Lattice and PT symmetries in tensor-network renormalization group: Case study of a hard-square lattice gas model

Abstract

The tensor-network renormalization group (TNRG) is an accurate numerical real-space renormalization group method for studying phase transitions in both quantum and classical systems. Continuous phase transitions, as an important class of phase transitions, are usually accompanied by spontaneous breaking of various symmetries. However, the understanding of symmetries in the TNRG is well-established mainly for global on-site symmetries like U(1) and SU(2). In this paper, we demonstrate how to incorporate lattice symmetries (including reflection and rotation) and the PT symmetry in the TNRG in two dimensions (2D) through a case study of the hard-square lattice gas with nearest-neighbor exclusion. This model is chosen because it is well-understood and has two continuous phase transitions whose spontaneously-broken symmetries are lattice and PT symmetries. Specifically, we write down proper definitions of these symmetries in a coarse-grained tensor network and propose a TNRG scheme that incorporates these symmetries. We demonstrate the validity of the proposed method by estimating the critical parameters and the scaling dimensions of the two phase transitions of the model. The technical development in this paper has made the 2D TNRG a more well-rounded numerical method.

21 pages, 9 figures, and 3 tables; open source code published on GitHub. For the revised version, typos are fixed and some details are clarified as per the referees' suggestion

References in corpus (38)

• Entanglement in quantum critical phenomena
• Entanglement entropy and conformal field theory
• 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 network states and algorithms in the presence of a global U(1) symmetry
• Tensor network decompositions in the presence of a global symmetry
• Loop optimization for tensor network renormalization
• Tensor network states and algorithms in the presence of a global SU(2) symmetry
• Renormalization of tensor networks using graph independent local truncations
• Algorithms for tensor network renormalization
• Developments in the Tensor Network -- from Statistical Mechanics to Quantum Entanglement
• Monte Carlo simulations of 2d hard core lattice gases
• Gauge fixing, canonical forms and optimal truncations in tensor networks with closed loops
• Renormalization group flows of Hamiltonians using tensor networks
• Columnar order and Ashkin-Teller criticality in mixtures of hard-squares and dimers
• High-Activity Perturbation Expansion for the Hard Square Lattice Gas
• Local scale transformations on the lattice with tensor network renormalization
• Categorical Tensor Network States
• Entanglement branching operator
• From PT-symmetric quantum mechanics to conformal field theory
• PT Symmetry in Classical and Quantum Statistical Mechanics
• Series expansions from the corner transfer matrix renormalization group method: the hard squares model
• Nuclear norm regularized loop optimization for tensor network
• Tensor RG approach to high-temperature fixed point
• Scaling dimensions from linearized tensor renormalization group transformations
• Comment on `Series expansions from the corner transfer matrix renormalization group method: the hard-squares model'
• Tensor Renormalization Group at Low Temperatures: Discontinuity Fixed Point
• Complex entanglement entropy for complex conformal field theory
• Rotations, Negative Eigenvalues, and Newton Method in Tensor Network Renormalization Group
• Integrability vs non-integrability: Hard hexagons and hard squares compared
• Three-dimensional real space renormalization group with well-controlled approximations
• Transfer Matrix and Lattice Dilatation Operator for High-Quality Fixed Points in Tensor Network Renormalization Group
• Tensor-Ring Decomposition with Index-Splitting
• 3D Tensor Renormalisation Group at High Temperatures
• Lattice-reflection symmetry in tensor-network renormalization group with entanglement filtering in two and three dimensions
• Essential difference between 2D and 3D from the perspective of real-space renormalization group