An Origin of Matrix Product Statein Statistical Mechanics
西野友年 (神戸大学)

Tomotoshi Nishino (Kobe University)
Andrej Gendiar (Slovak Academy of Sciences)
Based on R.J.Baxter: “Dimers on a Rectangular Lattice” J. Math. Phys. 9, 650 (1968)a prelude --> There was researchers who wrote the following article in 1941: Phys. Rev. 60, 263 (1941)

Kramers-Wannier Approximation(a) They tried to evaluate the maximal eigenvalue of the Transfer Matrix of the 2D Ising model.
(b) They assume a Variational state as a PRODUCT of local factors.
(*) idea: to use Boltzmann weight for 1D Ising model under external magnetic fieldas the variational state of the transfer matrix of the 2D Ising model.

V( S1, S2, ..., SN ) = M( S1; S2 ) M( S2; S3 ) ..... M( SN-1; SN )S is the Ising Spin, and M is the 2 x 2 Matrix, which contains 4 parameters.
(Actually there are only 2 INDEPENDENT parameter, c and d.)(c) They tune the parameter to obtain the minimum of the Free Energy per row.

(Such a variational ratio is quite common in (quantum) Monte Carlo Simulations.)(d) They established the World Record in numerical (by hand?!) evaluation of Tc.

Specific Heat of the 2D Ising Model

Exact
K-W approx.

Temperature

Precision: ~7% in Tc(*) Note that precision in Free Energy is much better.

... their scheme is not old at all.

K-W Approx. applied to 3D Ising Model(cond-mat/9909097)
Magnetization

Variational state:
2D Ising under External Field

Precision of Tc: 1.5%(*) one can extend theformulation to any dimension,
where the efficiency might
increase with the dimension.
奥西 and 西野
Kramers-Wannier approximation inINFINITE DIMENSION ???
---> MPS in infinite D.

= J/kT2010: Odyssey Two

The Blue Book:
Free Online http://tpsrv.anu.edu.au/Members/baxter/bookAlso I love the red books.

An Origin of Matrix Product Statein Statistical Mechanics
R.J.Baxter: “Dimers on a Rectangular Lattice” J. Math. Phys. 9, 650 (1968)
There was (and is) a man who wrote the following phrase in 1967.

(*) The problem he treated is the thermal equilibrium state ofdimers on a m x n square lattice under Gibbs statistics.
Energy of a bond: E = 0 when empty.
E = -2 kT ln t when occupied vertically.
E = -2 kT ln s when occupied horizontally.
Corresponding Boltzmann Weight W = exp(-E/kT)
W = 1 when empty.
W = t * t when occupied vertically.
w = s * s when occupied horizontally.

It is convenient to attribute the Boltzmann weight for each Vertex.
We denote horizontal bond variable (= 0 or 1) by ʻaʼ and vertical one by ʻbʼ.
There are 5 vertex configurations:bʼaVertex Weight K:aʼ

K = 1 for empty vertex.

b
K=sK=tPartition Function of the system can be represented as a configuration sum for theproduct of local Boltzman weights.

... we use images from Baxter’s paper to remind his originality .... (and to reduce typing task)

Row-to-Row Transfer Matrix: a horizontal product of Boltzmann weights (Periodic BC)bʼ1bʼ2bʼ3bʼ4bʼ5bʼ6bʼ7bʼ8b1b2b3b4b5b6b7b8

Partition function obtained from the transfer matrix V

Variational evaluation of the largest eigenvalue

How to obtain the variational state x?
(This is the Time Evolution Scheme that we know.)

The Matrix Product Ansatz
Eq. (3.3)bHalf-Column Transfer Matrix = “Local Tensor” in Tensor Network State

Variational Freedom, Exactness, and Expected Efficiencybʼ=1Fixed BC

Column-to-Column Transfer Matrix
Extended C-T-C Transfer Matrix

Variational Ratio
XK

2.53.103.101. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp.... this is an origin of MPS in statistical mechanics .... (another one is by Derrida.)

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

Eigenvalue of theC-T-C Transfer Matrix

Eigenvectors; X for R and P for SGKP=PG

192.53.103.101. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp(**) these equations are written in thematrix form. Please look and guess!
GX
G=X

Variational Estimate of the Partition Function per Vertex

GX

X

PK

P

GGP

P

X

X
G

The rest of the task is to find out the variational minimum, adjusting G andobtaining the eigenvectors of CTC matrices. Direct optimization is straightforward;
Baxter go beyond!

Introduction of the CORNER TRANSFER MATRIXThe CTM corresponds to a Quadrant of the system.

