0e9j/UtX17GEX04FXg7yLgPF/odf0paPTanpmPVEZZpfslfxfsGf2QD+wSm/61RpfP6vjRyMvQRUfUZuSP5vyG2QnxpOE1ybjqbmwYREMYAgj1JUppLCIVWTolUOc4xo30qC0IC1LK7VUqSHU9OFbSLlPPdiW4Q==</latexit><latexit sha1_base64="ZmJz+9hwVlIqSoO49dQGJOpBP9k=">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</latexit>...

Loop gas state: a vortex free state

H.-Y. Lee, R. Kanako, T.O. and N. Kawashima, PRL 123, 087203 (2019)

A simple vortex free state corresponding to the isotropic Kitaev model:<latexit sha1_base64="PLbYnl2wyXjTl9jmYZaUmXEfx4w=">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</latexit>

Ferromagnetic state pointing (1,1,1) direction.

<latexit sha1_base64="fWqO2LGaLc/mVAyJ4LDcQ7ogyi8=">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</latexit><latexit sha1_base64="5gbO9cY4lCCZw1hOcrAmczdTUNY=">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</latexit><latexit sha1_base64="iGoO2UUsxY3zAvBhwG/e0/0mJCM=">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</latexit>

D=2, TPS

Properties of the LGS

H.-Y. Lee, R. Kanako, T.O. and N. Kawashima, PRL 123, 087203 (2019)<latexit sha1_base64="PLbYnl2wyXjTl9jmYZaUmXEfx4w=">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</latexit>

Symmetries:
From the symmetries of Q and |111>, LGS is symmetric under•••

Lattice translation
C6 lattice rotation (+ spin rotation)
Reflection + Times reversal*

Magnetism

Single reflection or time reversal symmetry is broken due tounderlying |111> state, although it can be recovered byconsidering a linear combination of Q|111> and Q|-1-1-1>.

Vortex free condition ensures that the LGS is non-magnetic.
Qualitatively very similar to the Kitaev spin liquid.

Criticality of the LGS<latexit sha1_base64="PLbYnl2wyXjTl9jmYZaUmXEfx4w=">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</latexit>

H.-Y. Lee, R. Kanako, T.O. and N. Kawashima, PRL 123, 087203 (2019)

Criticality of the gapless KSL:•It belongs to so called conformal quantum point.
cf. E. Ardonne, P. Fendley, and E. Fradkin, Ann. Phys. 310, 493 (2004).

The wave function itself shows criticality (in 2d).
•It belongs c=1/2 Ising universality class.
(eg. K. Meichanetzidis et al, Phys. Rev. B 94, 115158 (2016))

If a wave function | ϕ⟩ is adiabatically connected to the Kitaev spin liquid,⟨ϕ | ϕ⟩ should show critical behavior which belongs to the Ising universality.

Criticality of the LGS<latexit sha1_base64="PLbYnl2wyXjTl9jmYZaUmXEfx4w=">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</latexit>

H.-Y. Lee, R. Kanako, T.O. and N. Kawashima, PRL 123, 087203 (2019)

LGS is mapped to classical loop gas:<latexit sha1_base64="AJS0I3glTRD295hYU89mhK9ZN5w=">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</latexit>:product of σγ corresponding to the graph G<latexit sha1_base64="WQRNTTtTjZgmESmAh39ZYJdhrik=">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</latexit><latexit sha1_base64="5/CKfxq6WdFd20FpPwmiWVovGLI=">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</latexit>:loop length<latexit sha1_base64="iGoO2UUsxY3zAvBhwG/e0/0mJCM=">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</latexit><latexit sha1_base64="jrJjXEXo3/ormzC9twCkRDn1/34=">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</latexit>

Identical wit the partition function of the classical loop gas modelwith fugacity 1/

3.

On the honeycomb lattice, it is exactly solvable.
(B. Nienhuis Phys. Rev. Lett. 49 1062 (1982).)

It is actually the critical point of the loop gas model,
ant its criticality belongs to the Ising universality class.

LGS : a simple Kitaev spin liquid like state?

