Renormalization of Tensor Network States I. Coarse Graining Tensor Renormalization Tao Xiang Institute of Physics Chinese Academy of Sciences txiang@iphy.ac.cn
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 = N total a i i i=1 N N total b k k k=1 To find a small and optimized set of basis states k to represent accurately a wave function refine the wavefunction by local RG transformations
Numerical Renormalization Group = N total a i i N N total b k k i=1 k=1 To find a small and optimized set of basis states k to represent accurately a wave function Physics: Mathematics: compression of basis space (phase space) or compression of information low rank approximation of matrix or tensor
Is Quantum Wave Function Compressible? N total = 2 L2 B A L The answer: Entanglement Entropy Area Law S L ln N N ~ 2 L << 2 L2 = N total L Ising model Minimum number of basis states needed for accurately representing a ground state
Tensor Network States 1. Faithful representation of the partition functions of all classical and quantum lattice models 2. Variational ansatz of the ground state wave function of quantum lattice models
Virtual Bond Dimension D: How Large Needed? Entanglement entropy S = L L ln D D ~ e (independent of L) PEPS is exact ground state wavefunction in the limit D Projected Entangled Pair State (PEPS) D Local tensor Virtual basis Physical basis
2D Interacting Fermions: How Large D Needed? Entanglement entropy S = L lnl L ln D D ~ L α D grows in some power law with the system size Projected Entangled Pair State (PEPS) D Local tensor Virtual basis Physical basis
Comparison between DMRG and Tensor RG Stoudenmire and White, Annu. Rev. CMP 3, 111(2012) PEPS S=1/2 AF Heisenberg model on infinite square lattice Reference energy: VMC extrapolation Sandvik PRB 56, 11678(1997)
Problems to be solved by tensor renormalization group Classical statistical model How to trace out all tensor indices? Quantum lattice model Approach I: Directly evaluate the (2+1) partition function Approach II: Find the ground state wavefunction (PEPS) Evaluate the physical observables
Tensor representations of classical statistical models H. H. Zhao, et al, PRB 81, 174411 (2010)
What Are Tensor Network States? 1. Faithful representation of the partition functions of all classical and quantum lattice models 2D quantum systems are equivalent to 3D classical T xi x i y i y i ones
Example: one dimensional Ising model H = S i S i+1 S 1 S 2 S 3 S N-1 S N i Z exp SiSi 1 S... S i S 1 1 N... S N Tr A A A... A A S S S S S S S S 1 2 2 3 N 1 N N 1 A A e e e e N max N 1D: partition function is a matrix product
Two-Dimensional Ising model H = ij S i S j Z = Tr exp H = Tr = Tr {S} exp H T Si S j S k S l S i S j = T Si S j S k S l = S i S j = exp H S l S k S l S k
Tensor-network representation is not unique H= -JSiS j ij Z Tr exp H ij Tr Ty x y ' x ' ij T U U U U 1 i i i i i 1 2 3 4 S1 1 S1 2 S1 3 S1 4 1 2 3 4 S Singular Value Decomposition 2 3 1 S 1 S 2 4 exp M exp H JS S S S ij i j i j M U U S S S S 1 2 1 1 1 2 1
Tensor-network representation in the dual lattice H= -JSiS j ij Z Tr exp H Tr T y x y ' J 1 2 3 4 / 2 T e 1 2 3 4 1 1 2 3 4 i x ' i i i i Duality transformation 1 S 1 S 2 3 2 S 4 S 3 4 SS 1 1 2 SS 2 2 3 SS 3 3 4 SS 4 4 1 / H J 2 1 2 3 4 S S S S S S S S 1 1 2 3 4 1 2 2 3 3 4 4 1
Gauge Invariance PP 1 T 1 T 2 T 1 T 1 P T 2 P 1 T 2 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 d 3 D 3 L d 2 D 3 TEBD d 3 D 3 L d 2 D 3 TRG D 6 lnl D 4 SRG D 6 lnl D 4 HOTRG D 7 lnl D 4 HOSRG D 8 lnl D 6 TNR D 7 lnl D 5 Loop-TNR D 6 lnl D 4 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
Coarse grain tensor renormalization group Levin, Nave, PRL 99 (2007) 120601 Step I: Rewiring Step II: decimation M kj, il mji mlk m D n1 T T U V kj, n n il, n Singular value decomposition
Singular value decomposition of matrix Singular value decomposition ij i, n n j, n n1 n1 N f U V D U V i, n n j, n Schmidt decomposition n n n n sys env n2 is the eigenvalue of reduced density matrix i sys System j env, Environment f i j i j ij sys env
Coarse grain tensor renormalization group Step II: decimation Txyz SxikS yjiszkj ijk
Accuracy of TRG D = 24 Ising model on a triangular lattice
Second Renormalization of Tensor Network Model (SRG) TRG: truncation error of M is minimized by the singular value decomposition Z=Tr MM env But, what really needs to be minimized is the error of Z! SRG: The renormalization effect of M env to M is considered Xie et al, PRL 103, 160601 (2009) Zhao, et al, PRB 81, 174411 (2010) system environment
Poor-Man SRG: Entanglement Mean Field Approximation env env 1/ 2 1/ 2 1/ 2 1/ 2 Z=Tr MM Mkl, ij k l i j Mean field (or cavity) approximation M U V kj, il kj, n n il, n 4 n1... D Bond field measures the entanglement between U and V = 1/2 1/2 From environment From system
Accuracy of Poor Man s SRG T c = 4/ln3 D = 24 Ising model on a triangular lattice
SRG Evaluate the environment contribution M env using TRG TRG M env
( n 1) M ijkl 1. Forward iteration M M (0) (1) M ( N ) ( n) M i ' j ' k ' l ' 2. Backward iteration M M ( N) ( N1) (0) M M env M M S S S S ( n1) ( n) ijkl i' j' k ' l ' k ' jp j ' pi i' lq l ' qk i' j' k ' l ' pq
Accuracy of SRG D = 24 Ising model on a triangular lattice
Coarse graining tensor renormalization by HOSVD HOSVD Higher-order singular value decomposition M (n) Lower-rank approximation D D 2 D Z. Y. Xie et al, PRB 86, 045139 (2012)
Coarse graining tensor renormalization by HOSVD Step 1: To contract two local tensors into one x = (x 1, x 2 ), x = (x 1, x 2 ) D D 2 D
Coarse graining tensor renormalization by HOSVD Step 2: determine the unitary transformation matrices by the HOSVD M (n) D D 2 D
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
Higher order singular value decomposition (HOSVD) Generalization of the singular value decomposition of matrix to tensor Core tensor all-orthogonal: pseudo-diagonal / ordering: Tucker decomposition L. de Latheauwer, B. de Moor, and J. Vandewalle, SIAM, J. Matrix Anal. Appl, 21, 1253 (2000).
