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Computational
Science
Alliance
T h e Un ive r s i t y of To k yo

15mm

テンソルネットワークによる情報圧縮と物性物理への応用
東大理, JST さきがけ

大久保毅

Contents•

Huge data in physics•

Information compression•

Basics: singular value decomposition•

Tensor network renormalization•

Tensor network quantum states•

Applications•

Summary and outlook

Contents•

Huge data in physics•

Information compression•

Basics: singular value decomposition•

Tensor network renormalization•

Tensor network quantum states•

Applications•

Summary and outlook

Huge data in physics
Many-body problems in physics• Celestial movement• Gases, Liquids• Molecules, Polymers (eg. Proteins), ...
• Electrons in molecules and solids• Elemental particles (Quantum Chromo Dynamics)
In these problems, "systems" contain huge degrees of freedoms:
6N-dimensional phase space for classical mechanics
O(eN)-dimensional Hilbert space for quantum systems

(Classical) statistical mechanics
Canonical ensemble:：State (e.g. {S1, S2, ... SL} ): Probability to appear state Γ: Inverse temperature：Hamiltonian (Energy)

Partition function（分配関数）＝Normalization factor of the canonical ensemble

Relation to the free energy in thermodynamics

If we can calculate Z, we can easily estimate thermodynamic properties.

Expectation value in canonical ensemble
Expectation value of O:
Expectation value of physical quantity⟷Macroscopic physical quantities observed in thermodynamics
We can calculate thermodynamic quantities form microscopic model,
if we can calculate the sum of all states

Real problems：

is too huge to calculate exactly<latexit sha1_base64="wPSeXFfxsbQoBSwLoEdTbQxlwN0=">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</latexit>(Even if we use super computer)Standard procedures: MD or MC samplings

Quantum systems
Quantum system: governed by Schrödinger equation:Hamiltonian (Energy):Wave function (state vector)

Inner product:

Nature: Elementary particles, e.g. electrons, obey quantum mechanics.
Static problems: Time-independent Schrödinger equation

Energy= Eigenvalue problem

Quantum many-body systems
Example of quantum system: Array of quantum bits
1 bit

A quantum bit is represented by two basis vectors.
or

2 bits<latexit sha1_base64="b0TSKdinjwk9T/dIa6/LeFDtppE=">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</latexit>

The Hilbert space is spanned by four basis vectors.

Simple notation::complex number
The Hamiltonian for 2 bits system can be represented in these bases.

Matrix element:

Quantum many-body systems
Example of quantum system: Array of quantum bits
N bits: Dimension of the Hilbert space = 2N
Hamiltonian is 2N × 2N matrix

Need to solve eigenvalue problem of huge matrix!
In physics,• We often interested in the "ground state" (smallest eigenvalue)
基底状態

We can concentrate to a special state.
•

Typical system only has "short range" interactions

Hamiltonian matrix becomes sparse.

Information compression by tensor network
How can we treat and calculate such eN data in numerics?
One of the methods is an approximate information
compression by tensor network representations
Calculation of the partition function:••

Tensor network representation of Z
Approximated contraction of it
trough a coarse graining<latexit sha1_base64="uhUm8tL2oC7dG1LOwKGrsT7xTTE=">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</latexit>

Eigen value problem:••

Tensor network representation ofan eigenvector
Variational optimization of it<latexit sha1_base64="XkC8uXv0uT1Qilx+79HHlBouX2k=">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</latexit><latexit sha1_base64="AN3KQi0sZodhQve/LudWDvR12b0=">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</latexit>

Contents•

Huge data in physics•

Information compression•

Basics: singular value decomposition•

Tensor network renormalization•

Tensor network quantum states•

Applications•

Summary and outlook

Singular value decomposition (SVD)
Singular value decomposition (SVD):
Any matrices are uniquely decomposed as<latexit sha1_base64="breXh8DXLtNA8Ewjr1L2O8OCXTI=">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</latexit>

Unitary

Unitaryr=rank(A)Diagonal matrix with
non-negative real elements
Singular values

Amount of data in SVD representation,neglect zero
singular values

If rank(A) is much smaller than M and N,
we can reduce the data to represent A.
(At this stage, no data loss)

Low rank approximation by SVD
Consider a matrix obtained by neglecting smaller singular values

Keep the largest k singular values(and corresponding singular vectors).

This approximation is one of the best low rank approximation.
<latexit sha1_base64="ZDInn7UeZY5fko9ApPg7p1gOAL4=">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</latexit>

Image compression: grayscale image
Image: 1024 × 768 pixels

768 x1024 matrix A
Amount of data=786,432
Perform SVD of A:
approximation
Amount of data=(768 +1024 + 1)×χ

Image compression:
compression grayscale image

Rank:
Data: 786,432(Original)

179,300

179,30Scalar, Vector, Matrix, Tensor,...
Scalar:<latexit sha1_base64="xjTtQZt/S06bAHzsr4SBpd4jRhs=">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</latexit>

Vector:<latexit sha1_base64="wlWQUpOEA9Mmb6ql6UkCS7qzJwA=">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</latexit>

Matrix:i

NumberOne dimensional array of numbers
Two dimensional array of numbers<latexit sha1_base64="PUy+WJIK1cOmBYjLGbwuDqULCAk=">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</latexit>

Tensor:Higher dimensional array of numbersi

Scalar: 0-dim. tensor
Vector: 1-dim. tensor
Matrix: 2-dim. tensorjkGraphical representations for tensor network•

Vector：•Matrix：Tensor：•* n-rank tensor = n-leg object

When indices are not presented in a graph, it represent a tensor itself.

Graphical representations for tensor network
Matrix product
C

＝

A

B

C

＝

A

B

Generalization to tensors
C
B

A

Contraction of a network = Calculation of a lot of multiplications

Low rank approximation: generalization to tensor
Tensor:ij

Naive application of SVD:
Make a matrix by dividing indices into two parts.

kThen apply SVD (and low rank approximation).
i

j

i

l

k

l

j

i

=

T

T

i

j
k

l

k

k
Note: The result depends on the initial mapping to a matrix.

TjContents•

Huge data in physics•

Information compression•

Basics: singular value decomposition•

Tensor network renormalization•

Tensor network quantum states•

Applications•

Summary and outlook

Tensor network representation of partition function
Classical Ising model on a chain

Partition function:
Transfer matrix(転送行列)

+1

-1

+1
-1

Tensor network representation in two dimension
Classical Ising model on the square lattice

We can use a tensor instead of the transfer matrix.
Tensor?

