Vertical Density Matrix Algorithm: A Higher-Dimensional Numerical Renormalization Scheme based on the Tensor Product State Ansatz

Abstract

We present a new algorithm to calculate the thermodynamic quantities of three-dimensional (3D) classical statistical systems, based on the ideas of the tensor product state and the density matrix renormalization group. We represent the maximum-eigenvalue eigenstate of the transfer matrix as the product of local tensors which are iteratively optimized by the use of the ``vertical density matrix'' formed by cutting the system along the transfer direction. This algorithm, which we call vertical density matrix algorithm (VDMA), is successfully applied to the 3D Ising model. Using the Suzuki-Trotter transformation, we can also apply the VDMA to two-dimensional (2D) quantum systems, which we demonstrate for the 2D transverse field Ising model.

Unnecessary files are removed. 8 pages, 7 figures, submitted to Phys.Rev.E

Cited by in corpus (59)

• The density-matrix renormalization group
• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
• Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
• Classical simulation of infinite-size quantum lattice systems in two spatial dimensions
• The iTEBD algorithm beyond unitary evolution
• Coarse-graining renormalization by higher-order singular value decomposition
• Renormalization and tensor product states in spin chains and lattices
• New Trends in Density Matrix Renormalization
• PEPS as ground states: degeneracy and topology
• Simulation of two-dimensional quantum systems using a tree tensor network that exploits the entropic area law
• Fermionic Projected Entangled Pair States
• Loop optimization for tensor network renormalization
• Isometric Tensor Network States in Two Dimensions
• Implementing global Abelian symmetries in projected entangled-pair state algorithms
• Faster Methods for Contracting Infinite 2D Tensor Networks
• Exploring corner transfer matrices and corner tensors for the classical simulation of quantum lattice systems
• Entanglement renormalization and gauge symmetry
• Decorated tensor network renormalization for lattice gauge theories and spin foam models
• Monte Carlo simulation with Tensor Network States
• Infinite boundary conditions for matrix product state calculations
• First order phase transition in the anisotropic quantum orbital compass model
• Developments in the Tensor Network -- from Statistical Mechanics to Quantum Entanglement
• Efficient variational diagonalization of fully many-body localized Hamiltonians
• A canonical form for Projected Entangled Pair States and applications
• Optimized contraction scheme for tensor-network states
• Automatic differentiation applied to excitations with Projected Entangled Pair States
• Renormalization group contraction of tensor networks in three dimensions
• Simulation of three-dimensional quantum systems with projected entangled-pair states
• Variational treatment of the Shastry-Sutherland antiferromagnet using Projected Entangled Pair States (PEPS)
• Application of the density matrix renormalization group method to finite temperatures and two-dimensional systems
• Assessing the accuracy of projected entangled-pair states on infinite lattices
• Gauging tensor networks with belief propagation
• Concatenated tensor network states
• The area law and real-space renormalization
• Tensor-network study of correlation-spreading dynamics in the two-dimensional Bose-Hubbard model
• Numerical Study of Spin-1/2 XXZ Model on Square Lattice from Tensor Product States
• Stripe Ansatzs from Exactly Solved Models
• Sudden expansion and domain-wall melting of strongly interacting bosons in two-dimensional optical lattices and on multileg ladders
• Quantum phase transition between symmetry enriched topological phases in tensor-network states
• Density of Yang-Lee zeros in the thermodynamic limit from tensor network methods
• Entanglement Properties of Gauge Theories from Higher-Form Symmetries
• Construction of a matrix product stationary state from solutions of finite size system
• Dynamics of correlation spreading in low-dimensional transverse-field Ising models
• Efficient tensor network algorithm for layered systems
• Monte Carlo simulations of two-dimensional fermion systems with string-bond states
• Matrix product decomposition for two- and three-flavor Wilson fermions: Benchmark results in the lattice Gross-Neveu model at finite density
• Tensor Product State Formulation for the Spin 1/2 Antiferromagnetic XXZ Model on the Checkerboard Lattice
• The Stochastic State Selection Method Combined with the Lanczos Approach to Eigenvalues in Quantum Spin Systems
• Simulation of Anyons Using Symmetric Tensor Network Algorithms
• A constrained stochastic state selection method applied to quantum spin systems
• Entanglement Perturbation Theory for Antiferromagnetic Heisenberg Spin Chains
• Efficient optimization of variational tensor-network approach to three-dimensional statistical systems
• Diagonal Isometric Form for Tensor Network States in Two Dimensions
• Optimized Implementation for Calculation and Fast-Update of Pfaffians Installed to the Open-Source Fermionic Variational Solver mVMC
• Scalable projected entangled-pair state representation of random quantum circuit states
• Measurement-based quantum computation using two-component BECs
• Finite Correlation Length Scaling of Disorder Parameter at Quantum Criticality
• Ground-state phase diagram of the SU($4$) Heisenberg model on a plaquette lattice
• Studies of braided non-Abelian anyons using anyonic tensor networks