Unitary Transformation Matrix Only horizontal bonds need to be cut if ε 1 < ε 2, U (n) = U L if ε 1 > ε 2, U (n) = U R truncation error = min(ε 1, ε 2 )
How to do HOSVD HOSVD can be achieved by successive SVD for each index of the tensor For example
Nishino Diagram of HOTRG
Second renormalization of tensor network states system environment Z=Tr MM env M env 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 M env to M is included
How to Determine the Environment Tensor? SRG: forward iteration + backward iteration Forward iterations: use TRG to determine U (n) and T (n) Backward iterations : evaluate the environment tensors
HOSRG: Bond Density Matrix
HOTRG at 3D (or 2+1D)
Higher order singular value decomposition 3D HOTRG
Computational Cost 2D 3D Memory CPU time Memory CPU time HOTRG D 4 D 7 D 6 D 11 HOSRG D 5 D 8 D 7 D 12
Magnetization of 3D Ising model Z. Y. Xie et al, PRB 86, 045139 (2012) HOTRG (D=14): 0.3295 Monte Carlo: 0.3262 Series Expansion: 0.3265 Relative difference is less than 10-5 MC data: A. L. Talapov, H. W. J. Blote, J. Phys. A: Math. Gen. 29, 5727 (1996).
Specific Heat of 3D Ising model D = 14 Solid line: Monte Carlo data from X. M. Feng, and H. W. J. Blote, Phys. Rev. E 81, 031103 (2010)
Critical Temperature of 3D Ising model Bond dimension
Critical Temperature of 3D Ising model method year T c 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 Quantum Ising model 2D QuantumTransverse Ising Model at T = 0K Z. Y. Xie et al, PRB 86, 045139 (2012)
Thermodynamics of the 2D Quantum Ising Model Internal Energy Magnetization
RG Flow of Local Tensors
Critical Behavior of Tensor Network Model fixing point ordered phase critical point disordered phase fixing point How does the tensor change with the RG steps?
Fix Point Tensor 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
RG Flow of the Tensors 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. T 1111 = 1 T 2222 = 1 T 1111 = 1 fixing point ordered phase critical point disordered phase fixing point
Central Charge at the Critical Point 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 n are eigenvalues of c = 6 ln max π M ud = r T r,r,u,d
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)
Potts model i = 1,,q Antiferromagnetic: J > 0 q < q c q = q c q > q c 1st/2nd phase transition at finite temperature critical at 0K no phase transition
Critical q for the antiferromagnetic Potts model Can q c > 4 in certain lattices? Lattice Coordination number q c honeycomb 3 <3 square 4 3 diced 4 3<q c <4 kagome 4 3 triangular 6 4 union-jack 6? centered diced 6?
Phase Transition with Partial Symmetry Breaking q=4 Potts Model on the UnionJack Lattice i = 1,,4 8 neighbors 4 neighbors Is there any phase transition?
Full versus partial symmetry breaking random orientation full symmetry breaking Entropy = 0 partial symmetry breaking Entropy is finite
Ground states and their entropies S = (N/2) ln 2 + 2 * (3N/4) ln 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 2 N/2 -fold degenerate: S = (N/2) ln 2
The red or green sublattice is ordered Entropy and Partial Order
Conjecture: there is a finite temperature phase transition There is a partial symmetry breaking at 0K q = 4 Potts model There is a finite T phase transition with two singularities: 1. ordered and disordered states 2. Z 2 between green and red
Phase Transition: Specific Heat Jump
Green or Red Sub-lattice Magnetization 1/16 q = 4 Potts model on the Union-Jack lattice
Partial order phase transition in other irregular lattices Checkerboard Lattice Centered Diced Lattice Diced Lattice
Critical q for the antiferromagnetic Potts model Lattice Coordination number q c honeycomb 3 <3 square 4 3 diced 4 3<q c <4 kagome 4 3 triangular 6 4 union-jack 6 >4 centered diced 6 >4
Summary 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