Tensor network representation in two dimension
A
A
A

A

A
A

A
A

A
A

Partition function = Tensor network of tensor A
Square lattice Ising model→Square lattice tensor network rotating 45 degrees.
*We can construct a tensor network where tensors are on the nodes of original lattice.

Outline of tensor network renormalization
Scaler represented
by L×L tensors(L×L)/2 tensors

Corse graining (Renormalization)
into √２times longer scale.

: D×D×D×D

Approximation

Reduce the number of tensors
keeping their size constant: D×D×D×D

Recipe of Tensor Renormalization Group (TRG)
M. Levin and C. P. Nave, Phys. Rev. Lett. 99, 120601 (2007)
Z.-C. Gu, M. Levin and X.-G. Wen, Phys. Rev. B 78, 205116 (2008)

1. Decomposition
Regard a tensor as a matrix
l

i

i

i

j

l
j

j
k

: D×D×D×D

D-rank approximation
by SVDlkk: D2×D2

Approximation

Recipe of Tensor Renormalization Group (TRG)
M. Levin and C. P. Nave, Phys. Rev. Lett. 99, 120601 (2007)
Z.-C. Gu, M. Levin and X.-G. Wen, Phys. Rev. B 78, 205116 (2008)

2. Coarse graining
Contraction of
inner indices

In total, two originaltensors are coarse
grained into a new tensor.

: D×D×D×D

Recipe of Tensor Renormalization Group (TRG)
M. Levin and C. P. Nave, Phys. Rev. Lett. 99, 120601 (2007)
Z.-C. Gu, M. Levin and X.-G. Wen, Phys. Rev. B 78, 205116 (2008)

Calculation cost:

SVD= O(D6)Contraction= O(D6)(per tensor)*By one TRG step, # of tensors is reduced by 1/2.

We can calculate the contraction in polynomial cost!

HOTRG and Anisotropic TRG
Coarse-graining tensors anisotropically:

This approach can be easily generalized to high dimensions.
HOTRG

Z. Y. Xie et al, Phys. Rev. B 86, 045139 (2012)⌘<latexit sha1_base64="6fFfiOA5K65pfNgux7h1VXJdczM=">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</latexit>

ATRG<latexit sha1_base64="Jd45KrQGZigKLvdVxV6pbP7tCsg=">AAACzXichVFLL8RQGD1T7/dgI2ExMSEsyK1IiIVIWNgxwyDxmLSdOzQ6bdPemaDGwkbiD1hYkUiIFX/BxsbSwk8QSxIbC9/caeIV3Ka95577ndNz76e7lukLxh4iSkVlVXVNbV19Q2NTc0u0tW3Bd/KewVOGYznekq753DJtnhKmsPiS63Etp1t8Ud+cLO0vFrjnm449L7ZdvprT1m0zaxqaICod7Zrpm1oLVgTfEtIs8HimGAxnBtRisT8djbNBJkfsJ1BDEEc4Zp3oHVaQgQMDeeTAYUMQtqDBp2cZKhhc4lYREOcRMuU+RxH1pM1TFacKjdhN+q7TajlkbVqXPH2pNugvFr0eKWPoYffsgj2zW3bJHtnbr16B9Chl2aZZL2u5m2457Jh7/VeVo1lg40P1Z2aBLEZlVpOyu5IpncIo6ws7R89zY8meoJedsifKf8Ie2A2dwC68GGcJnjz+I49OWaSLRL/Xfbolaqb6vXU/wcLQoEo4MRyfGA/bWotOdKOPejeCCUxjFiny3cc5rnCtzCh5ZVfZK5cqkVDTji9DOXgHms2k3w==</latexit><latexit sha1_base64="J5GUG7uZLoR7+CVTZKb0Xbm7JpI=">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</latexit>

D. Adachi, T. Okubo, and S. Todo, arXiv:1906.02007⌘<latexit sha1_base64="6fFfiOA5K65pfNgux7h1VXJdczM=">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</latexit>

Application to a classical partition function
Partition function(Periodic boundary condition)
Repeat TRG step
until only a few
tensors remain.

We can easily calculate physical quantities from Z.
Free energy:
Energy:Specific heat:(Use difference approximation
or auto differentiation)
H.-J. Liao et al, Phys. Rev. X 9, 031041(2019)

Example of calculation
Error of free energy for 2D Ising model

XIE, CHEN, QIN, ZHU, YANG, AND XIANG

energy, the accuracy of the specific heat data is much less thanIsing model in infinite sizethat of the internal energy, especially around the critical point.
This causes a big error in the determination of the exponent α
with the above formula. This problem can be solved by directly
evaluating this exponent from the temperature dependence of
the internal energy. From the temperature integration of thespecific heat, it is simple to show that the internal energy
shouldZ.exhibit
following
criticalbehavior:
Y. Xiethe
et al,
Phys. Rev.
B 86,
045139 (2012)
U = Uc + at + bt 1−α ,Energy and specific heat of 3D Ising model
PHYSICAL REVIEW B 86, 045139 (2012)

(a)
(n)

z

(b)

(n)

T y'1
x1
x'1
y'2
y1 i
x2
x'2
(n)
T
y2
z'

T

z

(n)

V

(n)

M

is already less than 10−7 even at the critical temperature,much more accurate than the TRG result.7,8 The HOSRG also

U x'

x U
y

(n)

(n)

V

y'
(n)

(n)

FIG. 4. (Color online) Comparison of the relative errors of freeenergy with respect to the exact results for the 2D Ising modelobtained by various√ methods with D = 24. The critical temperature
Tc = 2/ ln(1 + 2).

