Renormalization of Tensor Network States

Tao Xiang

Institute of Physics

Chinese Academy of Sciences

txiang@iphy.ac.cn

I. Coarse Graining Tensor Renormalization

Numerical Renormalization Group

brief introduction

Renormalization of Tensor Network States: Brief History

1975, Wilson proposed the Numerical Renormalization Group

(RG) method to solve the single impurity Kondo model (0

dimensional problem)

1992, White proposed the Density Matrix Renormalization Group

(DMRG), which becomes the most powerful method for studying

1D quantum lattice models

Starting from 2000s, various tensor-based renormalization group

methods were developed to solve 2D or 3D quantum or classical

statistical models

Difference between RG and Numerical RG

Renormalization Group (analytical)

Renormalization of charge, mass, critical exponents

and other few physical parameters

System must be scaling invariant

Numerical Renormalization Group

Direct evaluation of quantum wave function/partition

function

The system not necessary to be scaling invariant

Basic Idea of Numerical Renormalization Group

| =

𝒊=𝟏

𝑵𝒕𝒐𝒕𝒂𝒍

𝒂𝒊 | 𝒊

𝒌=𝟏

𝑵≪𝑵𝒕𝒐𝒕𝒂𝒍

𝒃𝒌 | 𝒌

To find a small and optimized set of basis states | 𝑘to represent accurately a wave function

refine the wavefunction by local RG transformations

Numerical Renormalization Group

Physics： compression of basis space (phase space)

or compression of information

Mathematics: low rank approximation of matrix or tensor

| =

𝒊=𝟏

𝑵𝒕𝒐𝒕𝒂𝒍

𝒂𝒊 | 𝒊

𝒌=𝟏

𝑵≪𝑵𝒕𝒐𝒕𝒂𝒍

𝒃𝒌 | 𝒌

To find a small and optimized set of basis states | 𝑘to represent accurately a wave function

Is Quantum Wave Function Compressible?

𝑁𝑡𝑜𝑡𝑎𝑙 = 2𝐿2

L

L

B

A

𝒍𝒏𝑵

𝑵 ~ 𝟐𝑳 << 𝟐𝑳𝟐

= Ntotal

Minimum number of basis

states needed for accurately

representing a ground state

S 𝑳

Entanglement Entropy Area Law

Ising model

The answer:

2. Variational ansatz of the ground state wave function

of quantum lattice models

1. Faithful representation of the partition functions of

all classical and quantum lattice models

Tensor Network States

Virtual Bond Dimension D: How Large Needed?

Physical

basis

Local

tensor

Virtual

basis

Projected Entangled Pair State (PEPS)

D

PEPS is exact ground state wavefunction in the limit D

Entanglement entropy

S = L ≈ L ln D D ~ 𝒆 (independent of L)

2D Interacting Fermions: How Large D Needed?

D grows in some power law with the system size

Entanglement entropy

S = L lnL ≈ L ln D D ~ 𝑳𝜶

Physical

basis

Local

tensor

Virtual

basis

Projected Entangled Pair State (PEPS)

D

Stoudenmire and White, Annu. Rev. CMP 3, 111(2012)

S=1/2 AF Heisenberg model on infinite square lattice

Reference energy: VMC extrapolation Sandvik PRB 56, 11678(1997)

Comparison between DMRG and Tensor RG

PEPS

Quantum lattice model

Approach I: Directly evaluate the (2+1) partition function

Approach II: Find the ground state wavefunction (PEPS)

Evaluate the physical observables

Classical statistical model

How to trace out all tensor indices?

Problems to be solved by tensor renormalization group

Tensor representations

of classical statistical models

H. H. Zhao, et al, PRB 81, 174411 (2010)

1. Faithful representation of the partition functions of

all classical and quantum lattice models

What Are Tensor Network States?

𝑇𝑥𝑖𝑥′𝑖𝑦𝑖𝑦′𝑖

2D quantum systems are

equivalent to 3D classical

ones

1

1 2 2 3 1 1

1

1

...

...

max

exp

...