AX“Gauge” degree of freedom,“Gauge” fixing by the diagonalization of CTM, which isequivalent to the diagonalization of the Density Matrix.

=AA
P

=

G*A

Downloaded 15 Feb 2010 to 192.53.103.101. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/coTo obtain the diagonalized CTM is to solve the variational problem, since A isconstructed as a product of tensors G.

Evolution (or extension) Scheme for CTM itselfGAGA... GKG induces an evolution for CTM.
Every after this extension, the matrix
ubject to AIP licensedimension
or copyright;
http://jmp.aip.org/jmp/copyright.jsp
is seedoubled.
Diagonalization of the extended CTM

Cut off the irrelevant parts

From the eigenvectors in Q, obtain the “improved” G. (This is a bit different from CTMRG.)

Baxterʼs numerical result on the dimer problem

One finds a careful statement on his numerical result.

192.53.103.101. 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 ...

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.
As far as I know, this is the first statement about the applicability of MPS toquantum problems. (N-B performed variational study based on Kramers-Wannierapproximation.) Sometimes, people do not try to study something straightforward.

Homework for DMRG Experts:
Consider the application of DMRG to the Dimer Problem, more generally, to the(symmetric) Vertex Models.
Apply the Infinite System Algorithm to the Row-to-Raw transfer matrix.
Prove that this is equivalent to iTEBD for the Column-to-Column transfer matrix.

Homework for TERG/SRG/MERA Experts:
Perform TERG for hexagonal lattice, leaving the row-spin at the bottom unrenormalized. Youʼll obtain a MERA-like tensor network. (MeraZe?)
1st step2nd step

3rd stepHomework for TNS Experts:
Repeating measurement on MPS, obtain a snapshot of spin configuration of athermal equilibrium state. Be careful to the treatment of the environment.

How about 3D? Baxter stated (in his textbook 1982)

Now in 2010, we have an approach; Tensor Network State.

(*) our contribution: cond-mat/0011103, 0101360, 0303376,
http://quattro.phys.sci.kobe-u.ac.jp/nishi/NiDMRG2000.html(**) Verstraete and Cirac formulated 2D PEPS: cond-mat/0407066.
(***) see also S.G. Chung: arXiv:1008.0362 (published in 2006.)

m = 2: 0.6%m = 3: 0.3%

does not goto zero. This constraint prevents D to havenegative eigenvalues. (We don't know the reason yet.)sApplication to 3D Ising Model (trial)
Spontaneous magnetization
as a function of 1/T → M

Uniform TPS
m= 2lkji

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

Precision in Tc
m = 2: 0.6%m = 3: 0.3%

Vertex type with m= 2
Kc = 0.2207 (in 2000)
cond-mat/0011103.... MC is superior(for the moment?!)

not shown:... It depends on thescaling with m.
(Tagliacozzo et al.)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

See also cond-mat/9909097

Application to 3D ANNNI Model

H = -J Σi,j,k Si,j,k ( Si+1,j,k + Si,j+1,k + Si,j,k+1 ) + J’ Σi,j,k Si,j,k Si+2,j,k
3D Ising Model + uniaxial frustration
4.5
4.0

B1
B2
B3

Para

PL

3.5

There is a wideparameter area,
where the finestructure inside it isnot fully determined.

C1kBT/J1

3.0A1A2

2.5
2.0

Ferro

C2A3
A4

1.5
1.0
0.2

B00.3

0.4

Antiphase

B4
0.5

0.6

J2/J1

0.7

0.8

0.9

1.0

Variational Statem i,jm i,j+1m i+1,jm i+1,j+1 m i+2,j+1

mi+2,j^Vi,jWe have chosen the position dependent IRF-type TPS
with NO auxiliary variable with overlap. This is a
simplest choice, that fits to the structure of the transfer
matrix. (... now we think that inclusion of auxiliary
variable might affect the calculated result ...)
m i,j+1m i+1,j+1

mi+2,j+1

m i,jo [m|m]m i,jm i+1,jWi,jB

m i+2,jFree energy minimization by sweeping (N = 100-site)ï5.45

Local tensors are
optimized placeby place.

ï5.521

ï5.46

ï5.522F

Convergence of the free energy F

ï5.520

ï5.47

ISM

ï5.52311RL

ï5.524

ï5.48

ï5.525

0200

400

ï5.49

600

800

1000

1LR1RLï5.50

i

2LR2RL

ï5.51
ï5.52
ï5.53

0100

200

300

400

500

600

700

800

900 1000i

The convergence is relatively slow because
there is frustration.