(17)

z'

M

FIG. 8. (Color online) The internal energy and the specific heat
for the 3D Ising model obtained by thez HOTRG with D = 14.
The Monte Carlo result (black curve) obtained
y' from an empirical(n+1)
x'
fit formula given in Ref.
T 27 is shownx for comparison.
(n+1)
T
y
z'
045

Interesting topics in tensor network renormalization•

Try to find efficient algorithm to remove "short range" entanglement•

TNR, Loop-TNR, GILT, Gauge fixing
TNR: G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015)
Loop-TNR: S. Yang, Z.-C. Gu and , X.-G. Wen, Phys. Rev. Lett. 118, 110504 (2017)
GILT: M. Hauru, C. Delcamp. S. Mizera Phys. Rev. B 97, 045111 (2018)
Gauge fixing: G. Evenbly, Phys. Rev. B 98, 085155 (2018)•

Application to lattice QCD••

TRG with Grassmann algebra Z.-C. Gu, F. Verstraete, and X.-G. Wen, arXiv:1004.2563
S. Takeda, and Y. Yoshimura PTEP 2015, 043B1 (2015).

Property at the criticality•

Relation to Conformal invariance
G. Evenbly and G. Vidal,Phys. Rev. Lett. 115, 200401 (2015)
G. Evenbly, Phys. Rev. B 95, 045117 (2017)

Contents•

Huge data in physics•

Information compression•

Basics: singular value decomposition•

Tensor network renormalization•

Tensor network quantum states•

Applications•

Summary and outlook

Information compression by tensor networks
We can not treat entire data in the present computers.
Try to reduce the "effective" dimension of(Hilbert) space
By considering proper subspace of the Hilbert space,
we can represent a quantum state efficiently.
Hilbert space
Tensor network quantum states!
SubspaceTensor network state
G.S. wave function:=

Vector (or N-rank tensor):``Tensor network”
decomposition
Matrix Product State(MPS)# of Elements＝aN
i1

i2 i3

i1

i2

i4 i5

=
：Matrix for state m

i3
i5

i4

i5
i3

General network
X,Y : Tensors
Tr : Tensor network contraction

i1

i4

i2

By choosing a ``good” network, we can express G.S. wave function eﬃciently.
ex. MPS: # of elements ＝2ND2

D: dimension of the matrix A

Exponential→ Linear*If D does not depend on N…

Area law of the entanglement entropy
Entanglement entropy:
Reduced density matrix of a sub system (sub space):A

B

Entanglement entropy = von Neumann entropy of ρA

B

General wave functions:
EE is proportional to its volume (# of spins).

LA(c.f. random vector)

Ground state wave functions:
For a lot of ground states, EE is proportional to its area.
J. Eisert, M. Cramer, and M. B. Plenio, Rev. Mod. Phys, 277, 82 (2010)

Ground state are in a small part of the huge Hilbert space!

Good reviews:

Matrix product state (MPS)(U. Schollwöck, Annals. of Physics 326, 96 (2011))(R. Orús, Annals. of Physics 349, 117 (2014))

MPS：Matrix for state i

Note:
•

MPS is called as "tensor train decomposition" in applied mathematics(I. V. Oseledets, SIAM J. Sci. Comput. 33, 2295 (2011))•

A product state is represented by MPS with 1×1 "Matrix" (scalar)Matrix product state without approximation
General wave function (or vector) can be represented by MPS exactly
through successive Schmidt decompositions
SVD

SVD of<latexit sha1_base64="6m/gbzqlDKBTUkwq2Gwhy8OFn50=">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</latexit><latexit 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sha1_base64="sUfwMe83hWkdG+Tm2dWYlfxOS9Q=">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</latexit>

(row)

…

(column)(row)(column)

In this construction, the sizes of matrices
depend on the position.
Maximum bond dimension = aN/2
At this stage, no data compression.

Matrix product state: Low rank approximation=

Maximum bond dimension = aN/2

Constant bond dimension = χ

If the entanglement entropy of the system is O(1) (independent of N),
matrix size "χ" can be small for accurate approximation.
MPS is good for gapped 1d systems.
On the other hand, if the EE increases as increase N,"χ" must be increased to keep the same accuracy.

Tensor Product State

N-rank tensor:=

T
m1 m2 m3 m4 m5

TPS (Tensor Product State) (AKLT, T. Nishino, K. Okunishi, …)
PEPS (Projected Entangled-Pair State)(F. Verstraete and J. Cirac, arXiv:cond-mat/0407066)

d-dimensional tensor network representation
for the wave function of a d-dimensional quantum system

TPS on square lattice

Tr: tensor network “contraction”: Rank 4+1 tensor

D = “bond dimension”

x,y,x’,y’ = 1,2, ... D
mi= 1,2, ... m

m = dimension of the local Hilbert space*D can be larger than m. “Virtual state “It satisfies the area low!

Enough large, but finite, D, a lot of
G.S. can be represented by TPS＊Finite D even for infinite system!

Variational calculation using TPS(Typical) Optimization: Imaginary time evolution
Approximatin

Costinformation Accuracy

Simple update

O(D5)

local

bad

Full update

O(D10)

globalbetterWe repeat updates about 103~ 105 steps

Evaluation: Contraction of the whole network
We use the corner transfer matrix method.
(R. J. Baxter (1968), T. Nishino, et al (1998), R. Orus et al (2009) ...)

Cost ~ O(D10)
Only a few calculations

Environment

C

e

e

e
C

C

e

C

TeNeS: Tensor Network Solver
We are developing a open source software for massively parallel
tensor network solver for 2D quantum lattice system.
(c++)
https://github.com/issp-center-dev/TeNeS
So far, it is version 0.1.
Update to version 1.0 is scheduled on March.

Contents•

Huge data in physics•

Information compression•

Basics: singular value decomposition•

Tensor network renormalization•

Tensor network quantum states•

Applications•

Summary and outlook

Quantum spin system: typical behaviors（Quantum) spin system:
Spin degree of freedoms defined ona lattice and interact each other
Example: "Transverse field Ising model"

Disorder(poing +x)

Ferro(pointing ±z)<latexit sha1_base64="WGPOYK4P8+EtP+RG4T2pDeTgX5o=">AAACj3ichVG7SgNBFD1Z389EbQQbMUSsZCJCxEICFmqXqImBRGR2HXXMvtidBDT4A1Z2olYKFuIf2Nr4Axb5BLFUsLHw7mZBVNQ7zMyZM/fcOcPVXVP6irFmTGtr7+js6u7p7esfGIwnhoaLvlPzDFEwHNPxSjr3hSltUVBSmaLkeoJbuik29OpicL9RF54vHXtdHbhi0+K7ttyRBldEFStL3LL4ViLJplkY4z9BOgJJRJFzEneoYBsODNRgQcCGImyCw6dRRhoMLnGbaBDnEZLhvcAReklboyxBGZzYKq27dCpHrE3noKYfqg16xaTpkXIcKfbIbtgLe2C37Im9/1qrEdYIvBzQrre0wt2KH4+uvf2rsmhX2PtU/elZYQdzoVdJ3t2QCX5htPT1w9OXtfnVVGOSXbFn8n/JmuyefmDXX43rvFi9+MOPTl7CKgGiNqW/N+UnKM5MpwnnZ5PZhahh3RjDBKaoKxlksYwcCvTKPk5whnNtSMtoC1q2larFIs0IvoS28gH5vJMP</latexit><latexit sha1_base64="ueUTnz6aYO08gQxt2hihgPUxD+g=">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</latexit>

Usually, the ground states have (magnetic) long range orders:
They may appear as a result of spontaneous symmetry breaking.
or
They may be induced by external fields.