N

N N N

N

i i

S S i

S S S S S S S S

S S

N

Z S S

A A A A

Tr A A

N

ee

eeA

1D: partition function is a matrix product

Example: one dimensional Ising model

S1 S2 S3 … … SN-1 SN𝐻 = −

𝑖

𝑆𝑖𝑆i+1

Two-Dimensional Ising model

𝑍 = Tr exp −𝐻

= Tr

∎

exp −𝐻∎

= Tr

{𝑆}

𝑇𝑆𝑖𝑆𝑗𝑆𝑘𝑆𝑙

𝐻 = −

𝑖𝑗

𝑆𝑖𝑆𝑗

𝑆𝑖

𝑆𝑘𝑆𝑙

𝑆𝑗𝑆𝑖

𝑆𝑘𝑆𝑙

= exp −𝐻∎𝑆𝑗

= 𝑇𝑆𝑖𝑆𝑗𝑆𝑘𝑆𝑙=

i j

ij

H= -J S S

' 'expi i i i

1 2 3 4 1 1 1 2 1 3 1 4 1 2 3 4

1

ij y x y x

ij i

S S S S

S

Z Tr H Tr T

T U U U U

exp expi j

1 2 1 1 1 2 1

S S ij i j

S S S S

M H JS S

M U U

Singular Value Decomposition

S1 S2

4

3 1

2

Tensor-network representation is not unique

i j

ij

H= -J S S

' '

/

expi i i i

1 2 3 4

1 2 3 4

y x y x

i

J 2

1 2 3 4

Z Tr H Tr T

T e 1

/

1 1 2

2 2 3

3 3 4

4 4 1

1 2 3 4

1 2 3 4 1 2 2 3 3 4 4 1

S S

S S

S S

S S

H J 2

S S S S S S S S 1

S1 S2

S4 S3 4

3 2

1

Duality transformation

Tensor-network representation in the dual lattice

Gauge Invariance

T1 T2

𝑃𝑃−1

T2 → 𝑃−1𝑇2

T1 → 𝑇1𝑃

To redefine the local tensors by inserting

a pair of inverse matrices on each bond

does not change the partition function

Coarse Graining Tensor Renormalization

RG Methods for Evaluating Partition Function

Transfer matrix renormalization group (TMRG, Nishino/classical 1995,

Xiang et al/quantum 1996)

Corner transfer matrix renormalization group (CTMRG, Nishino 1996)

Time evolving block decimation (TEBD, Vidal 2004)

Tensor renormalization group (TRG, Levin, Nave, 2007)

Second renormalization group (SRG, Xie et al 2009)

TRG with HOSVD (HOTRG, HOSRG Xie et al 2012)

Tensor network renormalization (TNR, Evenbly, Vidal 2015)

Loop TNR (Yang et al 2016)

Which Method Should We Use?

Accuracy

Efficiency or cost (CPU and Memory)

Applicability in 3D

Scaling invariance at the critical point

Computational Cost

Method CPU Time Minimum Memory

TMRG/CTMRG 𝑑3𝐷3𝐿 𝑑2𝐷3

TEBD 𝑑3𝐷3𝐿 𝑑2𝐷3

TRG 𝐷6ln𝐿 𝐷4

SRG 𝐷6ln𝐿 𝐷4

HOTRG 𝐷7ln𝐿 𝐷4

HOSRG 𝐷8ln𝐿 𝐷6

TNR 𝐷7ln𝐿 𝐷5

Loop-TNR 𝐷6ln𝐿 𝐷4

𝑑: physical dimension 𝐷: bond dimension 𝐿: lattice size

Applicability in 3D

In principle, all methods can be generalized to 3D.

But most of the methods are less efficient, the cost (both

CPU time and memory) is very high.

By far, the most efficient method in 3D is HOTRG and

HOSRG

Removing Local Entanglement

NTR and loop-NTR tend to remove the local

entanglements, and work better than the other coarse

graining RG methods at the critical regime

Disentangler

Step I: Rewiring

,

, ,

1

kj il mji mlk

m

D

kj

n

nn il n

M T T

U V

Singular value decomposition

Step II: decimation

Coarse grain tensor renormalization group

Levin, Nave, PRL 99 (2007) 120601

Singular value decomposition Schmidt decomposition

n2 is the eigenvalue of reduced

density matrix

n sys envn

n n , ,

1

, ,

1

N

ij i n n j n

n

i n n j

D

n

n

f U V

U V

System Environment

|jenv

,

ij sys envi j

f i j |isys

Singular value decomposition of matrix

Step II: decimation

xyz xik yji zkj

ijk

T S S S

Coarse grain tensor renormalization group

Accuracy of TRG

Ising model on a triangular lattice

D = 24

envZ=Tr MM TRG:

truncation error of M is

minimized by the singular

value decomposition

But, what really needs to be

minimized is the error of Z!

