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Lattice-reflection symmetry in tensor-network renormalization group with entanglement filtering in two and three dimensions

Xinliang Lyu, Naoki Kawashima

TL;DR

This work develops a general framework to incorporate lattice-reflection symmetry into entanglement-filtered tensor-network renormalization group (TNRG) in $2$D and $3$D. It introduces a transposition trick and SWAP-gauge construction to preserve symmetry through projective truncations and graph-independent EF, enabling sector-resolved linearization of the RG map to extract scaling dimensions. The authors present 2D and 3D EF-enhanced TNRG algorithms that preserve lattice-reflection symmetry, including substantial reductions in EF parameter count in 3D (from 24 to 3) and a formal treatment of symmetry-aware isometries and filtering matrices. Numerical demonstrations on the Ising model in both dimensions show partial success in resolving scaling dimensions by symmetry sectors and highlight challenges for higher-descendant operators, while validating the sectorized RG framework. The framework paves the way for studying lattice-rotation symmetry in TNRG and offers practical tools for symmetry-preserving real-space RG in higher dimensions with entanglement filtering.

Abstract

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 like rotation or reflection 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 operators: 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.

Lattice-reflection symmetry in tensor-network renormalization group with entanglement filtering in two and three dimensions

TL;DR

This work develops a general framework to incorporate lattice-reflection symmetry into entanglement-filtered tensor-network renormalization group (TNRG) in D and D. It introduces a transposition trick and SWAP-gauge construction to preserve symmetry through projective truncations and graph-independent EF, enabling sector-resolved linearization of the RG map to extract scaling dimensions. The authors present 2D and 3D EF-enhanced TNRG algorithms that preserve lattice-reflection symmetry, including substantial reductions in EF parameter count in 3D (from 24 to 3) and a formal treatment of symmetry-aware isometries and filtering matrices. Numerical demonstrations on the Ising model in both dimensions show partial success in resolving scaling dimensions by symmetry sectors and highlight challenges for higher-descendant operators, while validating the sectorized RG framework. The framework paves the way for studying lattice-rotation symmetry in TNRG and offers practical tools for symmetry-preserving real-space RG in higher dimensions with entanglement filtering.

Abstract

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 like rotation or reflection 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 operators: 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.
Paper Structure (42 sections, 11 theorems, 173 equations, 5 figures, 4 tables)

This paper contains 42 sections, 11 theorems, 173 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Projective truncations
  3. 2-to-1 isometric tensors
  4. Block-tensor transformation using projective truncations
  5. HOTRG-like block-tensor transformation
  6. Graph-independent entanglement filtering
  7. Formulation of the entanglement filtering
  8. Assembly of entanglement filtering and block-tensor map
  9. Exploiting the lattice-reflection symmetry
  10. Definition of the reflection symmetry
  11. An 1D toy example
  12. In 2D
  13. In 3D
  14. Origin of the SWAP-gauge matrix
  15. Transposition trick
  16. ...and 27 more sections

Key Result

Theorem 1

When $A_*$ is symmetric, the linearization $\mathcal{R}\bigr|_{A_*}$ in Eq. eq:linearbkten1D has two invariant subspaces: $\mathcal{M}^{(0)}_{nn}(\mathbb{R})$ and $\mathcal{M}^{(1)}_{nn}(\mathbb{R})$.

Figures (5)

  • Figure 1: Redundant entanglement structures in 3D
  • Figure 2: Location of redundant entanglement for a simple block-tensor transformation and the principle for incorporating the EF process into a block-tensor map. The tensors marked with a block solid dot are the anchor points with their position labeled as $(++)$ in Eqs. \ref{['eq:tsptrick2D']} and \ref{['eq:2dEFapproxSym']}.
  • Figure 3: The principle for incorporating the EF process into a block-tensor map in 3D.
  • Figure 4: Scaling dimensions of the 2D Ising model organized by the spin-flip $\mathbb{Z}_2$ and the lattice-reflection symmetry sectors. Bond dimensions in the 2D TNRG are $\chi=36, \chi_s = 10$.
  • Figure 5: Scaling dimensions of the 3D Ising model organized by the spin-flip $\mathbb{Z}_2$ and the lattice-reflection symmetry sectors. The bond dimensions in the 3D TNRG are $\chi =6, \chi_s = \chi_m=4$.

Theorems & Definitions (25)

  • Theorem 1
  • proof
  • Corollary 1.1
  • proof
  • Corollary 1.2
  • Theorem 2
  • proof
  • Remark
  • Remark
  • Remark
  • ...and 15 more