Frustration in spin system
Frustration：Competition among several optimization conditions
Optimization：minimization of the total energy
Antiferromagnetic
local energy minimization ： anti-parallel spin pair

Frustration in spin system
Frustration：Competition among several optimization conditions
Optimization：minimization of the total energy
Antiferromagnetic
local energy minimization ： anti-parallel spin pair

Ising spins
SquareAll pairs can be anti-parallel
No frustrationFrustration in spin system
Frustration：Competition among several optimization conditions
Optimization：minimization of the total energy
Antiferromagnetic
local energy minimization ： anti-parallel spin pair

Ising spins
SquareAll pairs can be anti-parallel
No frustration

Triangle

Frustration in spin system
Frustration：Competition among several optimization conditions
Optimization：minimization of the total energy
Antiferromagnetic
local energy minimization ： anti-parallel spin pair

Ising spins
SquareAll pairs can be anti-parallel
No frustration

Triangle

Frustration in spin system
Frustration：Competition among several optimization conditions
Optimization：minimization of the total energy
Antiferromagnetic
local energy minimization ： anti-parallel spin pair

Ising spins
SquareAll pairs can be anti-parallel
No frustration

Triangle

Frustration in spin system
Frustration：Competition among several optimization conditions
Optimization：minimization of the total energy
Antiferromagnetic
local energy minimization ： anti-parallel spin pair

Ising spins
SquareTriangle?
All pairs can be anti-parallel
No frustration

One of three pairs isnecessarily parallel
Frustration!

Frustration in spin system
Frustration：Competition among several optimization conditions
Optimization：minimization of the total energy
Antiferromagnetic
local energy minimization ： anti-parallel spin pair

Ising spins

Huge degeneracy in the ground state.
Large fluctuations!
Triangle
Square?
All pairs can be anti-parallel
No frustration

One of three pairs isnecessarily parallel
Frustration!

posing a ‘spinon Fermi surface’ coupled to a U(1) gauge field — ha
had some success in explaining data from experiments on κ-(BEDT
TTF)2Cu2(CN)3 (refs 65, 66). Related theories have been proposed fo
ZnCu3(OH)6Cl2 (ref. 67) and Na4Ir3O8 (ref. 68). However, comparison

Quantum spin liquida

If interactions of the (quantum) spin systems contain frustrations:
Their ground states might not have any long range order.
b

Quantum spin liquid
There are a lot of spin liquids based on the mean field theory.
c

•
•
•
•

•

•

Z2 spin liquid
Chiral spin liquidU(1) spin liquids...

Spin liquid (RVB)

Figure 3 | Valence-bond states of frustrated antiferromagnets. In a VBS
state (a), a specific pattern of entangled pairs of spins — the valence bonds(L. Balents, Nature (2010))— 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

We want to find novel states of the matter•
Quantum spin liquid
204•
Topological phase
We want to investigate phase transitions between them•
(Quantum) critical phenomena•
Topological phase transition
199-208 Insight - Balents NS.indd 204© 2010 MacmillanPu blposing a ‘spinon Fermi surface’ coupled to a U(1) gauge field — ha
had some success in explaining data from experiments on κ-(BEDT
TTF)2Cu2(CN)3 (refs 65, 66). Related theories have been proposed fo
ZnCu3(OH)6Cl2 (ref. 67) and Na4Ir3O8 (ref. 68). However, comparison

Quantum spin liquida

If interactions of the (quantum) spin systems contain frustrations:
Their ground states might not have any long range order.
b

QuantumAspin
lot ofliquidThere are a lot of spininteresting
liquids based onthingsthe meanoccurfield theory.
c

23
Z2the
spin liquidin
Avogadro
scale
~
10
•
Chiral spin liquid
•
→We
need
U(1) spin
liquids large scale calculations!
•

•

•

•

...

Spin liquid (RVB)

Figure 3 | Valence-bond states of frustrated antiferromagnets. In a VBS
state (a), a specific pattern of entangled pairs of spins — the valence bonds(L. Balents, Nature (2010))— 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

We want to find novel states of the matter•
Quantum spin liquid
204•
Topological phase
We want to investigate phase transitions between them•
(Quantum) critical phenomena•
Topological phase transition
199-208 Insight - Balents NS.indd 204© 2010 MacmillanPu bl

Kagome lattice Heisenberg model
Kagome lattice

Hamiltonian•

Ground state at zero field
Classical GS:

All states satisfying “120 degree structure”
Macroscopic degeneracy

Quantum fluctuation:
S=1/2 quantum spin：

Spin liquid?
•••

KagomeZ2 spin liquidU(1) spin liquid…

Taken from http://koharu2009.blogspot.jp/

Results：Magnetization curve(R. Okuma, D. Nakamura, T. Okubo, et al, Nat. Commun. 10, 1229 (2019))

Several magnetization plateaus are stabilized・Almost converged data up to D=7
1/9, 1/3, 5/9 :clear plateaus
7/9: weak anomaly

Magnetization curve by iTPS

1

D=4
D=5
D=6
D=7

0.8

7/9

0.6

5/9

0.4

1/3

0.2
0

1/9
0

0.5 1

1.5 2

2.5 3

Results：Magnetization curve(R. Okuma, D. Nakamura, T. Okubo, et al, Nat. Commun. 10, 1229 (2019))

Several magnetization plateaus are stabilized・Almost converged data up to D=7
1/9, 1/3, 5/9 :clear plateaus
7/9: weak anomaly

Magnetization curve by iTPS

1

D=4
D=5
D=6
D=7

0.8

7/9

0.6
1/3 plateau state is a resonated state.