H.-Y. Lee, R. Kanako, T.O. and N. Kawashima, PRL 123, 087203 (2019)

Qualitative properties of LGS:
<latexit sha1_base64="PLbYnl2wyXjTl9jmYZaUmXEfx4w=">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</latexit>

It satisfies the symmetries common with (gapless) KSL.
•••

Lattice translation
C6 lattice rotation (+ spin rotation)
Reflection + Times reversal

It is vortex free, and therefore nonmagnetic.
It shows the same criticality with KSL.
One can consider LGS as the simplest example of KSL.
(It might be similar to the case of AKLT sate against the Haldane phase.)Systematic improvement of LGS

H.-Y. Lee, R. Kanako, T.O. and N. Kawashima, PRL 123, 087203 (2019)

Energy of LGS for the Kitaev model:
When we calculate the energy of LGS, it is<latexit sha1_base64="Dn+JfhwY/IvihB9tPqlAgE35MbQ=">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</latexit><latexit sha1_base64="VQuQs46fG3Cre6MPm/O77St2o1E=">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</latexit>

Large discrepancy

Is is possible to improve the energy without spoiling nice properties of LGS?
Yes! We can systematically construct
a family of LGS by using tensor network.

Dimer gas operator

H.-Y. Lee, R. Kanako, T.O. and N. Kawashima, PRL 123, 087203 (2019)<latexit sha1_base64="itf8JOCM3DmJhnPtXWU6sNRHkck=">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</latexit>

0<latexit sha1_base64="GeBdsvXIfKYSrDYt1IZk+j1NyxU=">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</latexit>

ss
Rijk
=

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mappedl modelddition,centralical LG
HM[36–<latexit sha1_base64="wujF11inCiVgFG9Hi3Yqm4lFvOc=">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</latexit>

useful insee this,a gaugeijk , anda twonvariantof all dimer configurations:<latexit sha1_base64="5pGKh0Kh2GqTRcauZLwn9JCYp/Y=">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</latexit><latexit sha1_base64="o9yYeoDgANMt6DrpOFay0A5u3AA=">AAACinichVG7SgNBFD1ZXzFGE7URbIJBsQqzIvgCCWhhmYdRQUPYXSc6uC92N4EY/AFtFQsrBQvxD2xt/AGLfIJYRrCx8O5mQVSMd5iZM2fuuXOGq9q6cD3GWhGpp7evfyA6GBuKD48kkqNjW65VczRe0izdcnZUxeW6MHnJE57Od2yHK4aq8231aM2/365zxxWWuek1bF42lANTVIWmeETl5UoyzTIsiNRvIIcgjTByVvIBe9iHBQ01GOAw4RHWocClsQsZDDZxZTSJcwiJ4J7jBDHS1iiLU4ZC7BGtB3TaDVmTzn5NN1Br9IpO0yFlCtPsmd2xNnti9+yFffxZqxnU8L00aFc7Wm5XEqcTxfd/VQbtHg6/VF09e6hiMfAqyLsdMP4vtI6+fnzZLi4Xppsz7Ia9kv9r1mKP9AOz/qbd5nnhqosflbwEVXxEbZJ/NuU32JrLyITz8+nsatiwKCYxhVnqygKy2EAOJXqF4wznuJDi0py0JK10UqVIqBnHt5DWPwHOWZDP</latexit><latexit sha1_base64="5pGKh0Kh2GqTRcauZLwn9JCYp/Y=">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</latexit><latexit sha1_base64="v28wKHpGxIqLekH28hdYcVpqnVs=">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</latexit><latexit sha1_base64="qrJcrTO3paSOoWfHcCkot3kviRM=">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</latexit><latexit sha1_base64="5pGKh0Kh2GqTRcauZLwn9JCYp/Y=">AAACinichVHLSsNAFD3Gd61adSO4EUvFVbktgi+Qgi5ctta2gpaSxKmGpklI0oIWf0C3igtXCi7EP3Drxh9w0U8QlxXcuPAmDYgW6x1m5syZe+6c4SqWrjkuUbNH6u3rHxgcGg6NhEfHxiMTk3nHrNmqyKmmbtq7iuwIXTNEztVcXexatpCrii4KSmXDuy/Uhe1oprHjHluiWJUPDa2sqbLLVIZKkSjFyY/ZTpAIQBRBpM3II/ZxABMqaqhCwIDLWIcMh8ceEiBYzBXRYM5mpPn3AqcIsbbGWYIzZGYrvB7yaS9gDT57NR1frfIrOk+blbOI0QvdU4ue6YFe6fPPWg2/huflmHelrRVWafxsOvvxr6rKu4ujb1VXzy7KWPa9auzd8hnvF2pbXz+5amVXt2ONebqlN/Z/Q0164h8Y9Xf1LiO2r7v4UdiLX8VD3KbE76Z0gnwynmCcWYym1oOGDWEGc1jgriwhhS2kkeNXBM5xgUspLCWlFWmtnSr1BJop/Ahp8wvMM5DO</latexit><latexit sha1_base64="o9yYeoDgANMt6DrpOFay0A5u3AA=">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</latexit>.
(8)

