Tensor Network Formulations
from the view point of entanglement and RSRG
Tomotoshi Nishino (Kobe Univ.)

Realistic physical systems are normally entangled weakly.
This property enables us to apply RG transformation.

pixels indicate elements of small magnitude and lighter pixels indicate elements with larger magnitude. (a) Startingat a sub-critical temperature, T = 0.9 TC , the coarsegrained tensors quickly converge to the Z2 fixed-point tensor B Z2 ≡ B triv ⊕ B triv . (b) Starting at the critical temperature, T = TC , the coarse-grained
tensors converge to a
8 critnon-trivial fixed-point tensor B . Notice that the diﬀerence
between coarse-grained tensors, |B (s) − B (s+1) |, which is displayed with the same color intensity as the plots of |B (s) |, isalready very small (as compared to the magnitude of the elements in the individual tensors) for s = 1. (c) Starting at thesuper-critical temperature, T = 1.1 TC , the coarse-grained
tensors quickly converge to the disordered fixed point B triv ,that has only one non-zero element.

to scale-invariance, but to hold it for more RG ste

Recent? Progresses

Evenbly and Vidal
arXiv:1412.0732

Real Space Renormalization Group is of use!

T = 1.1 TC , we obtain a trivial fixed-point tensor B triv.
that has only a single significant element [B triv ]1111 = 1
with all other elements zero or arbitrarily small, whichis representative of the infinite temperature, disorderedphase.

FIG. 6. The precision with which T
invariant fixed point tensor for the 2
critical temperature Tc is examinedence between tensors produced by s
Appendix B.– Non-critical RG fixed points: TRG
δ (s) ≡ ∥A(s) − A(s−1) ∥, where tenso
versus TNR.
1. Fors (inthat∥A(s) ∥of=Eq.
FIG. 7. (a) CDL suchtensor
ACDL
6. small(b) Tens
limitation
torealizing
scale-invarian
made of CDL tensors, which contains correlations olattice Hamiltonian includes RG irr
In this appendix we discuss certain aspects of the flow
each plaquette.
scale-invariance at short-distance sthat TRG and TNR generate in the space of tensors,at larger distances. On the otherwhich emphasizes one of the main diﬀerences between
number of RG steps) the main obstr
is the numerical truncation errors, w
introducing RG relevant terms, eﬀefrom criticality and thus scale inva
FIG. 5. Plots of the elements of tensors [B (s) ]ijkl , when relarger bond dimension χ, which redu
shaped as 16 × 16 matrices, after s iterations of the TNRlows TNR to not only achieve a mocoarse-graining transformation, for several values of s. Dark
to scale-invariance, but to hold it fo
pixels indicate elements of small magnitude and lighter pixels indicate elements with larger magnitude. (a) Startingat a sub-critical temperature, T = 0.9 TC , the coarsegrained tensors quickly converge to the Z2 fixed-point ten-

RG flow is accurately traced at criticality.
Fixed point is numerically captured.

from recent numerical results by means of Tensor Network
2D Ising Model: Transition temperature can be obtained with 8 digits.
3D Ising Model: obtained Tc = 4.511546 (Monte Carlo: 4.511544)(T. Xiang et al: arXiv:1201.1144)(Evenbly and Vidal: arXiv:1412.0732)

estimation of conformal
weights for 2D Ising Model

A Big Step in
Real-Space RG

exact TRG(64) TRG+env(64) TEFR(64) TNR(24)c 0.5 0.49982
0.49988
0.499420.50001
σ 0.125 0.12498
0.12498
0.12504 0.1250004
ϵ11.00055
1.000400.99996
1.000091.125 1.12615
1.12659
1.12256
1.12492
1.125 1.12635
1.12659
1.12403
1.1251022.002432.00549
1.9992222.00579
2.00557
1.999862
2.00750
2.00566
2.0000622.010612.005672.00168
TABLE I. Exact values and numerical estimates of the central
charge c and lowest scaling dimensions of the critical Isingmodel. TRG results are obtained using the original Levin andNave’s algorithm [4]. TRG+env results are obtained using an
improved TRG method proposed in Ref. [5] under the nameof “poor-man’s SRG”. TEFR results are taken from Ref.
[7]. The first three numerical columns use bond dimensionχ ≡ Dcut = 64 and 1024 spins, while the TNR data uses
χ = 24 and 262,144 spins.