5/9

0.4

1/3

0.2
0

1/9
0

0.5 1

1.5 2

2.5 3

Eq. (3), the spin coupling, e.g., on a (x)-bond, is of S i S j% þ s2sin theverytype,
formpreciselyof the exchange
Hamiltonian
dependsonbondand excit
third
as in the Kitaev model. The honeycomb
geometry
profileof Kramersstates,as we pies, withlattice throughprovidesa adensity
particularly
striking
exampleof newHunddemonstratephysics below.
introduced by strong SO coupling: the[For ! ¼
andHeisenbergcouplings
model is ofconverted
into theKitaev states.—model
the 180!
Exchangeneighboring
KramersHamwith a spin-liquid
We consider
the limitgroundof the state.
strong spin-orbit coupling, i.e., canting aWhila physicalex-the is indepe
Theiscompound
Li2 RuO
3 [28] represents
when !
larger thanexchange
interaction
betweenfrom
distortion
ample of the A2 BO3 structure. By replacement of spin-one
isospins.
for isospins are then4þ The exchange Hamiltonians
4þ! intoSranwith spin-one-half Ir ions, one may realize aRu
2I
obtainedbyprojecting
thecorresponding
superexchangeAsstrongly spin-orbit-coupled Mott insulator with low-energy
theinmspin-orbital
modelsonto
theisospinstatesEq.
(1).
First,
we
Hamiltonphysics described by the Kitaev model.
mode
present‘‘Weak’’
thepresults
for
the
case
of
cubic
symmetry
(!
¼
0,
IrO
.—As
an
exampleofferromagnetism
of
Sr
2
4
S~~i
model
ﬃﬃﬃ
enou
Mott insulator,
discuss
layeredsin"a¼spin-orbit-coupled
1= 3), and discusslater the weeffectsofthe
a tetragonal
gered
rot
pling
of theundopedhigh-Tc
compoundSr
2 IrO4 , a ttwo
2g analog
distortion.
We
considercommoncases
of
TM-O-TMtanð2#ÞStrong spin-orbit couplingcally¼
Effectivebond``spin”
moment:" square lattice of Ir4þ ions is
cuprate
La
CuO
.
In
Sr
IrO
,
a
2
4
2
4
geometries: (A) a 180 -bond formed by corner- anisotrop! whformed by corner-shared IrO6 octahedra, elongated along
shared octahedra as in Fig. 2(a), and! (B) a 90" -bond Hamilton(B
the c-axis and rotated about it by ! ’ 11 [19] (see Fig. 4).
formedbyedge-shared
ones,
Fig.
2(b).
on a
a magnetic transition at # 240 K disSr2 IrO4 undergoes
G.Jackeli, et al., PRL 102, 017205 (2009) (A)A 180" bond: For this geometry, the nearest2
3
2 x
the
z z
S 1S 3 is diagonal in thezzorbital space
J. Chaloupka, et al., PRL 105, 027204 (2010)neighbor t2g
nond(a) hopping matrix(b)3and, on a given bond,only two orbitals are active, e.g., jxyi
a cha
z
xx
1.5
1
and jxzi orbitals along a bond in x-direction [Fig. 2(a)].
ThisThe spin-orbital exchange
Hamiltonian
for such a system
y
Ham
y y
Ir ions form an honeycomb lattice.
S 2S 3x x
x
S1S2
has alreadybeen reported: see Eq. (3.11) in Ref. [12]. After
omet
1
yy
projecting it onto the ground state doublet, we find an
gen i
2exchange Hamiltonian for isospins in a form ofof th
Ir - Ir direct exchange:
Heisenberg interaction
0.5
inter
FIG. 3 (color online). Examples of the structural units formed
ture(a)90! TM-O-TM bonds and corresponding(b)
spin-coupling patbyoterns. Gray circles
stand for magnetic ions, and small open
180
of0 tho
Ir - O - Ir exchange:
Anisotropic Kitaev interaction
circles denote oxygen sites. (a) Triangular 90unit
cell oftion
ABO2 -type layered compounds, periodic sequence of this unit
plane
forms a triangular lattice of magnetic ions. The model (3) on thisstructure is a realization of a quantum compassp model on a
FIG.With
4. T

A2IrO3

Iridium Oxides

φ/α

Na IrOz

triangular lattice: e.g., on a bond 1-2, laying perpendicular to
x-axis, the interaction is S x1 S x2 . (b) Hexagonal unit cell ofyz
A2 BO3 -type layered compound, in whichmagneticxzions(B-sites) form a honeycomb lattice. (Black dot: nonmagnetic
xy On an pxx-bond,xy
A-site).
the interaction is S xi S xj , etc. For thisystructure, the model (3) is identical to the Kitaev model.

Depending on the bond direction, only
specific spin component interact.

