i. coarse graining tensor renormalization...2016/07/12 · critical temperature of 3d ising model...
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Renormalization of Tensor Network States
Tao Xiang
Institute of Physics
Chinese Academy of Sciences
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