SRG:

The renormalization effect of

Menv to M is considered

system

Xie et al, PRL 103, 160601 (2009)

Zhao, et al, PRB 81, 174411 (2010)

environment

Second Renormalization of Tensor Network Model (SRG)

envZ=Tr MM / / / /

,

env 1 2 1 2 1 2 1 2

kl ij k l i jM

Mean field (or cavity) approximation

4

, , ,

1...

kj il kj n n il n

n D

M U V

= 1/2 1/2

From environment

From system

Bond field – measures the

entanglement between U and V

Poor-Man SRG: Entanglement Mean Field Approximation

Accuracy of Poor Man’s SRG

Ising model on a triangular lattice

D = 24

Tc = 4/ln3

TRG

Menv

Evaluate the environment contribution Menv using TRG

SRG

( 1) ( )

' ' ' ' ' ' ' '

' ' ' '

n n

ijkl i j k l k jp j pi i lq l qk

i j k l pq

M M S S S S

( 1)n

ijklM

( )

' ' ' '

n

i j k lM

1. Forward iteration

(0) (1)

( )N

M M

M

2. Backward iteration

( ) ( 1)

(0)

N N

env

M M

M M

Ising model on a triangular lattice

D = 24

Accuracy of SRG

Coarse graining tensor renormalization by HOSVD

DD2

D

M(n)

HOSVD

Higher-order singular

value decomposition

Lower-rank

approximation

Z. Y. Xie et al, PRB 86, 045139 (2012)

Step 1: To contract two local tensors into one

x = (x1, x2), x’ = (x’1, x’2)

DD2

D

Coarse graining tensor renormalization by HOSVD

Step 2: determine the unitary transformation matrices by the HOSVD

DD2

D

M(n)

Coarse graining tensor renormalization by HOSVD

Step 2: determine the unitary transformation matrices

By the higher order singular value decomposition

Higher order singular value decomposition

Coarse graining tensor renormalization by HOSVD

Step 3: renormalize the tensor

cut the tensor dimension according to the norm of the core tensor

Coarse graining tensor renormalization by HOSVD

Core tensor

all-orthogonal:

pseudo-diagonal / ordering:

L. de Latheauwer, B. de Moor, and J. Vandewalle, SIAM, J. Matrix Anal. Appl, 21, 1253 (2000).

Higher order singular value decomposition (HOSVD)

Generalization of the singular value decomposition of matrix to tensor

Tucker decomposition

Only horizontal bonds need to be cut

if ε1 < ε2 , U(n) = UL

if ε1 > ε2 , U(n) = UR

truncation error = min(ε1 , ε2 )

Unitary Transformation Matrix

HOSVD can be achieved by successive SVD for each index of the tensor

For example

How to do HOSVD

Nishino Diagram of HOTRG

envZ=Tr MM

TRG: truncation error of M is minimized

But, what really needs to be minimized is the error of Z!

SRG:

minimize the error of the partition function

The renormalization effect of Menv to M is included

system environment

M env

Second renormalization of tensor network states

Forward iterations: use TRG

to determine U(n) and T(n)

SRG: forward iteration + backward iteration

Backward iterations : evaluate

the environment tensors

How to Determine the Environment Tensor?

HOSRG: Bond Density Matrix

HOTRG at 3D (or 2+1D)

3D HOTRG

Higher order singular

value decomposition

2D 3D

Memory CPU time Memory CPU time

HOTRG D4 D7 D6 D11

HOSRG D5 D8 D7 D12

Computational Cost

Relative difference is less than 10-5

HOTRG (D=14): 0.3295

Monte Carlo: 0.3262

Series Expansion: 0.3265

MC data: A. L. Talapov, H. W. J. Blote, J. Phys. A: Math. Gen. 29, 5727 (1996).