of the tetr
an IrO2 -pl
about z-ax
% ’ with'=5,
the in-planquan
z-axisocc

017205-3xz

Ground state of the pure Kitaev model: Spin liquid

pz

xz

xz

yz
pz

intro
Jahn
studi
dege
How

Eq. (3), the spin coupling, e.g., on a (x)-bond, is of S i S j% þ s2sin theverytype,
formpreciselyof the exchange
Hamiltonian
dependsonbondand excit
third
as in the Kitaev model. The honeycomb
geometry
profileof Kramersstates,as we pies, withlattice throughprovidesa adensity
particularly
striking
exampleof newHunddemonstratephysics below.
introduced by strong SO coupling: the[For ! ¼
andHeisenbergcouplings
model is ofconverted
into theKitaev states.—model
the 180!
Exchangeneighboring
KramersHamwith a spin-liquid
We consider
the limitgroundof the state.
strong spin-orbit coupling, i.e., canting aWhila physicalex-the is indepe
Theiscompound
Li2 RuO
3 [28] represents
when !
larger thanexchange
interaction
betweenfrom
distortion
ample of the A2 BO3 structure. By replacement of spin-one
isospins.
for isospins are then4þ The exchange Hamiltonians
4þ! intoSranwith spin-one-half Ir ions, one may realize aRu
2I
obtainedbyprojecting
thecorresponding
superexchangeAsstrongly spin-orbit-coupled Mott insulator with low-energy
theinmspin-orbital
modelsonto
theisospinstatesEq.
(1).
First,
we
Hamiltonphysics described by the Kitaev model.
mode
present‘‘Weak’’
thepresults
for
the
case
of
cubic
symmetry
(!
¼
0,
IrO
.—As
an
exampleofferromagnetism
of
Sr
2
4
S~~i
model
ﬃﬃﬃ
enou
Mott insulator,
discuss
layeredsin"a¼spin-orbit-coupled
1= 3), and discusslater the weeffectsofthe
a tetragonal
gered
rot
pling
of theundopedhigh-Tc
compoundSr
2 IrO4 , a ttwo
2g analog
distortion.
We
considercommoncases
of
TM-O-TMtanð2#ÞStrong spin-orbit couplingcally¼
Effectivebond``spin”
moment:" square lattice of Ir4þ ions is
cuprate
La
CuO
.
In
Sr
IrO
,
a
2
4
2
4
geometries: (A) a 180 -bond formed by corner- anisotrop! whformed by corner-shared IrO6 octahedra, elongated along
shared octahedra as in Fig. 2(a), and! (B) a 90" -bond Hamilton(B
the c-axis and rotated about it by ! ’ 11 [19] (see Fig. 4).
formedbyedge-shared
ones,
Fig.
2(b).
on a
a magnetic transition at # 240 K disSr2 IrO4 undergoes
G.Jackeli, et al., PRL 102, 017205 (2009) (A)A 180" bond: For this geometry, the nearest2
3
2 x
the
z z
S 1S 3 is diagonal in thezzorbital space
J. Chaloupka, et al., PRL 105, 027204 (2010)neighbor t2g
nond(a) hopping matrix(b)3and, on a given bond,only two orbitals are active, e.g., jxyi
a cha
z
xx
1.5
1
and jxzi orbitals along a bond in x-direction [Fig. 2(a)].
ThisThe spin-orbital exchange
Hamiltonian
for such a system
y
Ham
y y
Ir ions form an honeycomb lattice.
S 2S 3x x
x
S1S2
has alreadybeen reported: see Eq. (3.11) in Ref. [12]. After
omet
1
yy
projecting it onto the ground state doublet, we find an
gen i
2exchange Hamiltonian for isospins in a form ofof th
Ir - Ir direct exchange:
Heisenberg interaction
0.5
inter
FIG. 3 (color online). Examples of the structural units formed
ture(a)90! TM-O-TM bonds and corresponding(b)
spin-coupling patbyoterns. Gray circles
stand for magnetic ions, and small open
180
of0 tho
Ir - O - Ir exchange:
Anisotropic Kitaev interaction
circles denote oxygen sites. (a) Triangular 90unit
cell oftion
ABO2 -type layered compounds, periodic sequence of this unit
plane
forms a triangular lattice of magnetic ions. The model (3) on thisstructure is a realization of a quantum compassp model on a
FIG.With
4. T

A2IrO3

Iridium Oxides

φ/α

Na IrOIs there Kitaev spin liquidinthevicinityofNa
2IrO3?
Depending on the bond direction, onlyz

triangular lattice: e.g., on a bond 1-2, laying perpendicular to
x-axis, the interaction is S x1 S x2 . (b) Hexagonal unit cell ofyz
A2 BO3 -type layered compound, in whichmagneticxzions(B-sites) form a honeycomb lattice. (Black dot: nonmagnetic
xy On an pxx-bond,xy
A-site).
the interaction is S xi S xj , etc. For thisystructure, the model (3) is identical to the Kitaev model.

specific spin component interact.