We can show

Due to [Ĥ↵ , Ŵp ] = 0, it is obvious that R̂G commuteswith Ŵp for any G, and hence R̂DG does; [R̂DG , Ŵp ] = 0.
In fact, we can easily prove [R̂DG , Q̂LG ] = 0 and thatthe DG operator respects all symmetries of the KSL [33].
Therefore, its multiplication to | 0 i does not contaminate
the features of the KSL regardless of z. Moreover, it
can be expressed as the polynomial function of the KHM
Hamiltonian, which may be the reason why it improves
the energy of the ansatz quite efficiently. The first key
observation is that we can graphically represent Eq. (1)<latexit sha1_base64="bAxLKRRpvtQytVT3k4XkS6yvWzc=">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</latexit>, and it satisfies all symmetries same with LGO.

So, application of DG operator does not spoil the properties of LGO.

String gas states: energies

H.-Y. Lee, R. Kanako, T.O. and N. Kawashima, PRL 123, 087203 (2019)

nth-order string gas state (SGS)<latexit sha1_base64="B8x7BCltld7WHGgehROinPXWhhI=">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</latexit><latexit sha1_base64="5Qb/LsEo95dpbmpA1zYQW0ILSz0=">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</latexit>D# of parameters

E/J
ΔE/Eex<latexit sha1_base64="FsSOBLRzWV/lDt7HmBTkGr05vnQ=">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</latexit>

20-0.163490.17

41-0.19643
0.02<latexit sha1_base64="win7U/ct4AKxMxdb3ItB7Ici4U8=">AAACl3ichVG7SgNBFD2ur/iO2ig2waBYhdkQSGJjQJCUiZpEMBp21zEObnaX3U0gRn/AH1CwSsBC/ANbG3/Awk8QSwUbC+8+RCzUO8zMmTP33DnDVS1dOC5jT31S/8Dg0HBkZHRsfGJyKjo9U3bMpq3xkmbqpr2jKg7XhcFLrnB1vmPZXGmoOq+ox+vefaXFbUeYxrbbtvheQ6kb4lBoikvU/mnVckQtWbUVo67zWjTOElk/YgFIp0KQlWNygvkRRxgFM3qHKg5gQkMTDXAYcAnrUODQ2IUMBou4PXSIswkJ/57jDKOkbVIWpwyF2GNa63TaDVmDzl5Nx1dr9IpO0yZlDEvskd2wV/bAbtkz+/i1Vsev4Xlp064GWm7Vps7ntt7/VTVod3H0rfrTs4tDZHyvgrxbPuP9Qgv0rZOL163VzaXOMuuxF/LfZU/snn5gtN606yLfvPrDj0pe/CoeojZ99SL2OygnEzLhYiqey4QNi2ABi1ihrqSRQx4FlOgVG5fooifNS2vShpQPUqW+UDOLHyEVPwGJmJc3</latexit>

82-0.19681
0.0007

Exact-0.19682-

By using only two variational parameters,
we can obtain very accurate energy.