Magnetization of 3D Ising model

Z. Y. Xie et al, PRB 86, 045139 (2012)

Solid line: Monte Carlo data from X. M. Feng, and H. W. J. Blote, Phys. Rev. E 81, 031103 (2010)

D = 14

Specific Heat of 3D Ising model

Critical Temperature of 3D Ising model

Bond dimension

Critical Temperature of 3D Ising model

method year Tc

HOTRG D = 16

D = 23

2012

2014

4.511544

4.51152469(1)

NRG of Nishino et al 2005 4.55(4)

Monte Carlo Simulation 2010 4.5115232(17)

2003 4.5115248(6)

1996 4.511516

High-temperature expansion 2000 4.511536

S. Wang, et al, Chinese Physics Letters 31, 070503 (2014).

2D QuantumTransverse Ising Model at T = 0K

2D Quantum Ising model

Z. Y. Xie et al, PRB 86, 045139 (2012)

Internal Energy Magnetization

Thermodynamics of the 2D Quantum Ising Model

RG Flow of Local Tensors

critical

point

fixing

pointfixing

point

ordered phase disordered phase

How does the tensor change with the RG steps?

Critical Behavior of Tensor Network Model

• After a RG iteration, the scale is enlarged (the system size is

reduced) and the entanglement between tensors is reduced

• The local tensor T(n) converges after many steps of iterations,

and the converged tensor is completely disentangled

Fix Point Tensor

critical

point

fixing

pointfixing

point

ordered phase disordered phase

The fixing point tensor is diagonal up to gauge uncertainty

At high symmetric point, it is a rank-1 tensor.

At low symmetric point (symmetry breaking), it is direct sum of two

or more rank-1 tensors.

𝑇1111 = 1𝑇1111 = 1

𝑇2222 = 1

RG Flow of the Tensors

The fixing point tensor at the critical point contains the

information on the central charge and scaling dimensions

When the system size is smaller than the correlation length, it

behaves like a critical system

c=6 ln 𝑚𝑎𝑥𝜋

n are eigenvalues of

𝑀𝑢𝑑 =

𝑟

𝑇𝑟,𝑟,𝑢,𝑑

Central Charge at the Critical Point

Application: Potts Model on Irregular Lattices

Partial Symmetry Breaking and Phase Transition

QN Chen et al, PRL 107, 165701 (2011)

M. P. Qin, et al, PRB 90, 144424 (2014)

i = 1,…,q

Antiferromagnetic: J > 0

q < qc 1st/2nd phase transition at finite temperature

q = qc critical at 0K

q > qc no phase transition

Potts model

Lattice Coordination number qc

honeycomb 3 <3

square 4 3

diced 4 3<qc<4

kagome 4 3

triangular 6 4

union-jack 6 ?

centered diced 6 ?

Can qc > 4 in certain lattices?

Critical q for the antiferromagnetic Potts model

i = 1,…,48 neighbors

4 neighbors

q=4 Potts Model on the UnionJack Lattice

Is there any phase transition?

Phase Transition with Partial Symmetry Breaking

full symmetry breaking

Entropy = 0

partial symmetry breaking

Entropy is finite

random

orientation

Full versus partial symmetry breaking

If red or green sublattice is ordered, the ground states are

3N/4-fold degenerate S = (3N/4) ln

both red and green sublattices are ordered, the ground

states are 2N/2-fold degenerate: S = (N/2) ln 2

S = (N/2) ln 2 + 2 * (3N/4) ln

Ground states and their entropies

The red or green sublattice

is ordered

Entropy and Partial Order

There is a partial symmetry breaking at 0K

There is a finite T phase transition with two singularities:

1. ordered and disordered states

2. Z2 between green and red

q = 4 Potts model

Conjecture: there is a finite temperature phase transition

Phase Transition: Specific Heat Jump

q = 4 Potts model on the Union-Jack lattice

1/16

Green or Red Sub-lattice Magnetization

Diced Lattice

Centered Diced Lattice

Checkerboard Lattice

Partial order phase transition in other irregular lattices

Lattice Coordination number qc

honeycomb 3 <3

square 4 3

diced 4 3<qc<4

kagome 4 3

triangular 6 4

union-jack 6 >4

centered diced 6 >4

Critical q for the antiferromagnetic Potts model

In the past decade, various coarse graining RG methods have

been developed to compute tensor network models

These methods provide a powerful tool for studying 2D/3D or

2+1D lattice models

More applications of these methods can and should be done

in future

Summary