of the tetr
an IrO2 -pl
about z-ax
% ’ with'=5,
the in-planquan
z-axisocc

017205-3xz

Ground state of the pure Kitaev model: Spin liquid

pz

xz

xz

yz
pz

intro
Jahn
studi
dege
How

ies experdiagonalization.
IrO3 , which includes
coux, y, and isotropicz axes in superexchange
defining the t2g -orbitals
areillustrated
rder[7, 8]. z axes in defining the t2g -orbitals,gs inandare illustrated
out of thehoneycomb
plane. The honeycomb
addition as
to directions
the Kitaev-type
anisotropic
nearestoneycomb
planeis thenperpendicular
toaxes(x, y,dependz) = plane.
(1, 1, 1). TheThe
dashed honeycomb
hbor Ising interactions
whose
anisotropy
irections
out
of
the
honeycomb
additional
anisotropic couplings asboundary
represents
a 24-siteout
clusterhedescribebond directions.
However,
it turned
that used
this later for the exact
eel isthenbe perpendicular
to (x, y, z) = (1, 1, 1). The!
dasheddiagonalization.
ange
cou! ⃗T ⃗cannotstraightforwardly consistent with theĤ = the exact Ŝ ℓ JΓ Ŝ m ,(1)
c nearestndary
a 24-siteclusteragorder represents
either. This discrepancy
inspiredfurtherused later for
Γ=X,Y,Z ⟨ℓ,m⟩∈Γesiesdepend
on suitableadditional
low-energy
eﬀective hamiltonians
anisotropic
couplings as foronalization.
thatwith
this A =Na or Li. First, models with further(Y. Yamaji et al. Phys. Rev. Lett. 113, 107201(2014))
rO
T
3
⃗
T
!
!
y
x
z
with couplingsthe⃗Addi-⃗= (Ŝℓ , Ŝℓ , Ŝℓ ) is a vector of SU(2) spin opwhere Ŝ ℓ (1)hbor[7, 8, 12,Ĥ13]
were studied. Ŝ
Ŝ
,
=
J
Γ
m
ℓ
d Isingfurthererators. The exchange couplings are given in matricesalanisotropy[14] due to Γ=X,Y,Z
a strong⟨ℓ,m⟩∈Γ
trigonal disonians forJΓ . The summations are over the nearest-neighbor pairs
on, which actually contradicts the distortions in the(1, 1, 1)
h furtherT⟨ℓ, m⟩. The group of bond Γ with Γ=X, Y and Z is de⃗ ab initio
wasalso
eriments [8] andintheyx treatments,zwhere Ŝ ℓ = (Ŝℓ , Ŝℓ , Ŝℓ ) is a vector of SU(2) spin opx-bondd. Addifined in Fig. 1. The exchange matrices are parametrized
z-bondmined. Quasimolecularorbitals[15],insteadofthezerators. The exchange couplings are given in matrices
gonal disasTmic orbitals assumed
intheKitaev-Hubbard
model
JΓ . The summations are over the nearest-neighbor pairs ⎡ons in the⎤⎡ ′ ′′ ′ ⎤xe claimed as a ⟨ℓ,properchoiceofthestarting
point.
So
J
I
I
K
I
I
y-bondm⟩. The group of bond Γ with Γ=X,
Y
and
Z
is
de1
2
Γ
m
, was also
22ℓyhe origin of the
zigzagtype
antiferromagnetic
order
′′
′′
⎣
⎦
⎣
JZ = I1 J I2 , JX = I2 J I1′ ⎦ ,fined in Fig. 1. The exchange matrices are parametrized
ad of therved for Na2 IrO
the possible⟨ℓ,m⟩∈Γ
route to realize
as 3 andΓ=X,Y,Z
I2 I2 KI2′ I1′ J ′rd model⎡ ′′ ′′ ′ ⎤quantum spin liquid are controversial.
⎡⎤⎡ ′ ′′ ′ ⎤J2point. So
J I2 I1
J I1 I2
K I2 I2
T
thisorder
Letter,we derive an ⎣ab initio spin
etic
′′
′′
′ ⎦ J = ⎣ I ′′ K ′ I ′ ⎦ ,⎦ , Jmodel
⎣for
(2)
J
=
I
J
I
=
I
J
I
, Y
Z
1
2
X
2
2
2
1
y
x
z
IrO
and
show
that
trigonal
distortions
presentin′′′to realize
3
I1 I2 J
I2 ℓI2 K
I2′ I1′ J ′ℓℓℓ
IrO3 in addition to the spin-orbit
holds the⎡ ′′ couplings⎤
J modelI2′′ forI1′ A2 IrO3 will
J3
where we choose a real and symmetric parameterizationThe
simplest
andrealistic
spin
model for
′′
′
′
⎦and
KitaevHeisenberg
likeJ(2) coupling
JYcoupling= ⎣ I2 K hamiltonian
IK
,
2
by
using
U(1)and
SU(2)-symmetryof electron wave
out to modify
the Kitaev-Heisenberg
bypresent
inI1′ I2′ J ′+holds theOﬀ-diagonal
couplings
I1 and I2
where we choosea real and symmetricparameterization
2 IrO3 willby using U(1)- and SU(2)-symmetry of electron wave
tonian by
It also contains J2 and J3 interaction term.

ab initio Hamiltonian of Na2IrO3itional anisotropic couplings as

ab initio Hamiltonian
Ĥ =

!

! ⃗
⃗
Ŝ J Ŝ ,(1)⃗
re Ŝ = (Ŝ , Ŝ , Ŝ ) is a vector of SU(2) spin opors. The exchange couplings are given in matrices
The summations are over the nearest-neighbor pairsm⟩. The group of bond Γ with Γ=X, Y and Z is ded in Fig. 1. The exchange matrices are parametrized⎡

J2 : only “z-bond” which is perpendicular to NN z-bond.

⎤

⎡

⎤

′
′′
′
three
third
neighbors
JJ3 : Iall1 ofIthe
K
I
I
2
2
2
′′
′′
JZ = ⎣ I1 J I2 ⎦ , JX = ⎣ I2 J I1′ ⎦ ,I2 I2 KI2′ I1′ J ′to the trigonal
distortion, the ab initio Hamiltonian contains strong
⎡ Due
⎤
′′
J ′′ I2oﬀ-diagonal
I1′
couplings, together with J2 and J3 interaction(2)
JY = ⎣ I2′′ K ′ I2′ ⎦ ,-6.19
-6.2
-6.21
-6.22
-6.23
iPEPS
-6.24
DMRG: Lx × 6 T. Okubo et al, PRB 96, 054434 (2017).
-6.25
ED
-6.26
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35

Results: comparison with other methods
Energy (per sites)-6.17 (a)
-6.18
-6.19
-6.2
-6.21
-6.22
-6.23
iPEPS
-6.24
DMRG: Lx × 6
-6.25
ED
-6.26
0
0.05
0.1

0.15

0.2

0.25

Order parameter

0.3

0.35

0.4
(b)
0.35
0.3 of iTPS, DMRG and ED are consistent.
Energies
0.3
0.25
iPEPS:
0.2
0.2• For 4 × 6 lattice, DMRG and(1,1,0)ED
0.15(1,-1,0)
iPEPS(0,0,1)
0.1
0.1givealmost
DMRG:
Lx × 6same energy.
0.05
0• Finite ED
D
of
iTPS
and
finiteof DMRG
0.1Lx 0.2
0.3
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35

are overlapped.

0.4
(b)
0.35
0.3
0.25
0.2
0.15
iPEPS
0.1 DMRG: Lx × 6
0.05
ED
0
0
0.05
0.1

0.3
0.2
0.1
0

0.15

0.2

0.1

iPEPS:(1,1,0)(1,-1,0)(0,0,1)
0.2

0.25

0.3

0.3

0.35

Zigzag(Z) order parameters are consistent.
•

Extrapolations of them are <M> ~ 0.3•

Spins are almost along (1,1,0) direction,
which is consistent with the experimental
observations.

-6.19
-6.2
-6.21
-6.22
-6.23
iPEPS
-6.24
DMRG: Lx × 6 T. Okubo et al, PRB 96, 054434 (2017).
-6.25
ED
-6.26
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35

Results: comparison with other methods

Stabilityof the Zigzag(Z) state inOrdertheparameter
thermodynamic
Energy(per sites)-6.17 (a)
-6.18
-6.19
-6.2
-6.21
-6.22
-6.23
iPEPS
-6.24
DMRG: Lx × 6
-6.25
ED
-6.26
0
0.05
0.1

0.4
(b) iTPS calculation.
limit is confirmed0.35
by

0.15

0.2

0.25

0.3

0.35

0.4
(b)
0.35
0.3 of iTPS, DMRG and ED are consistent.
Energies
0.3
0.25
iPEPS:
0.2
0.2• For 4 × 6 lattice, DMRG and(1,1,0)ED
0.15(1,-1,0)
iPEPS(0,0,1)
0.1
0.1givealmost
DMRG:
Lx × 6same energy.
0.05
0• Finite ED
D
of
iTPS
and
finiteof DMRG
0.1Lx 0.2
0.3
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35

are overlapped.