By LGS and SGS, we can accurately represent thegapless Kitaev spin liquid qualitatively and quantitatively!

found toare
reflectthe most qualitative features of the KSL, andloops
suppressed.
nSG provides a quantitatively accurate approximation to theInterestingly, the variational energy becomes also exact
he
KSL whereas keeping the qualitative features intact. In whatcy(withinmachineas LG
approaching
follows, wereinterpretprecision)
the CSLs as theand SG states theand opposite limits,providedirectevidence
identifying
the gap
topologicalorderasin J #2 /J [22,29].
φ
=
0
where
the
excitation
closes√aeach phase.
LEE,
KANEKO,OKUBO,
Note that ZO(1) (x) in Eq. (4) becomes critical at xc = 1/ 3 AND KAWASHIMA
0]
H.-Y.
Lee,
R. Kanako,
T.O.
and
N. Kawashima, PRB 101, 035140 (2020)
m
[25],
which
leads
to
the
critical
LG
state
at
(q
,
r
)
=
c c
$ √
√II. MODEL
LEE, KANEKO, OKUBO, AND KAWASHIMAngStar lattice
( The
3).
Remarkably,
the
LG
state
is
optimized
2 KSM
3
−
3,
2
−
ss Kitaev
model
on astheis defined[22]star latticedswith (qc , rc ) at φ!= 0 which
implies that the ground state is
% !
re
J
J
γ γ
γ% γ%
the criticalmacroscopic
Ĥ =LG− stateσ̂i exhibitingσ̂ j −σ̂i σ̂ j ,(1)loops. Furtherso
4 "i j#∈γ
4 "i j#∈γ %
more,
its
low-energy
physics
is described by the Ising CFT
he
1]
[25]
is forconsistent
with withthe expected[22,30–32].
where whichσ̂ γ stands
the Pauli matrix
γ , γ % = x, y, onez,

LGS for chiral spin liquidsi
Groundstate isa chiral spin liquid.
%

whereas "i j#γ accordingand "i j#γ , respectively,
denote the nearestTherefore,
to this circumstantial
evidence,
we may%neighbor pair on the intratriangle (γ ) and intertriangle (γ )concludethat the LG ansatz at φ = 0 is the exact ground state.
bonds connecting sites i and j as defined in Fig. 1(a). Note thatthe Hamiltonian commuteswith two types of flux operators
Energydefined on the triangle plaquette V̂p = σ̂1x σ̂2y σ̂3z and dodecagon
y Abelianplaquette Ŵp non-Abelian= σ̂1x σ̂2z σ̂3y σ̂4x · · · σ̂12[22] where the site indices1
are defined inexact
Fig. 1(a). Therefore, the Hamiltonian is block-0.38
diagonalized,
each block is characterized
by the(b)set of
(a)
| and
0.8
0

-0.4

*
r*
q*
qc

©2020
American Physical Society
dE[%]
0.6

0.3

E

035140-1| 1

Topological
EE of the star lattice where
FIG. 1. (a) Schematics

0.2

-0.42

0.4

0.1
0
0

-0.44
0

0.1

0.2

0.3

/

0.2
0.25 0.5

0.4

0.5

0

0

0.1 0.2 0.3 0.4/

bonds defined in the model [Eq. (1)] are specified byand yellow colors, respectively. (b) Four local confi
corresponding magnetic states of the LG state [Eq. (3is defined in Eq. (2), and x, y, z denote each bond of th

the flux numbers or eigenvalues of flux ope±1, Wp = ±1}. Since the operator V̂p consists omatrices, the time-reversal (TR) transformation
number, i.e., T V̂p T −1 = −V̂p , and hence, the T
is spontaneously broken in eigenstates of Ĥ.
[22] that the ground states do not break any latti(that is, the CSLs) and live in the vortex-free
0.5 {Ŵp = 1, V̂p = 1}. In addition, the model exhibi
cal phase transition√ between the non-Abelian an#
CSLs at J /J = 3 [22].

LGS as initial states: KGG' model

H.-Y. Lee, R. Kanako, et al, Nat. Commun. 11, 1639 (2020).

By using LGS as the initial states,
we determined the phase diagram of KGG model accurately.
Although the nature of NP1 and NP2 states have not been clear.
(They break rotational symmetry spontaneously.)
Cf. M. Gohlke et al., Phys. Rev. Research 2, 043023 (2020).

