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This paper advances the tensor-network renormalization group by formally incorporating lattice and PT symmetries, significantly improving numerical accuracy for studying phase transitions in 2D systems. By extending TNRG beyond standard global on-site symmetries to include reflection, rotation, and parity-time constraints, it provides a more robust framework for analyzing critical phenomena in both classical and quantum models. The release of open-source code alongside these theoretical developments offers immediate utility for researchers working at the intersection of statistical mechanics and computational physics.
Lattice and PT symmetries in tensor-network renormalization group: Case study of a hard-square lattice gas model
arXiv:2603.25492v2 Announce Type: replace-cross 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.
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
Statistical Mechanics (cond-mat.stat-mech); High Energy Physics - Theory (hep-th); Computational Physics (physics.comp-ph); Quantum Physics (quant-ph)
arXiv:2603.25492 [cond-mat.stat-mech]
(or arXiv:2603.25492v2 [cond-mat.stat-mech] for this version)
Phys. Rev. B 114, 034111 (2026)
https://doi.org/10.1103/vl23-xsyr DOI(s) linking to related resources
Full text under the publisher's open licence · 249 words
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