Renormalization group flows of Hamiltonians using tensor networks

Abstract

A renormalization group flow of Hamiltonians for two-dimensional classical partition functions is constructed using tensor networks. Similar to tensor network renormalization ([G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015)], [S. Yang, Z.-C. Gu, and X.-G Wen, Phys. Rev. Lett. 118, 110504 (2017)]) we obtain approximate fixed point tensor networks at criticality. Our formalism however preserves positivity of the tensors at every step and hence yields an interpretation in terms of Hamiltonian flows. We emphasize that the key difference between tensor network approaches and Kadanoff's spin blocking method can be understood in terms of a change of local basis at every decimation step, a property which is crucial to overcome the area law of mutual information. We derive algebraic relations for fixed point tensors, calculate critical exponents, and benchmark our method on the Ising model and the six-vertex model.

arXiv commentaccepted version for Phys. Rev. Lett, main text: 5 pages, 3 figures, appendices: 9 pages, 1 figure

References in corpus (19)

• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• Tensor renormalization group approach to 2D classical lattice models
• Coarse-graining renormalization by higher-order singular value decomposition
• Tensor Network Renormalization
• Tensor-entanglement renormalization group approach to 2D quantum systems
• String order and symmetries in quantum spin lattices
• Second Renormalization of Tensor-Network States
• Topological Defects on the Lattice I: The Ising model
• Renormalization of tensor-network states
• Loop optimization for tensor network renormalization
• Anyons and matrix product operator algebras
• Tensor network renormalization yields the multi-scale entanglement renormalization ansatz
• Diagonalizing transfer matrices and matrix product operators: a medley of exact and computational methods
• Real Space Renormalization in Statistical Mechanics
• Sparse and Unique Nonnegative Matrix Factorization Through Data Preprocessing
• Algorithms for tensor network renormalization
• Matrix Product Density Operators: Renormalization Fixed Points and Boundary Theories
• Topological conformal defects with tensor networks
• Local scale transformations on the lattice with tensor network renormalization

Cited by in corpus (62)

• Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
• Tensor networks for complex quantum systems
• Hyper-optimized tensor network contraction
• Neural Network Renormalization Group
• Faster Methods for Contracting Infinite 2D Tensor Networks
• Renormalization of tensor networks using graph independent local truncations
• Anisotropic Tensor Renormalization Group
• Tensor lattice field theory with applications to the renormalization group and quantum computing
• Mapping topological to conformal field theories through strange correlators
• Gauge fixing, canonical forms and optimal truncations in tensor networks with closed loops
• Berezinskii-Kosterlitz-Thouless transition in lattice Schwinger model with one flavor of Wilson fermion
• Tensor network and ($p$-adic) AdS/CFT
• Fast counting with tensor networks
• Bond-weighted Tensor Renormalization Group
• Automatic Differentiation for Second Renormalization of Tensor Networks
• Anomalies and entanglement renormalization
• Boundary Tensor Renormalization Group
• Finite-size and finite bond dimension effects of tensor network renormalization
• Entanglement branching operator
• Computing the renormalization group flow of two-dimensional ${\varphi }^4$ theory with tensor networks
• Hyper-optimized approximate contraction of tensor networks with arbitrary geometry
• Resolving the Berezinskii-Kosterlitz-Thouless transition in the 2D XY model with tensor-network based level spectroscopy
• Efficient variational contraction of two-dimensional tensor networks with a non-trivial unit cell
• Examples of symmetry-preserving truncations in tensor field theory
• Nuclear norm regularized loop optimization for tensor network
• Discrete aspects of continuous symmetries in the tensorial formulation of Abelian gauge theories
• Tensor Renormalization Group Algorithms with a Projective Truncation Method
• Boundary conformal spectrum and surface critical behaviors of the classical spin systems: a tensor network renormalization study
• Scaling dimensions from linearized tensor renormalization group transformations
• Tensor RG approach to high-temperature fixed point
• Continuous tensor network renormalization for quantum fields
• Variational corner transfer matrix renormalization group method for classical statistical models
• Network Renormalization
• Triad second renormalization group
• Automatic Contraction of Unstructured Tensor Networks
• Matrix product state fixed points of non-Hermitian transfer matrices
• Tensor network calculation of the logarithmic correction exponent in the XY model
• Tensor Renormalization Group Centered About a Core Tensor
• Efficient Decomposition of High-Rank Tensors
• Tensor Network Renormalization Study on the Crossover in Classical Heisenberg and $\mathrm{R}{\mathrm{P}}^2$ Models in Two Dimensions
• Tensor network method for reversible classical computation
• Phase boundary location with information-theoretic entropy in tensor renormalization group flows
• The Kibble Zurek Mechanism of Topological Defect Formation in Quantum Field Theory with Matrix Product States
• Three-dimensional real space renormalization group with well-controlled approximations
• Rotations, Negative Eigenvalues, and Newton Method in Tensor Network Renormalization Group
• Entanglement Entropy of the Two-Dimensional +-J Ising Model on the Nishimori Line
• Transfer Matrix and Lattice Dilatation Operator for High-Quality Fixed Points in Tensor Network Renormalization Group
• Tensor network simulations for nonorientable surfaces
• Finite-size scaling on the torus with periodic projected entangled-pair states
• Tensor Renormalization Group for interacting quantum fields
• p-adic CFT is a holographic tensor network
• The Tangent Space to the Manifold of Critical Classical Hamiltonians Representable by Tensor Networks
• Virasoro Generators in the Fibonacci Model Tensor Network -- Tackling Finite Size Effects
• Essential difference between 2D and 3D from the perspective of real-space renormalization group
• Tensor network renormalization approach to antiferromagnetic 6-state clock model on the Union Jack lattice
• Triad representation for the anisotropic tensor renormalization group in four dimensions
• A Practical Introduction to Tensor Network Renormalization with TNRKit.jl
• Implicitly disentangled renormalization
• Variational boundary based tensor network renormalization group
• Lattice and PT symmetries in tensor-network renormalization group: Case study of a hard-square lattice gas model
• Minimizing couplings in renormalization by preserving short-range mutual information
• Noise-tolerant Detection of $Z_N$ Topological Orders in Quantum Many-body States