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

TL;DR

This paper extends tensor-network renormalization group methods to include lattice and PT symmetries, demonstrated through a case study of the hard-square lattice gas model with two continuous phase transitions.

Contribution

It introduces a scheme to incorporate lattice and PT symmetries into 2D TNRG, enhancing its capability to study symmetry-breaking phase transitions.

Findings

Successfully estimated critical parameters and scaling dimensions for the model's phase transitions.

Validated the incorporation of lattice and PT symmetries in TNRG through numerical results.

Enhanced the robustness of 2D TNRG as a numerical method for symmetry-related phase transitions.

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.