間で、例えば次のように和を取りことから、ければ幸いだ。
いは 4 脚のテンソル
り、脚を示す文字すら足らなくなる。そこで
和は縮約と呼ばれる。なお、本稿ではベクト“Tensor” intensorformulation
m networkn系に含まれテンソルの間では縮約!
!
!
丸や四角などの閉じた図形と、そこから突き
c complex=
Vi Aij Wj =Aij Vi Wj*
Elements
arescalars.
グルメントのダイアグラム表示
Bの素を、誤解がない限り大文字で表す。
だけが、縮約されずに
ijkℓ
i=1 j=1ijてテンソルを視覚的に表し、様々な計算を
ダ
*
1-leg
(vector),2-leg(matrix),
3-leg,
4-leg,….
取り扱う点
ラムは 8 脚テンソル
以上の脚、つまり足あるいは添え字を持つもを指すのか、まずは教科書的に
2 つのテンソル
表現する慣習がある。図中で
*
Not
all
of
the
legs
are
(purely)physical.
スカラー量cを表すことができる。このように、揃
典型的な問
行、横に
n 列の行列A を考えてfつのベgは、対応する脚についての縮約を表すわけだ(の要素)だ。このテンソル
Bと、
4ル
え字で取る和は縮約と呼ばれる。なお、本稿ではベ*Pleaseforgetabout
covarianceandsymmetries.
含む、幅広でも複素数でも良い。ベクトル(2) のように、スカラーを与える数式のダイ
Bij
や行列の要素を、誤解がない限り大文字で表す。
(occasionally
there are symmetries, though.)W,X,
Yとの縮約もスカラー量になる。
ーク形式に
び W *=Aうに、(WW· · Wn ) とのつ以上の脚、つまり足あるいは添え字を持
線が全て結ばれる。部分的な縮約、例えば次
tensormaya certain area in the system.
1,3
2 , ·represent
したい。 !
!
B と、 4 つを取り ′がテンソル (の要素) だ。このテンソル
す Contractionダイア
c トル= V, W,
Bijkℓ
V
W
X
Y
among
tensors
D
=
ij
i
j
k
ℓ Bijkℓ Akℓ
X, Y との縮約もスカラー量になる。
! ijkℓkℓ(MPS) にAij Vi Wj ′ !
(1)
ij Wj =る テンソij 残った脚 i と
c =j に対応する
Bijkℓ Vi W
Xk Y ℓ2 j本の「結ばれてijkℓ V 、行列 A、テンソ
ではお行儀よく、ベクトル
4 節では、 ダイアグラムに現れる。

* We obtain a scalar or tensor, as a result.
できる。このように、揃った添

(2)のように、スカラーを与える数式のダ!
運
ワーク形式にDiagrams
横にn 列の行列Aを考えて
fg(2) hc′ =線が全て結ばれる。部分的な縮約、例え
Bijkℓ Vi Wj Xk Yℓ..
伝えしたい。
複素数でも良い。ベクトル
ijkℓ
!
考
W
に表す
ダイア
j
D
=
B
A
ij
ijkℓ kℓ
Wここまではお行儀よく、ベクトル= (W1 , W2 , · · · Wn ) とのV、行列 A、テンソル 書
j
kℓ
(MPS)
に
状態
j
i
り V
、それぞれの要素
i B
A W Vi V, Aiij ,BBkijkℓX と、用語や記号を使いである テンソ 残った脚 i と j に対応する 2 本の「結ばけて来た。煩雑さを避けるため、テンソルネットワーク
l
k
l
!
A
4
節では、
い。
ダイアグラムに現れる。
Y
=
Aij V
Vii W
やj Aij や B(1)式では要素
ijkl を記述に用い、脚の数に
5 節ではテン
もっと沢山のテンソルの間で縮約を取ij
テ
らず全てテンソルと呼ぶ。なお、相対性理論で考慮する
点から眺め、* Tensor6legsいたようにダイアグラムが網目になる
are represented by lines.
でテンソルの共変性」などは、全く必要としない。
る。このように、揃った添
雑に感じられネ̇ッ̇ト̇ワ̇ー̇ク̇という言葉の由来だ。こ* Connectedル
lines
represent
contraction.
る。なお、本稿ではベクトル形式の全体像
2indices.
脚 (= 添え字が2
図 2. テンソルネットワークの* Open lines構造を詳しく見よう。
correspond to remaining
り大文字で表す。Bijkℓ のよ
4脚のテンソルが碁石のように並んいは* Diagrams are more often used than equations.
あるいは添え字を持つもの
図 2 のテンソルネットテンソルの間では縮約が取られ、上端にExample: Tensors in Statistical Physics* Consider the square-lattice Ising Model.
il
lk

Q

(We consider the diagonal lattice.)jj* Black Dots: Ising spin variables.
i, j, k, l are either 1 or -1.

ki* Boltzmann Weight for the shaded region:
Qijkl = exp[ J(ij+jk+kl+li)/kT ]

i
l

j
k

… this is already a 4-leg tensor.
* In case we consider models with continuous
local degrees of freedom, tensor legs are also
continuous.

an example of the Tensor Network* Legs at the top of the diagram are “alive”.

f

g

h

i

j

k

l

m

* There are various interpretations on the network diagram.

f

g

h

i

j

k

l

m

2D Ising Model* Partition function of the squarelattice Ising model with open
boundary conditions for bothsides and the bottom, and fixed
ones at the top.

* Tensor network is NOT new at all. It has been known for
more than 70 years.

1D Quantum Ising Model* Through the quantum-classical correspondence(= discrete Path Integral), the above diagram canbe interpreted as a quantum wave function.

identified as a wave function

f

g

h

i

j

k

l

m

* Look at the bottom of the diagram.

Matrix Product State (MPS)
f

g
α

h

i

β

γ

j

k

δ

ε

l

m

ζ

η

* Roman letters: physical indices* Greek letters: auxiliary indices.

Matrix Product Operator (MPO) ~ Transfer Matrix
o

p

q

κ
f

λ
g

r

s

μ
h

t

ν
i

u

ρ
j

v

σ
k

ξ
l

m

←

←

real/imaginary time evolution

f

g

h

i

j

k

l

m

now →
where T = exp(-iΔtH)
or exp(-ΔτH)←

Time evolution by MPO →past: Initial MPS →←Extended MPS

IT IS NOT EASY TO CONTRACT TENSORS!!
* Number of terms increases exponentially with respect tothe number of contracted bonds.
* Dimension of wave function increases exponentiallywith respect to the size of the system.
* It is hopeless to handle Hilbert space directly.