ARTICLEICATIONS | https://doi.org/10.1038/s41467-020-15320-x
They seem to be adiabatically connected to classical state...
NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-15320-xabaMagnetization

h

Flux
h

0.4

NP1

0.15

0.3

Zig-zag

N

0.1

KSL
0.1

0.2
Γ/ K 

0.3

0

0

0.1

0.1

Γ/ K

0.2 c
6-siteKSLKSL
0

0

N

0.2
Γ/
K 

6

Γ/ K 0.4

Zig-zag

P2

0.05

0.05

18

0.5

0.6
NP2

0.2

NP2

0

0.1

P2

P NP1

0.8

NP1

P

0.1

0

18-site

P NP1
0.15

P

b (Γ ′, h) = (0, 0.15)

Γ′ = 0

0.2

0.3

0
0.3

–0.20

(Γ ′, h) = (–0.03, 0.15)
P

NP1
NP2

ZigzagΓ/ K 0.5

Fig. 4 Comparison to the phase diagram of theK & Γ model. a Phase diagram at Γ0 ¼ 0 as functions of Γ=jKj and h. NP1 and NP2

Phase diagram of ED（24-site）(Y. Yamaji et al, PRL 113, 107201(2014))

LGS as initial states

FIG. 3: (color online): (a) ∆-dependence of matrix elemen
of JZ , JX as functions of ∆. Around the ab initio valu
at ∆ (∼ −28 meV) listed in Table II, K < 0, K ! <J > 0, J ! > 0, and J !! > 0 are stably satisfied with gra
ual dependences on ∆. (b) Ground state pahse diagram f
Na2 IrO3 with lattice distortions represented by changes inThephasemagnetization
boundaris are determined by anomalies in secoLocal
derivatives of the exact energy for the 24-site cluster with
spect to ∆. Around the ab initio parameter ∆ = −28 me
the zigzag order appears. By increasing ∆, a 6-site unit corder (or 120◦ -structure[27])illustrated in the lower left pan
SL
and a 24-site unit cell long-period order[23], appear. (c)dependence of the ground state of the generalized Kitae
Heisenberg
model for “expanded lattices.” Here we negle
Ferro
the small hopping parameters other than t and take a larg
Hund coupling JH = 0.3 eV. Spin liquid phases compete wi
12-siteferromagnetic states and 12-siteunit cell orders illustrated
the lower right panel[23].

ab initio Hamiltonian for Na2IrO3 (with lattice expansion)
Energy and Phase diagram
-4.6

0.4

-4.7

SL

-4.8
-4.9

0.3

Ferro
12-site

-5

0.2

-5.1

0.1

-5.2
-5.3
-60

•

-40

-20

0

20

40

60

0

-60

-40

-20

0

20

40

60

Center,Institute for Solid State Physi
iTPS phase diagram is qualitatively consistentpercomputer
with
the
ED.
Univ. of Tokyo. This work was also supported by Gran•for Scientific Research (No. 22104010, and N
Around Δ=0, a Kitaev spin liquid phase is in-Aid
clearlystabilized.
223400901) from MEXT, Japan.

まとめ
テンソルネットワークは量子多体状態を研究できる有用なツール•••

iTPS.により、様々なフラストレート量子スピン系を解析できる

変分計算だけでなく、我々は（ギャップレス）スピン液体のコンパクトなテンソルネットワーク表現を提案••

このテンソルネットワーク状態は loop gas または string gas 構造を持つ•

この状態はキタエフ模型と共通の対称性を保持している•

この状態は臨界的であり、イジングユニバーサリティに属するテンソルネットワーク法は有限温度の計算にも適用可能•

有限系サイズの計算では、MPO表現を用いて高精度計算が行え、実験と比較可能な熱ホール伝導度も解析できる

物性研高度化プロジェクト：Tensor Netwok Solver (TeNeS)
https://github.com/issp-center-dev/TeNeS
無限系のTPS（iTPS）を用いた変分法による基底状態計算
虚時間発展法によるテンソルの最適化
MPI/OpenMPによる大規模並列計算に対応
mptensor（森田） によるテンソル演算の並列化

二次元の量子スピン系やボゾン系が簡単に計算可能

カゴメ格子模型の磁化曲線

D = 10@sekirei 72node
~4時間弱



mVMCやHPhiと類似のinput file
標準的な二次元格子にデフォルトで対応
原理的には任意の二次元格子に対応可能
開発チーム
•
•
•
•
•
•

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