Variational corner transfer matrix renormalization group method for classical statistical models

Abstract

In the context of tensor network states, we for the first time reformulate the corner transfer matrix renormalization group (CTMRG) method into a variational bilevel optimization algorithm. The solution of the optimization problem corresponds to the fixed-point environment pursued in the conventional CTMRG method, from which the partition function of a classical statistical model, represented by an infinite tensor network, can be efficiently evaluated. The validity of this variational idea is demonstrated by the high-precision calculation of the residual entropy of the dimer model, and is further verified by investigating several typical phase transitions in classical spin models, where the obtained critical points and critical exponents all agree with the best known results in literature. Its extension to three-dimensional tensor networks or quantum lattice models is straightforward, as also discussed briefly.

arXiv comment7 pages, 7 figures

References in corpus (35)

• Efficient classical simulation of slightly entangled quantum computations
• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• Tensor renormalization group approach to 2D classical lattice models
• Tensor networks for complex quantum systems
• Competing states in the t-J model: uniform d-wave state versus stripe state
• The iTEBD algorithm beyond unitary evolution
• Corner Transfer Matrix Renormalization Group Method
• Solutions of the Two Dimensional Hubbard Model: Benchmarks and Results from a Wide Range of Numerical Algorithms
• Coarse-graining renormalization by higher-order singular value decomposition
• Simulation of two dimensional quantum systems on an infinite lattice revisited: corner transfer matrix for tensor contraction
• Gapless spin-liquid ground state in the $S=1/2$ kagome antiferromagnet
• Tensor Network Renormalization
• Variational optimization algorithms for uniform matrix product states
• Second Renormalization of Tensor-Network States
• Renormalization of tensor-network states
• Lecture Notes of Tensor Network Contractions
• Spin-orbital quantum liquid on the honeycomb lattice
• Loop optimization for tensor network renormalization
• Gapped spin liquid with ${\mathbb{Z}}_2$-topological order for kagome Heisenberg model
• Faster Methods for Contracting Infinite 2D Tensor Networks
• Exploring corner transfer matrices and corner tensors for the classical simulation of quantum lattice systems
• A Density Matrix Algorithm for 3D Classical Models
• A tensor product state approach to spin-1/2 square $J_1$-$J_2$ antiferromagnetic Heisenberg model: evidence for deconfined quantum criticality
• Tensor renormalization group study of classical XY model on the square lattice
• Probability-Changing Cluster Algorithm for Two-Dimensional XY and Clock Models
• Critical properties of the two-dimensional $q$-state clock model
• Renormalization group flows of Hamiltonians using tensor networks
• Residual entropies for three-dimensional frustrated spin systems with tensor networks
• Optimized contraction scheme for tensor-network states
• Phase transition of the q-state clock model: duality and tensor renormalization
• Numerical study of the phase transitions in the two-dimensional Z(5) vector model
• Infinite Projected Entangled-Pair State algorithm for ruby and triangle-honeycomb lattices
• DMRG study of the Berezinskii-Kosterlitz-Thouless transitions of the 2D five-state clock model
• Phase transitions of the five-state clock model on the square lattice
• Cross derivative: a universal and efficient method for phase transitions in classical spin models

Cited by in corpus (11)

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• Corner transfer matrix renormalization group approach in the zoo of Archimedean lattices
• Tensor network Monte Carlo simulations for the two-dimensional random-bond Ising model
• Efficient calculation of three-dimensional tensor networks
• Exploring quantum phase transitions by the cross derivative of the ground state energy
• Signatures of quantum criticality in the complex inverse temperature plane
• The Green's function Monte Carlo combined with projected entangled pair state approach to the frustrated $J_1$-$J_2$ Heisenberg model
• Efficient optimization of variational tensor-network approach to three-dimensional statistical systems
• Single-layer framework of variational tensor network states
• Phase diagram of the single-flavor Gross--Neveu--Wilson model from the Grassmann corner transfer matrix renormalization group
• Grassmann tensor networks