From K-W approximation to TPS/PEPSa Pioneering work: Kramers-Wannier Approx. Phys. Rev. 60, 263 (1941)* Eigenvector of the transfer matrix of 2D Ising model is approximated by 2-state MPS.
* Tensor elements are adjusted so that variational partition function is maximized.
* One of the earliest example of “Numeircal Physics”.

Phys. Rev. 60, 263 (1941)

Kramers-Wannier Approximation

Estimate the partition function of 2D Ising model as Rayleighratio for the transfer matrix with respect to a variational state
written in terms of matrix (= 2-leg tensor) product.
V( S1, S2, ..., SN ) = M( S1, S2 ) M( S2, S3 ) ..... M( SN-1,SN )
S1

S2

S3

S4

S5

Rayleigh Ratio

Maximize ! adjusting M* V is equivalent to 1D Ising model under external field.

Numerical Physics (by hand!)During the calculation, one has to treat 6-leg Ising ladder, and to obtain the
maximal eigenvector of the transfer matrix. This was obtained by thePower method, performed by hand.H. A. KRAMERS AND G. H. WAN
TABLE I. Values of eigenvectors.
VEcToR A~

VEcTQR AS

1.000000

1.000000.249038

VECTOR A3

1.000000.186931.105677.231108

VECTOR A4

1.00DOOO.159434.103346.160827.099340.026148.079123.222893.145548.100846.129103.098804.019608.075422.148388.096230.020594equal to ~. Kith
may unite the two

VECTOR Ag

1.000000.137739.098916.112698.097652.016676.071758.114363, 096454=.094527 x(K) X(E) /cosh 2.017936.009995.017578.010148.002709.008510.022371

For our problem
direct interest, for
critical region. T.017673.008320agreeable criterion
proximate solution
development with

.065628.022060.023648.016343.n10661.016817.069010.016516.218677.141720.216286.011008.023617.070174.066754.062185.056461.022385.060217

6. THE SP… whoC
If our model
did
it? lead
only (which is theSpecific Heat
276

Phys. Rev. 60, 263 (1941)

Fr@. 8. Specific heat
curves.
2-screw approxiVariation
mation.
—
——
—
Bethe
method.
.
Kirkwood
method.
method.

/I

versus temperature

/

/

/
/

/

/

R

/

/

/

/
I
I

. i
/

l

I/K =2~T/J

2

K-W approximation andexact solution

Two ancestors ofmodern analyses
Note that differencein Free energy isvery small.

L. Onsager, Phys. Rev. 65, 117 ((1943))

Exact
K-W approx.
Error in Tc ~ 7%

These two view points are unified later by Baxter.

ation for the 3D Ising Model
545
K-W approx. applied to 3D Ising Model(cond-mat/9909097)

1 .0

Variational state is 2D
Ising model under finite
external field.

0 .8
0 .6

M

Spontaneous
Magnetization

0 .4
0 .2
0 .0
0 .2

The K-W approximation
works better in higher
dimensions.

Plots: K-W approx.
Curve: from M.C.
0 .3

0 .4

K = J/kT(So far, no application in
3+1 dimensional system.)
DownloadeFig. 2. The magnetization M as a functionof K. The solid line denotes M MC in
Eq. (3 .4) obtained from Monte Carlo simulations. 22), 23) In the vicinity of the calculatedtransition point, M is proportional to√
K − Kc .

Quantum Systems… a Similar variational state was considered for 1D quantum
spin chains.
Nightingale and Bloete, Phys. Rev. B 33, 659 (1986)

S1

S2

S3

S4

S5

This article is known as the numerical proof (?) of the existenceof the Haldane gap. (We return to this article again.)

Generalization by Baxter
introduction of auxiliary variable

Numerical!

J. Math. Phys. 9, 650 (1968)

Afα , Agαβ , · · ·, Aℓζη , Am
η が 1 次元的に並んだテンソルネッ
トワークだ。両端の Afα と Am
η は 2 脚、それ以外は 3 脚
J. Math. Phys. 9, 650 (1968)のテンソルで、図との対応から脚 f ∼ m はテンソルの右

Matrix Product (Variational) State
肩に置いた。このダイアグラムは、次の縮約
!
j
f g
h
i
φf ghijklm =Aα Aαβ Aβγ Aγδ Aδε Akεζ Aℓζη Am
η

(5fg h 2 度ずつ現れるテンソルの脚
i
j
k
l
m
を表している。総和はα∼η

つまり図 3 の太い水平線が示す脚について取る。このようα ∼ η (補に、テンソルを一列に並べ縮約を取ったものは、
α β γ δεζ η助変数) を行列の脚とみなして、行列積波動関数 と呼ばれGreek
letters
represent
D-stateauxiliary
variables.
る。そして、対応する状態
|Φ⟩

=

The
state is |fappliedto the dimer model. (6φf ghijkℓm
ghijkℓm⟩!

+(0)*/++(0)*-*.#"-/-&2
2/"*!"!-*.#"-/-&2

-&/&+*(/&+.101. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp/%&.&.*+-&$&*+#&*.//&./& ()" %*& .*+/%"-+*"&.3"--&!

stribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp

Expectation values can be calculated easily※ Variational ratio can be calculated from left to right.

←〈Φ|Φ〉^←〈Φ|H|Φ〉

Density Matrix Renormalization Group (DMRG) implicitly uses this superiority.

f…m run from 1 to d,
α…η run from 1 to D.