Spin modulation.... seems to be periodic, and is NOT sinusoidal.
<mi> at C11.0(There is nonlineareffect in modulation)

0.5
0.0
ï0.5
ï1.0120130140i150

160

170

0.5
0.0

4.5

ï0.5

4.0

ï1.0
120

3.5

130

140i

150

160

B1B2
B3Para

PL

170

C1
3.0

kBT/J1<mi> at C2

1.0

A1A2

2.5
2.0

Ferro

C2A3
A4

1.5
1.0
0.2

B0

0.3

0.4

Antiphase

B4
0.5

0.6

J2/J1

0.7

0.8

0.9

1.0

From the large scale structure it can be concluded that themodulation is NOT LOCKED at the parameter point C2.

0.5

FixedïFree

<mi> at C2

1.0

0.0
ï0.5

Fixedïparallel<mi> at C2

ï1.0
1.0
0.5
0.0
ï0.5

Fixed antiïparallel<mi> at C2

ï1.0
1.0
0.5
0.0
ï0.5
ï1.0

0100

200
i300

400

Modulation Period along a line in the parameter space(A4-B4, very low temperature)
8.0
3

7.5

<4 ,3> 2<4 ,3>
2<4,(4,3) >3<4,(4,3) >
<4,3>

7.0

2h

<4,3 >
3<4,3 >
4<4,3 >

6.5

<3>
6.0

5.5
0.498

0.499

0.500

0.501

0.502

0.503

0.504

0.505

J2/J1(Note that stable low temperature calculation is difficult.)

Modulation wave number,compared with the perturbation result
1.1
<2>
6

1.0

<3,2 >

0.9<3,2 >
2<3,2 >

3

4

<3,2 ><3,2>

q/q<2>

5<3,2 >

0.8
<3>

0.7

2

<3 ,2>
3<3 ,2>

2<4,3 >
0.6

3<4,3 >
<4,3>
0.5
ï1.0

ï0.5

0.0
0.5
1.0
1.5
2.0
ï1
(J2/J1ï0.5)/(kBTJ1 exp{ï8J1/kBT})

2.5

3.0

Phase diagram: The locked area is narrower than it waspredicted.
5.0
4.5Paramagnetic phase

4.0

PL

3.5

<6>
<5>
<4>

<3,2>
<3>

dateludMo ses
pha

kBT/J1

3.0
2.5
2.0

Ferromagneticphase

Antiphase <2>

1.5
1.0
0.5
0.0
0.0

0.1

0.2

0.3

0.4

0.5

J2/J1

0.6

0.7

0.8

0.9

1.0

◎ Tensor Network on Hyperbolic Lattices
Their Hausdorff dimension is infinite!

Ferromagnetic Ising Model on the (5, 4) lattice

H = - J Σ σi σj( i and j are NN sites )MC simulation is quite hard on this lattice !!

aeb Wdc◎ Recursive Structure of the (5,4) latticein general, (q,p=even) lattice has
the same property.

a1b Wdc1'24 3-PW-C-P

2'3'4'

Half `row’ transfer matrix P(after taking configuration sum for
spins inside, leaving those on theboundary)

Structureof theLattice
124 3

Corner transfer matrix Ce-PW-

P-C

3' 4'2'-P-C

◎ Renormalization Group Transformation
1

124 3-

P-PW-C-

P-

C(a)

P-PW
P

24 3-C-P

1'3' 4'2'-

CW-CW-

P-C(b)

-P

-P

2'
3'
4'

→ Corner transfer Matrix RG (CTMRG)◇ map the half-row spin to a m-state block spin.
◇ also apply the mapping to CTM and HRTM.
◇ extend the area of CTM and HRTM recursively◇ obtain the 1-point function at the center ofsufficiently large finite size system.
(T. Nishino and K. Okunishi, J. Phys. Soc. Jpn. 65 (1996) 891,cond-mat/9507087.)-PW

P-P-C(a)-C

W-P-C(b)-P

Mean-Field Universality: always on Hyperbolic Lattices???
0.05<mi >2

0.04
0.03
0.02
0.01
0.00
2.775

2.780

2.785

2.790

2.795

2.800

2.805

2.795

2.800

2.805

T
<mi +1 mi >

0.40
0.39
0.38
0.37
0.36
2.775

2.780

2.785

2.790

T◇ conjectured Mean field like transition is verified.
(H. Shima and Y. Sakiniwa, J. Phys. A 39 (2006) 4921.)

http://quattro.phys.sci.kobe-u.ac.jp/nishi/NiDMRG2000.html

An Origin of Matrix Product Statein Statistical Mechanics
R.J.Baxter: “Dimers on a Rectangular Lattice” J. Math. Phys. 9, 650 (1968)
There was (and is) a man who wrote the following phrase in 1967.