0.3
0.25
0.2
0.15
iPEPS
0.1 DMRG: Lx × 6
0.05
ED
0
0
0.05
0.1

0.3
0.2
0.1
0

0.15

0.2

0.1

iPEPS:(1,1,0)(1,-1,0)(0,0,1)
0.2

0.25

0.3

0.3

0.35

Zigzag(Z) order parameters are consistent.
•

Extrapolations of them are <M> ~ 0.3•

Spins are almost along (1,1,0) direction,
which is consistent with the experimental
observations.

Phase diagram varying the trigonal distortion
T. Okubo et al, PRB 96, 054434 (2017).
(a)Phase diagram-5

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

-5.2

IC

iPEPS
DMRG

-6

iPEPS
DMRG

-6.4
-20

0

20

40

(d)
120°/6-site(e-1) 16-site (XY)

0.01
0
60

DMRG: 6x8 cluster(e-1)
16-site(XY)
iTPS: 4x4, 2x6, 6x8, 8x12,
6x10
unit
cells
•
•

120°/3-site
120°/6-site

0.03

120°/3-site

-6.2

-40

(c) (d)

0.02
(c)

-6.6
-60

Zigzag (X)120°/3-site

0.04

120°

-5.6
-5.8

0.05

(c) (b)(e-2) 16-site (Z)Energies obtained by iTPS and DMRG are consistent
New phases are stabilized compared with the previous ED reports

(e-1)
(XY)
(e-2)16-site
16-site(Z)

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

Lattice expansion

FIG. 3: (color online): (a) ∆-dependence of matrix elemen
ab-initio Hamiltonian for Na2IrO3 (with lattice expansion)of J , J as functions of ∆. Around the ab initio valu
Z

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

X

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 in
The phase boundaris are determined by anomalies in secoderivativesof the exact energy for the 24-site cluster with
Local magnetization
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
and a 24-site unit SL
cell long-period order[23], appear. (c)dependence of the ground state of the generalized Kitae
Heisenberg model for “expanded lattices.” Here we negle
thesmall hopping parameters other than t and take a larg
Ferro
Hund coupling JH = 0.3 eV. Spin liquid phases compete wi
ferromagnetic states and 12-site unit cell orders illustrated
12-sitethe lower right panel[23].

0.1

-5.2
-5.3
-60

•

-40

-20

0

20

40

60

0

-60

-40

-20

0

20

60

percomputer Center, Institute for Solid State Physi
Univ. of Tokyo. This work was also supported by Granin-Aid for Scientific Research (No. 22104010, and N
223400901) from MEXT, Japan.

iTPS phase diagram is qualitatively consistent with the ED.
•

40

Around Δ=0, a Kitaev spin liquid phase is clearly stabilized.

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

Lattice expansion

FIG. 3: (color online): (a) ∆-dependence of matrix elemen
ab-initio Hamiltonian for Na2IrO3 (with lattice expansion)of J , J as functions of ∆. Around the ab initio valu

at ∆can(∼ −28meV) listed in Table II, K < 0, K <If weexpand
J > 0, J > 0, and J > 0 are stably satisfied with gra
ual dependenceson ∆. (b) Ground state pahse diagram f
the lattice
constant,
Na IrO with lattice distortions represented by changes in
The phase boundaris are determined by anomalies in secoweprobablyfindliquid material !
derivativesof the exact energy for the 24-site cluster with
Localspin
magnetization
Energy and Phase diagram
Z

X

′

′

2

-4.6

0.4

-4.7

SL

-4.8
-4.9

0.3

Ferro
12-site

-5

0.2

-5.1

′′

3

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
and a 24-site unit SL
cell long-period order[23], appear. (c)dependence of the ground state of the generalized Kitae
Heisenberg model for “expanded lattices.” Here we negle
thesmall hopping parameters other than t and take a larg
Ferro
Hund coupling JH = 0.3 eV. Spin liquid phases compete wi
ferromagnetic states and 12-site unit cell orders illustrated
12-sitethe lower right panel[23].

0.1

-5.2
-5.3
-60

•

-40

-20

0

20

40

60

0

-60

-40

-20

0

20

60

percomputer Center, Institute for Solid State Physi
Univ. of Tokyo. This work was also supported by Granin-Aid for Scientific Research (No. 22104010, and N
223400901) from MEXT, Japan.

iTPS phase diagram is qualitatively consistent with the ED.
•

40

Around Δ=0, a Kitaev spin liquid phase is clearly stabilized.

Contents•

Huge data in physics•

Information compression•

Basics: singular value decomposition•

Tensor network renormalization•

Tensor network quantum states•

Applications•

Summary and outlook

Summary•

By using tensor network representation, we can largely compress the information
into compact forms.
••

A partition function in the statistical physics can be represented by a tensor
network•

Its contraction can be done efficiently by tensor network renormalization
technique.

•By using MPI parallelization, 4d (3+1 d) calculation becomes realistic.

We can approximate low energy wave function by using tensor network
states.
•

Efficiency is guaranteed by the area law of the entanglement entropy•

We can represent infinite systems with finite bond dimensions D.

•

MPI parallel code TeNeS is available.

•

For 2d frustrated spin systems, iTPS is one of the most powerful methods.

Outlook••

Application to difficult problems•

Excitation spectrums•

Finite temperature properties•(Lattice QCD)

Nowadays, tensor network representations expand their application to other fields.
•

Machine learning for classification problem with MPS••

E. Miles Stoudenmire and D. J. Schwab, NIPS 2016

W

Unsupervised Generative Modeling•

Born machine instead of Boltzmann machine•

S. Cheng et al, Phys. Rev. B 99, 155131 (2019)<latexit sha1_base64="vOJX0Msp2QRw56wrP2TXE8H3N2E=">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</latexit>••

Validating quantu

Z.-Y. Han et al, Phys. Rev. X 8, 031012 (2018).

Quantum circuit simulation•

A. McCaskey et al, PLoS One 13 e0206704 (2018). (MPS)•

C. Guo et al, Phys. Rev. Lett. 123, 190501 (2019). (TPS)<latexit sha1_base64="uw8KgSzv2MgXTgPinGcdthpJmzE=">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</latexit>