Structure of MPS
f

g
α

h
β

i
γ

j
δ

k
ε

l
ζ

m
η

It contains large number of variational parameters that can be tuned.

Baxter’s numerical result on the dimer problem

One finds a careful statement on his numerical result.

03.101. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp... can anyone here can be as careful as Baxter? Apparently, I cannot ...

Baxter’s Corner Transfer Matrix (CTM) method
has been used since then.
108

S, K. Tsang

5. N U M E R I C A L

SOLUTION

For Latticefinite n, theproblem reduces
to evaluating theApplied
cj and mo from theSquareVariational
Approximations
equation (65). Numerically, we find that mo is always greater than
topolynomial
the
Ising
Model
2, so we can define a real parameter e, by
S. K. Tsang 1

mo = c~a + c~"a,

0 < c, < 1(74)(The c, defined here is different from that used previously.) The zeros of the5, 1978at cl, ca ..... c,, soright-handReceived
side of September
Eq. (65) occurc j ) f a ( i c ~ ) / p method( - i c ~ ) fdevelopeda ( - icj) = by- Baxter1,
jis =applied
1, 2 .....
(75)
Thep ( ivariational
to nthe zero-field
Ising modelofonboth
the square
The problem
is simplified
thatlowof
Taking logarithms
sides, lattice.
and usingconsiderations
fromtothe
solving a relatively small system of nonlinear equations. The estimates totemperature expansions of the equation (75) for the appropriate branch cut
the spontaneous magnetization and the critical temperature from thein the complex
plane,
we may write
the equationassequenceof variational
approximations
are obtained.
The results converge
rapidly
to
the
exact
ones.
They
exhibitacrossoverphenomenonn- ihih2 and satisfy
2 a~scalingt a n -relation.
lCl-l+c~+tan-17+tan-1

z=z

eTz

c~

ca z _

cs

c71 _ c~1/2)7r,= 1, 2 .....
n(76)
KEY=W(nO R-D Sj :+Isingmodel; jvariational
approximation; spinor representatives; crossover phenomena; scaling.

where hi and ha are given by (64).
Equation (76) has been solved numerically for a range of values of z
below the critical point, using the Newton-Raphson method. The computa1, I N T R O D U C T I O Ntion was done on a UNIVAC 1100/42 computer, using double-precisionWe
reportpoint
here arithmetic
an investigation
on significant
the convergencevariational
floatingwith 18
digits. ofA asolution
was method
assumedwhen
appliedto the zero-fieldmodel. change of each variable, throughto haveconverged
only whenIsing
the relativetechniqueappliedto the1968 byan
oneTheiteration,
was wasless than
10-1~.
As monomer-dimera measure of the systemdegree into which
Baxte61~andtothezero-field
Potts
model
in
1976
by
Kelland.
~2~
Recently,
equation was satisfied, we also calculated the difference between the rightBaxter
extendedsides
the divided
method byto thea fairly
generalon the terms
squareinand left-handabsolute
sumIsingof allmodel
the additive
lattice.
(3~ A set Thisof matrix
equationsobtainedthe equation.
was nevergreaterwas
thanI0-lo. and a rapidly convergent
sequenceof approximations
to thedeveloped.
Solutionswere obtained
forfreen energy= 2-20.was
Since
the n =It is
2 thecasepurpose
can beofsolved
this work
to test theit convergence
this toapproach.
We of
apply
the method
analytically,
serves as a ofguide
the degree
accuracyof theS.K. Tsang, J. Stat.
Pays. 20 (1979) 95

Cross-over in Critical behavior of expectation values / observables
S.K. Tsang, J. Stat.
Pays. 20 (1979) 95

RAPID COMMUNICATIONPHYSICAL REVIEW B 82, 060410!R" !2010"

SYMMETRY BREAKING AND CRITICALITY IN TENSOR-…

10-2

log 10(δ)

-10

-6
-15

-1

10

-8

Something important isfound several times.
1

0.5

0

0.4

% acceptedlog 10(∆m)

0

-4

m

log 10(∆E)

-5

-1

-2

0

50

100

step

150

-2

0.3

0.2

10

0.1

-3

10

D = 10
D=6
D=4
D=2

0

0

50

100

step

Liu et. al, Phys. Rev.

150

FIG. 2. !Color online" Stochastic energy minimization with 104
updates per step for a D = 4 MPS at h / J = 1.01432, using 64-bit
floating-point arithmetic. The relative energy and magnetization er-

B 82, 060410 R 2010

-6

-5

10

-4

10

10

-3

[hc(D)-h]/hc(D)

-2

10

-1

10

0

10

FIG. 4. !Color online" Demonstration of asymptotic MPS meanfield behavior and scaling crossover in one dimension. The
D-dependent critical fields are: h c / J = 1.0717967 !D = 2", 1.0143343!D = 4", 1.0063523 !D = 6", and 1.0021646 !D = 10". The lines haveslopes # = 1 / 8 and 1/2.

History... they put ancilla, and formed MPS ...

Exact MPS (of very large dimension)

Bethe Ansatz (1931), Onsager (1943)----> Gaudin, Lieb, Mattis, Yang, Baxter, etc.....
の観点 参考文献
MPS of[1]small
dimension(exact/approx.)
U. Schollwöck:
Ann.
Phys. 326 (2011) 96; arXiv: 1008.3477.
[2] H.A. Kramers and G.H. Wannier: Phys. Rev. 60 (1941) 263.
AKLT, Fannes, Zittartz, Derrida, etc.... (as we know quite well)[3] R.J. Baxter: J. Math. Phys. 9 (1968) 650.

