coarse graining tensor renormalization by the higher-order...
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Coarse Graining Tensor Renormalization by the Higher-Order Singular Value Decomposition
Tao Xiang
Institute of Physics/Institute of Theoretical Physics
Chinese Academy of Sciences
Acknowledgement
References H. C. Jiang et al, PRL 101, 090603 (2008) Z. Y. Xie et al, PRL 103, 160601 (2009) H.H. Zhao et al, PRB 81, 174411 (2010) Q. N. Chen et al, PRL 107, 165701 (2011) H. H. Zhao et al, PRB 85, 134416 (2012) Z. Y. Xie et al, arXiv:1201.1144
Zhiyuan Xie
Qiaoni Chen
Huihai Zhao
Jing Chen
Jinwei Zhu
IOP / ITP / ICM, CAS
Bruce Normand (Renmin University of China)
Guangming Zhang (Tsinghua University )
Cenke Xu (University of California, Santa Barbara)
Liping Yang (Beijing Computational Science Research
Center)
Strong correlated systems
Strong many-body effect:
Single particle approximation invalid
No good analytic tool: particles are
highly entangled
Weak coupling systems
Week interaction
Mean-field or single particle
approximation works
Particles are app. disentangled
Why should we study the renormalization of tensors?
Cold atoms 3He-superfluid 4He-superfluid CMR simple metal heavy fermions QHE high-Tc superconductor
10-6 10-4 10-2 100 102 104 T(K)
Energy
Weak coupling Strong coupling: non-perturbative
Can we solve the problem numerically?
Cold atoms 3He-superfluid 4He-superfluid CMR simple metal heavy fermions QHE high-Tc superconductor
10-6 10-4 10-2 100 102 104 T(K)
Energy
Weak coupling
Density Functional Theory
Walter Kohn 1997
Numerical Renormalization Group
K. Wilson 1982
Strong coupling: non-perturbative
1. Wilson NRG 1975 - 0 Dimensional problems (single impurity Kondo model)
2. Density Matrix Renormalization Group (1D algorithm) 1992 - 1 Dimensional quantum lattice models
3. Tensor Renormalization Group 2 or higher dimensional lattice models
Three Stages of Numerical Renormalization Group Study
( )1
1
~ ~ ln
~ exp
d
d
S L D
D L
−
−
Environment
system L
L D: minimal number of basis states needed
for describing the ground state
d: spatial dimension
What is the wavefunction that satisfies this area law?
Why is it more difficult to study higher-dimensional systems?
Area Law of Entanglement Entropy
The Answer: Tensor Network State
Tensor network state (projected entangled pair state) is a faithful
representation of the ground state of a quantum lattice model
Verstraete, Cirac, arXiv:0407066
x = 1 … D
Classical Statistical Models = Tensor-network Model
The partitions for all statistical models with local interactions
can be represented as tensor-network models
Tensor-Network Representation in the Original Lattice
i jij
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 xij i
S S S SS
Z Tr H Tr T
T U U U Uσ σ σ σ σ σ σ σ σ σ σ σ
β
Λ Λ Λ Λ
= − =
=
∏ ∏
∑
( ) ( )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 Uσ σ σ
β β
Λ
= − =
=
Singular Value Decomposition
S1 S2 σ4
σ3 σ1 σ2
Tensor-Network Representation in the Dual Lattice
i jij
H= -J S S∑
( )( ) ( )
' '
/
expi i i i
1 2 3 4
1 2 3 4
y x y xi
J 21 2 3 4
Z Tr H Tr T
T e 1β σ σ σ σσ σ σ σ
β
δ σ σ σ σ− + + +
= − =
= −
∏ ∏
( ) /
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 SS SS SS S
H J 2S S S S S S S S 1
σσσσ
σ σ σ σσ σ σ σ
====
= − + + +
= =
S1 S2
S4 S3 σ4
σ3 σ2
σ1
Duality transformation
Levin, Nave, PRL 99 (2007) 120601
Step I: Rewiring
2
,
, ,1
kj il mji mlkm
kj n il nn
D D
n
M T T
U V=
→
Λ
=
=
∑
∑Singular value decomposition: SVD best scheme for truncating a matrix
Step II: decimation
D D2
Coarse Grain Tensor Renormalization Group (TRG)
Accuracy of TRG
Ising model on a triangular lattice
D = 24
TRG is a good method, but it can be further improved
Second renormalization of tensor-network state (SRG)
( )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
Λ
I. Poor Man’s SRG: entanglement mean-field approach
( )envZ=Tr MM / / / /,
