Tensor-network renormalization group (TNRG) is an efficient real-space renormalization group method for studying the criticality in both classical and quantum lattice systems. Exploiting symmetries of a system in a TNRG algorithm can simplify the implementation of the algorithm and can help produce correct tensor RG flows. Although a general framework for considering a global on-site symmetry has been established, it is still unclear how to incorporate a lattice symmetry in TNRG. As a first step for lattice symmetries, we propose a method to incorporate the lattice-reflection symmetry in the context of a TNRG with entanglement filtering in both two and three dimensions (2D and 3D). To achieve this, we write down a general definition of lattice-reflection symmetry in tensor-network language. Then, we introduce a transposition trick for exploiting and imposing the lattice-reflection symmetry in two basic TNRG operations: projective truncations and entanglement filtering. Using the transposition trick, the detailed algorithms of the TNRG map in both 2D and 3D are laid out, where the lattice-reflection symmetry is preserved and imposed. Finally, we demonstrate how to construct the linearization of the TNRG maps in a given lattice-reflection sector, with the help of which it becomes possible to extract scaling dimensions in each sector separately. Our work paves the way for understanding the lattice-rotation symmetry in TNRG.Tensor-network renormalization group (TNRG) is an efficient real-space renormalization group method for studying the criticality in both classical and quantum lattice systems. Exploiting symmetries of a system in a TNRG algorithm can simplify the implementation of the algorithm and can help produce correct tensor RG flows. Although a general framework for considering a global on-site symmetry has been established, it is still unclear how to incorporate a lattice symmetry in TNRG. As a first step for lattice symmetries, we propose a method to incorporate the lattice-reflection symmetry in the context of a TNRG with entanglement filtering in both two and three dimensions (2D and 3D). To achieve this, we write down a general definition of lattice-reflection symmetry in tensor-network language. Then, we introduce a transposition trick for exploiting and imposing the lattice-reflection symmetry in two basic TNRG operations: projective truncations and entanglement filtering. Using the transposition trick, the detailed algorithms of the TNRG map in both 2D and 3D are laid out, where the lattice-reflection symmetry is preserved and imposed. Finally, we demonstrate how to construct the linearization of the TNRG maps in a given lattice-reflection sector, with the help of which it becomes possible to extract scaling dimensions in each sector separately. Our work paves the way for understanding the lattice-rotation symmetry in TNRG.

arXiv:2510.19428Physical review. B./Physical review. BMar 6, 20262 citations

Mathematical properties of renormalization-group transformations

Robert B. Griffiths

Physica A Statistical Mechanics and its ApplicationsMar 1, 198140 citations

Properties of renormalization group transformations

Peter Kunszt

arXiv:hep-lat/9706019Nuclear Physics BApr 1, 19984 citations

Some symmetry properties of renormalization-group transformations

Marko V. Jarić

For a Landau-Ginzburg-Wilson Hamiltonian of any given symmetry we show how one can find a group ${G}_{T}$ of orthogonal transformations in parameter space, which commute with renormalization-group transformations. Then a renormalization-group transformation may be expanded into covariants of ${G}_{T}$ . We also present a systematic procedure for finding fixed points; they are most likely to decouple the Hamiltonian or to increase its symmetry. The merit of the conclusions obtained is illustrated using an example of a system with ${C}_{4}$ symmetry. Agreement with the results of $\ensuremath{\epsilon}$-expansion calculations has been found.For a Landau-Ginzburg-Wilson Hamiltonian of any given symmetry we show how one can find a group ${G}_{T}$ of orthogonal transformations in parameter space, which commute with renormalization-group transformations. Then a renormalization-group transformation may be expanded into covariants of ${G}_{T}$. We also present a systematic procedure for finding fixed points; they are most likely to decouple the Hamiltonian or to increase its symmetry. The merit of the conclusions obtained is illustrated using an example of a system with ${C}_{4}$ symmetry. Agreement with the results of $\ensuremath{\epsilon}$-expansion calculations has been found.

Physical review. B, Condensed matterSep 1, 197814 citations

For a Landau-Ginzburg-Wilson Hamiltonian of any given symmetry we show how one can find a group

{G}_{T}

of orthogonal transformations in parameter space, which commute with renormalization-group transformations. Then a renormalization-group transformation may be expanded into covariants of

{G}_{T}

. We also present a systematic procedure for finding fixed points; they are most likely to decouple the Hamiltonian or to increase its symmetry. The merit of the conclusions obtained is illustrated using an example of a system with

{C}_{4}

symmetry. Agreement with the results of $\ensuremath{\epsilon}$-expansion calculations has been found.