ーによ

、最近

概念と

対応が、

ンソル

[4] B. Derrida, M.R. Evans, V. Hakim, and V. Pasquier: J. Phys.
A Math. Gen. 26 (1993) 1493.
[5] I. Aﬄeck, T. Kennedy, E.H. Lieb, and H. Tasaki: Phys. Rev.
Lett. 59 (1987) 799.
[6] M. Fannes, B. Nachtergale, and R.F. Werner: Europhys. Lett.
10 (1989) 633.
[7] S.R. White: Phys. Rev. Lett. 69 (1992) 2863 .

and Bloete, Phys. Rev. B 33, 659 (1986)
ークはNightingale[8] I. Aﬄeck, T. Kennedy, E.H. Lieb, and H. Tasaki: Commun.
den Nijs
andPhys.
Rommelse,
Math.
115 (1988)Phys.
477. Rev. B 40, 4709 (1989)

CTM method is related to Density Matrix (and entanglement)
Later in 1982, Baxter
states in his textbook,about the density matrix

In 1986, Nightingale and Bloete stated in Phys. Rev. B 33, 659 (1986),

This method (*) was formulated by Baxter for classical models instatistical mechanics. The generalization to quantum mechanicalsystems is straightforward.
Density matrix renormalization group (DMRG) was established just 6 years later.

Corner Transfer Matrix RG(Baxter CTM + DMRG)

arXiv:cond-mat/9507087

Okunishi and Nishino (1996)! when i was living in Sendai !

Ising model on (5, 4) lattice can be studied by CTMRG

H = - J Σ σi σj( i and j are NN sites )The space is negatively curved.
All the sites are equivalendThis is also an exampleof the Tensor Network
The lattice has recursive
nature, which enables us toapply CTMRG.
It turned out that the phasetransition is mean-field like.

a

e

b W
d
c

◎ Recursive Structure of the (5,4) lattice

a

1

b W
d
c

1'
2
4 3

-P

W
2'

-C

-P

e

3'
4'

Half `row’ transfer matrix P

Structureof theLattice12
4 3

Corner transfer matrix C

-P

-

W

-C

3' 4'

2'

P

-

P

-C

Lookinthat tot
tially inca multiprecursive
to the ar
total num(CM )6 cr
Entanglement 6
3.5

0.8

n→∞
n = 128n = 64n = 32n = 16
n=8

0.7

n→∞
n = 128n = 64n = 32n = 16n=8
n=4
n=2

Sn (Tn )0.7
FIG. 2: ◎(Coloronline)The 2Dtwo lattice
typical lattice geometries
forWeaklycurved3.0
n = 1 (left) and n = 2 (right) for the size M =0.65. The filled
2.5
circles denote the exceptional lattice sites with 0.5the coordina0.4
tion number seven, 2.0and number of the exceptionalsites is 90
0.3
in the left and 18 in1.5the right, c.f. Eq.(9).
0.2

0.6
0.5

0.4
1

cn

Sn

0.3

1.0

0.1
0.0

0.5
3.63

3.64

3.65

3.66

T

3.67

3.68

3.69

3.60

3.65

3.70

T

10

n

100

where w
(floor) fu
the follo
3.80
3.75

Baxter considered higher dimensional generalization. &*%&./"2/++'

This is nothing but the Tensor Product State, or PEPS
(cond-mat/0011103, 0101360, 0303376)(Two-dimensional extension of MPS)
http://quattro.phys.sci.kobe-u.ac.jp/nishi/NiDMRG2000.html

s(cond-mat/0011103
0101360m= 2
k0303376)

Application to 3D Ising Modell= Uniform Tensor
M
Network State(or uniform PEPS)ji

IRF type with m= 1
Kc = 0.2188 (in 1999)
cond-mat/0001083

Applied to 3D Ising:
Precision in Tc
m = 2: 0.6%m = 3: 0.3%

Vertex type with m= 2
Kc = 0.2207 (in 2000)
cond-mat/0011103
not shown:

Vertex type m = 3
Kc = 0.2210 (in 2003)
cond-mat/0303376
Monte Carlo
Kc = 0.2216544

2.0

2.5

3.0

1 /T
3.5
4.0

in Quantum Physics

51+!&)"*.&+*(/"*.+-*"/1+-'1.
,-+,+."!.*"2/"*.&+*+#/%" .//"
+#/%" )+!"(I. Affleck, T. Kennedy, E.H. Lieb, and H. Tasaki: Commun.Math. Phys. 115 (1988) 477

5&"&!,,(&"!/"*.+-*"/1+-'/+/%"!"#+-)"! .//"*!/-"/&/
*0)"-& ((33)"*.+#
Y. Hieida, K. Okunishi and Y. Akutsu: New. J. Phys. 1 (1999) 7.1.

MPS and Entanglement
tangleentangledisentangle(distangle?!)★Why MPS and TPS/PEPS are so accurate?
Singular Value Decomposition (SVD) is a key technique.

Singular
Value Decomposition《特異値分解》*%a 行
lineardecomposition
for arbitrary matrix.
& 列の行列' を考える．要素' は，縮約()!

#!"! = å λ!ξ $!ξ! % "ξ" & !

（$#）

ξ

の形で一意に表現できる．ξ
は「%
と are
& の小さい方」* ‘lambdas’ are the singular values,
which
non-negative. までの範囲を動く脚だ．行列
と + は，列ベクトル同士の直* U and V are orthogonal (or*unitary)matrix, which satisfy theorthogonal relations.

