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 $U\left(1\right)$ and $SU\left(2\right)$. In this paper, we demonstrate how to incorporate lattice symmetries (including reflection and rotation) and $\mathcal{PT}$ 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 $\mathcal{PT}$ symmetries. Specifically, we write 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.

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• Critical phenomena
• Phase transitions
• Renormalization group
• Tensor network methods
• Tensor network renormalization