env 1 2 1 2 1 2 1 2kl ij k l i jM Λ Λ Λ Λ≈
Mean field (or cavity) approximation
4, , ,
1...kj il kj n n il n
n D
M U V=
= Λ∑
Λ = Λ1/2 Λ1/2 From environment From system
Bond field – measures the entanglement between U and V
Λ
Accuracy of Poor Man’s SRG
Ising model on a triangular lattice
D = 24
Tc = 4/ln3
II. More accurate treatment of SRG
TRG Menv
Evaluate the environment contribution Menv using TRG
( 1) ( )' ' ' ' ' ' ' '
' ' ' '
n nijkl i j k l k jp j pi i lq l qk
i j k l pqM M S S S S− = ∑ ∑
( 1)nijklM −
( )' ' ' 'n
i j k lM
1. Forward iteration (0) (1)
( )N
M MM
→
→ →
2. Backward iteration ( ) ( 1)
(0)
N N
env
M MM M
−→
→ → =
Accuracy of SRG
Ising model on a triangular lattice
D = 24
• Efforts of Prof. Wen’s group:
Decompose a tensor into 3 parts
11 RG moves
Error ~ 1 %
Z. C. Gu, M. Levin, and X. G. Wen, unpublished
Extension to 3D is difficult
Corner Transfer Tensor Renormalization Group method
Nishino & Okunishi, JPSJ 67, 3066 (1998)
Variational wavefunction (transfer matrix) Okunishi, Nishino, Prog. Theor. Phys, 103, 541 (2000); Maeshima, Hieida, Akutsu, Nishino, Okunishi, PRE 64, 016705 (2001); Nishino, Hieida, Okunishi, Maeshima, Akutsu, Gendiar, Prog. Theo. Phys. 105, 409 (2001). Gendiar & Nishino, PRB 71, 024404 (2005)
Main problem: tensor dimension D = 2 ~ 5 error > 0.6 %
Efforts made by Nishino and collaborators
• Recent work of Garcia and Latorre
Dmax ~ 5 error ~ 2.7 %
A. Garcia-Saez, and J. I. Latorre, arXiv:1112.1412
TRG with Higher-Order Singular Value Decomposition of Tensors
Higher order singular value decompostion
D
D2 D
Z. Y. Xie et al, arXiv:1201.1144
Core tensor all-orthogonal: pseudo-diagonal / ordering:
Higher-Order Singular Value Decomposition(HOSVD)
L. de Latheauwer, B. de Moor, and J. Vandewalle, SIAM, J. Matrix Anal. Appl, 21, 1253 (2000).
Nearly optimal low-rank approximation
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
How to do the HOSVD
HOSVD can be achieved by successive SVD for each index of the tensor
For example
D = 24
Ising model on the square lattice
Accuracy of HOTRG
Hierarchical structure of HOTRG
Second Renormalization : HOSRG
Forward iterations: HOTRG to determine U(n) and T(n) backward iterations : evaluate the environment tensors
HOSRG: sweeping
Forward iterations:
Evaluate the bond density matrix
Find new U(n) and T(n)
D = 24
Ising model on the square lattice
Accuracy of HOSRG
D = 10
Ising model on the square lattice
HOSRG with forward-backward sweeping
Three dimensions
Benchmark result for the 3D Ising model
Critical exponent TRG: 0.3237 MC: 0.3262 HTSE: 0.3265
Tc = 4.511546
Relative Error
TRG < 10-6
other RG methods ∼ 10-2
3D Ising model
Thermodynamics of 2D QuantumTransverse Ising Model
T = 0K
σi = 1,…,4 8 neighbors 4 neighbors
Phase Transition with Partial Symmetry Breaking QN Chen et al, PRL 107, 165701 (2011)
The Potts model is a basic model of statistical physics
It has been intensively studied for more than 70 years
q=4 Potts Model on the UnionJack Lattice
Is there any phase transition?
Full versus Partial Symmetry Breaking
full symmetry breaking
Entropy = 0
partial symmetry breaking
Entropy is finite
random orientation
Ground States and Their Entropies
If red or green sublattice is ordered, the ground states are
ξ3N/4-fold degenerate S = (3N/4) ln ξ
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 ξ
The red or green sublattice is ordered
Entropy and Partial Order
Conjecture: There is a Finite T Phase Transition
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
Phase Transition: Specific Heat Jump
Partial Order Phase Transition in Other Irregular Lattices
Diced Lattice Centered Diced Lattice
Checkerboard Lattice
Summary
1. HOSRG provides an accurate and efficient numerical
method for studying 2D or 3D classical/quanutum lattice
models
2. AF Potts Model has an entropy driven partial ordering and
finite T phase transition on irregular lattices