交関係を満たし，一般化ユニタリー行列と呼ばれる．
!

!
!
#
=
%
%
δ
å #ξ #η ξη $
# ="

"

! #
!
=
&
&
δ
å $ξ $η ξη &!

（$$）

$ ="

λ&ξ*は特異値と呼ばれる非負の実数で，大きい順
≥ λbe
"≥&
Normally, singular values decays rapidly, and tiny onesλ$can
This is one of the key point in the tensor network
λ0 ≥omitted.
…に並べる慣習がある．このような行列' の分解を，
formulation.

特異値分解と呼ぶ．

compression of an image● Singular Values:##########

1.000
0.875
0.750
0.625
0.500
0.375
0.250
0.125
0.000

###

###
##
##
##
##
##
##
##
## #
# ##
##
##
##
##
##
##
#
#
##
##
##
##
##
##
########################

#
@
%
*
+
.

5.76078916114264
4.49306136234049
3.53367812591030
2.93100645840880
2.75155825714451
1.47701283908936
0.936697418652581
0.831935681254495
0.475256853905076
3.315897423129413E-016
1.400243040313679E-016
0.000000000000000E+000

ORIGINAL● 4 dominantsingular values are sufficient to recover rough image.
@@######@#.@#+ .
.
+@@.
.#+. -.- ..+#.
.#*+%%.....*%+*#.
#* -%#..*...*..%#-- *#
@#
...
...
#@
@#
...
...
#@
## .+%-%.. ##-#*%%.....+%-*#@+.. %%%%%%%@%+ ..+@.@#+ .
.
+@@.
@@######@###########
+##
#@
@#
##
%%
@%
##
##
@# #
# *#
##
##
##
##
##
##
-@ .
.. @
##
##
%%
%#
##
#+
######+###++############

1+2+3+4

1+2+3+4+5+6+7##+

Schmidt decompositionof a quantum state
perform SVD
at the center

corresponding state

Linear Combination

Schmidt decompositionf g h iUξλξ

j k l mV

4 ξ&〉right を観測する確率も /ξ になる．なお，4 Ψ〉が規格化さEntanglement

れていれば /ξ の総和も $ になる．
《エンタングルメント・エントロピー》

情報科学では，確率があれば，それに対応する情報エ* WhenSVD is applied
to the ψdistribution
P(fghijklm), dividing
図 2 波動関数,##g-().!% の特異値分解．
theindices into fghi and jklm, the singular values represent
ントロピー，つまりフォン・ノイマンエントロピーを考
some sort of FREQUENCY of linear combinations of states. "えるのが常道だ．規格化された確率分布/ξ＝（&λ&ξ）に対4量子状態の例として，式* The
von Neumann entropy （,）で考えた Ψ〉に着目し，波応するエントロピーを求めてみよう．
動関数
ψ,##g-().!% !を「脚 ,#g-( を含む左半分」と，「脚 ).!% を含
!

!
!
"
"
!
=−
"
#$%
"
=−
（
λ
）
#$%
（
λ
）
（$/）
ξ
ξ &!
!%
å ξ
å ξ g-( を行の脚と，).
む右半分」に分けて考えてみよう．,#
ξξ

を列の脚とみなせば，ψ,##g-().!% は行列の形 ψ（&,##g-(）（&).!%）に表せ
is often called as the Entanglement Entropy, since it quantifies
るので，特異値分解できる．
the quantum entanglement, when a quantum state is considered.
!
!
The terminology is also
field of statistical! used in the
"
&!
（$"）
ψ（ ! g"#（=λ'(!"#$%& 日本物理学会
ξ （ ! g"#）
$%
&
ξ
$%
&
ξ
）
!
）
（
!
）
å
mechanics.
ξCompression of informationf g h iU

ξλξ

Wave function and SVD

Approximation

This process do not spoil the entanglement drastically.
original
Entanglement Entropy

after the compression

j k l mV

How MPS is obtained?
perform SVD from the left

do SVD again.

f

g
α

h
β

i
γ

j
δ

k
ε

l
ζ

m
η

In this manner, one can transform an arbitrary state to MPS.
※ We state only the possibility of expression by use of MPS.
※ Greek indices are restricted to χ
※ If the system is critical, one has to use larger χ

Entanglement structure is a key point in TN formulation.

Renormalization Group picture

Diagram for a Tree
Tensor Network

Scale Invariant?

RG and Critical Phenomena

Snapshot of the 2D
Ising model

Scale invariance

Low T

T = TcReal Space Renormalization

Kadanoff,
Wilson, etc.

Circles are local degrees of freedom
RG Transformation

X

X

X

X

Block
Spins

(a) Treat several spins as an effective spin. (Block Spin Transformation)(b) Keep only relevant freedom.
(Renormalization)(c) Scale change, which is 1/2 in the above case. (Rescaling)(d) Consider interaction among blocks. (RG flow)

Conventional RSRG: Conceptually good, but not Quantitative
T > Tc Interaction decreases with RG steps.(disordered)T = Tc Interaction remains constant.
(critical)
T < Tc Interaction increases with RG steps. (ordered)★ when interaction parameter K is slightly smaller
than its critical value
2
3
K = Kc - δ → Kc - λδ → Kc - λ δ → Kc - λ δ → ....
M
Suppose that when λ δ = C, the correlation becomes
local. Then correlation length ξ can be obtained fromMM
2 = ξ and λ δ = C★ Result-νξ = const. ( Kc - K ) 、

ν = log 2 / log λReference: Kadanoff et al: Rev. Mod. Phys. 86 (2014) 647.

Block-spin transformations are chosen (partially) intuitively.

How to improve the situation?
※ Choice of block spin transformation is not automatic.
※ New site degrees of freedoms are identiﬁed as the block.

Let us reconsider from tensor view point.
i

il

iiʼ

(d=4)
(d=2)

lk

Q
ki

jj

l

j

Extend the block size

llʼ

jjʼ

k

kkʼ

l
※Block touches only through their side
→ those tensor legs that are on thesides should be considered as the newdegrees of freedom.

k

Q
i

j

長さが

2×

2 = 2 と、最初の領域 Q の 2 倍となる。粗Tensor Renormalization Group
視化のプロセスをもう一度行えば、次に得られるテンソル
k

0.9

k

0.95

1

l
k

l

l
(4)
Qµτ
の脚の自由度は
16 × 16= 256
ρσ steps:
RG
ﬁrst, perform
SVD
for となり、そろそろテζTNR 法に
図 13.
法と
ξ TERGF’
G’

し、充分に良い近似値を求める方向へと、計算の方針を変ji
この節で紹介したテンソ
ji
エーションが提案されている

Q

ンソルの要素を計算機に格納しておくことが、現実的では
the local tensor.
jiなくなる。そこで、分配関数 Z を厳密に求めることを放棄FG更するのだ。この段階で、特異値分解を ブロックスピン自 てからブロックの拡大を行Restrict
greek indices!
ブロックの拡大から入る計
由度の抽出 の目的で使うのだ。

り込み群 (Higher-Order T

It can be decomposed verticallyχ

図 12. 特異値分解の後にχ 個だけ自由度を残す
Contract
4-tensors thatare connected

3 次元イジング模型に対してF

χ = 24 程度まで行た計算がηG
ロ・シミュレーションで推定

jiと同様の精度を叩き出してξ 密な定量的計算として実用
ζ
F’
G’
F’
G’
(3)
テンソル Qαβδγ を、式 (23), (24) のように特異値分解をまだ満足してはいけないkl
通じて 2 つのテンソルに分解する際に、主要な特異値を χ を何十個も残すことに、相F個だけ考慮し、微小な特異値を無視してしまおう。
はないだろうか。伝統的なν
Repeat the above procedurχ
256
いぜい数̇Gえ̇る̇程̇度̇の自由度
!
!
(3)
(3)
(3)
(3)
(3)
Qαβδγ =F(αβ)µ G(δγ)µ ∼F(αβ)µ G(δγ)µ(26) た」ではないか。ブロック
µ=1

µ=1

There are several computational methods.
(TERG, HOTRG, TNR, Loop-TNR, etc.)
(d=4)

(d=4)

iiʼ

l

c

k

Q

j

llʼ

jjʼ >> SVD >>

a

b
d

i

kkʼ

What is common?
※Entanglement structure of the system is almost kept
※Focusing on bonds (links) other than sites.
※What is renormalized is not Hamiltonian but tensors.

※From the RG view point, it is curious to consider how therenormalized Hamiltonian looks like when it is written byequations, in particular when the system is critical.

Recent? numerical results by means of Tensor Network
2D Ising Model: Transition temperature can be obtained with 8 digits.
3D Ising Model: obtained Tc = 4.511546 (Monte Carlo: 4.511544)(~ 24 states are kept)(T. Xiang et al: arXiv:1201.1144)(Evenbly and Vidal: arXiv:1412.0732)

estimation of conformal
weights for 2D Ising Model

A Big Step in
Real-Space RG

exact TRG(64) TRG+env(64) TEFR(64) TNR(24)c 0.5 0.49982
0.49988
0.499420.50001
σ 0.125 0.12498
0.12498
0.12504 0.1250004
11.00055
1.000400.99996
1.00009ϵ1.125 1.12615
1.12659
1.12256
1.12492
1.125 1.12635
1.12659
1.12403
1.1251022.002432.00549
1.9992222.00579
2.00557
1.999862
2.00750
2.00566
2.0000622.010612.005672.00168
TABLE I. Exact values and numerical estimates of the central
charge c and lowest scaling dimensions of the critical Isingmodel. TRG results are obtained using the original Levin andNave’s algorithm [4]. TRG+env results are obtained using an
improved TRG method proposed in Ref. [5] under the nameof “poor-man’s SRG”. TEFR results are taken from Ref.
[7]. The first three numerical columns use bond dimensionχ ≡ Dcut = 64 and 1024 spins, while the TNR data uses
χ = 24 and 262,144 spins.

a milstone
Evenbly and Vidal
arXiv:0707.1454

conventional MPS
f

g
α

h
β

i
γ

j
δ

k
ε

l
ζ

m
η

MERA, the multi-scale entanglement renormalization Ansatz,
which has been applied to 1D quantum systems. This tensor
network efficiently describes system at criticality.

played with the same color intensity as the plots of |B |, isalready very small (as compared to the magnitude of the elements in the individual tensors) for s = 1. (c) Starting at thesuper-critical temperature, T = 1.1 TC , the coarse-grained
tensors quickly converge to the disordered fixed point B triv ,8thatEntanglement
has only one non-zeroStructureelement.
in the Lattice models,

Problem of CDL

Evenbly and Vidal
arXiv:1412.0732

in particular at criticality.
T = 1.1 TC , we obtain a trivial fixed-point tensor B triv.
that has only a single significant element [B triv ]1111 = 1
with all other elements zero or arbitrarily small, whichis representative of the infinite temperature, disorderedphase.

Appendix B.– Non-critical RG fixed points: TRG
versus TNR.

FIG. 7. (a) CDL tensor ACDL of Eq. 6. (b) Ten
made of CDL tensors, which contains correlations
each plaquette.

In this appendix we discuss certain aspects of the flowthat TRG and TNR generate in the space of tensors,
In TRG formulation, so called the Corner
which emphasizes one of the main diﬀerences between

Double Line (CDL) remains even after
the RG transformation. Since the original
degrees of freedom partially (?) remains,
the presence of CDL (drastically!) spoils
the numerical precision in RSRG.

a solution for CDL
9

Evenbly and Vidal
arXiv:1412.0732

Solution: apply thedisentangler before
performing RG
transformation by isometry.

This scheme was first
introduced in MERA, themulti-scale entanglement
renormalization Ansatz,
applied to 1D quantum
systems.
Evenbly and Vidal
arXiv:0707.1454

TRG v.s. TNR

Evenbly and Vidal
3
arXiv:1412.0732

numerical results by Tensor Network Renormalization
2D Ising Model: Transition temperature can be obtained with 8 digits.
3D Ising Model: obtained Tc = 4.511546 (Monte Carlo: 4.511544)(T. Xiang et al: arXiv:1201.1144)(Evenbly and Vidal: arXiv:1412.0732)

estimation of conformal
weights for 2D Ising Model

A Big Step in
Real-Space RG

exact TRG(64) TRG+env(64) TEFR(64) TNR(24)c 0.5 0.49982
0.49988
0.499420.50001
σ 0.125 0.12498
0.12498
0.12504 0.1250004
ϵ11.00055
1.000400.99996
1.000091.125 1.12615
1.12659
1.12256
1.12492
1.125 1.12635
1.12659
1.12403
1.1251022.002432.00549
1.9992222.00579
2.00557
1.999862
2.00750
2.00566
2.0000622.010612.005672.00168
TABLE I. Exact values and numerical estimates of the central
charge c and lowest scaling dimensions of the critical Isingmodel. TRG results are obtained using the original Levin andNave’s algorithm [4]. TRG+env results are obtained using an
improved TRG method proposed in Ref. [5] under the nameof “poor-man’s SRG”. TEFR results are taken from Ref.
[7]. The first three numerical columns use bond dimensionχ ≡ Dcut = 64 and 1024 spins, while the TNR data uses
χ = 24 and 262,144 spins.

(Evenbly and Vidal: arXiv:1412.0732)
exact TRG(64) TRG+env(64) TEFR(64) TNR(24)c 0.5 0.49982
0.49988
0.499420.50001
σ 0.125 0.12498
0.12498
0.12504 0.1250004
ϵ11.00055
1.000400.99996
1.000091.125 1.12615
1.12659
1.12256
1.12492
1.125 1.12635
1.12659
1.12403
1.1251022.002432.00549
1.9992222.00579
2.00557
1.999862
2.00750
2.00566
2.0000622.010612.005672.00168
TABLE I. Exact values and numerical estimates of the central
charge c and lowest scaling dimensions of the critical Isingmodel. TRG results are obtained using the original Levin andNave’s algorithm [4]. TRG+env results are obtained using an
improved TRG method proposed in Ref. [5] under the nameof “poor-man’s SRG”. TEFR results are taken from Ref.
[7]. The first three numerical columns use bond dimensionχ ≡ Dcut = 64 and 1024 spins, while the TNR data uses
χ = 24 and 262,144 spins.

theory [37]. This is best known in the context of critquantum spin chains, where Cardy’s formula relates
smallest scaling dimensions ∆α of the conformal theto the lowest eigenvalues Eα of the critical Hamilton
on a periodic chain according to

2π∆α + · · · ,(Lwhere L is the number of spins on the chain, Eα (is the energy in an infinite system, η (independenα) depends on the normalization of the Hamilton
and where the sub-leading non-universal corrections
O(1/L2 ) in the absence of marginal operators. Thus,
can estimate scaling dimensions ∆α by diagonalizin
critical Hamiltonian on a finite periodic chain, as i
often done using exact diagonalization techniques. I
two dimensional statistical system, the analogous reis the observation that even in a finite system, a critpartition function Z is organized according to the sca
dimensions of the theory as!
2πcaLx LyZ =ee−Ly Lx (∆α − 12 )+··· ) ,(Eα (L) − Eα (∞) = η

not known …

point tensor (and thus recovers scale invariance) at critα
icality.
As
demonstrated
in
Ref.
[7],
TEFR
produces
From the theoretical view point, it whereis curious
to consider
how
a is some non-universal
constant,
c is the cen
accurate estimates of the central charge c and scaling dicharge of the CFT, and we assumed isotropic coupli
mensions
∆
for
the
Ising
model,
and
one
might
indeedα
the renormalized Hamiltonian looksandlike
when
is ofwrittenbya squaretorus itmadeLx × Ly sites.
misinterpret those as being evidence for having generated
aequations,critical fixed-pointin
tensor.
However, thewhen
estimatesthere- system is critical.
particularported in Ref. [7] for c and ∆α are of comparable ac-Tensor Network Formulations
from the view point of entanglement and RSRG
Tomotoshi Nishino (Kobe Univ.)