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Chiral symmetry on the edge of 2D symmetry protected topological phases

Xie Chen, Xiao-Gang Wen

TL;DR

The paper shows that 2D bosonic SPTs with $\mathbb{Z}_N$ and $U(1)$ symmetry host gapless 1D edges where the symmetry acts chirally on the low-energy modes, protecting gaplessness against symmetric perturbations. Using explicit 1D rotor models and their matrix-product representations, it connects edge dynamics to elements of $\mathcal{H}^3[G,U(1)]$, clarifying how non-onsite symmetries enforce robust gapless edges. For $U(1)$ SPTs, the edge exhibits an even-integer quantized Hall conductance despite a non-chiral edge, signaling a chiral response at the boundary. The work thus ties the group-cohomology classification to concrete edge realizations and demonstrates the completeness of edge constructions for these 2D bosonic SPT phases.

Abstract

Symmetry protected topological (SPT) states are short-range entangled states with symmetry, which have symmetry protected gapless edge states around a gapped bulk. Recently, we proposed a systematic construction of SPT phases in interacting bosonic systems, however it is not very clear what is the form of the low energy excitations on the gapless edge. In this paper, we answer this question for two dimensional bosonic SPT phases with Z_N and U(1) symmetry. We find that while the low energy modes of the gapless edges are non-chiral, symmetry acts on them in a chiral way, i.e. acts on the right movers and the left movers differently. This special realization of symmetry protects the gaplessness of the otherwise unstable edge states by prohibiting a direct scattering between the left and right movers. Moreover, understanding of the low energy effective theory leads to experimental predictions about the SPT phases. In particular, we find that all the 2D U(1) SPT phases have even integer quantized Hall conductance.

Chiral symmetry on the edge of 2D symmetry protected topological phases

TL;DR

The paper shows that 2D bosonic SPTs with and symmetry host gapless 1D edges where the symmetry acts chirally on the low-energy modes, protecting gaplessness against symmetric perturbations. Using explicit 1D rotor models and their matrix-product representations, it connects edge dynamics to elements of , clarifying how non-onsite symmetries enforce robust gapless edges. For SPTs, the edge exhibits an even-integer quantized Hall conductance despite a non-chiral edge, signaling a chiral response at the boundary. The work thus ties the group-cohomology classification to concrete edge realizations and demonstrates the completeness of edge constructions for these 2D bosonic SPT phases.

Abstract

Symmetry protected topological (SPT) states are short-range entangled states with symmetry, which have symmetry protected gapless edge states around a gapped bulk. Recently, we proposed a systematic construction of SPT phases in interacting bosonic systems, however it is not very clear what is the form of the low energy excitations on the gapless edge. In this paper, we answer this question for two dimensional bosonic SPT phases with Z_N and U(1) symmetry. We find that while the low energy modes of the gapless edges are non-chiral, symmetry acts on them in a chiral way, i.e. acts on the right movers and the left movers differently. This special realization of symmetry protects the gaplessness of the otherwise unstable edge states by prohibiting a direct scattering between the left and right movers. Moreover, understanding of the low energy effective theory leads to experimental predictions about the SPT phases. In particular, we find that all the 2D U(1) SPT phases have even integer quantized Hall conductance.

Paper Structure

This paper contains 4 sections, 27 equations, 3 figures.

Table of Contents

  1. Appendix A: The third group cohomology $\cH^3[G,U(1)]$ for symmetry $G$
  2. Appendix B: Matrix Product Operator Representation of Symmetry
  3. Appendix C: Cohomology class of symmetry operator $U^{(M)}_N$ in Eqn.(\ref{['UMN']})
  4. Appendix D: Interpretation in terms of fermionization

Figures (3)

  • Figure 1: Low energy states of $XY$ model $H_2'$(Eqn.(\ref{['H_Z2_XY']})). $x$-axis is lattice momentum $k/(\pi/a)$, where $a$ is the lattice spacing. $y$-axis is energy with ground state energy set to $0$ and first excited state energy normalized to $1/4$. $+$ represents positive $\mathbb{Z}_2$ quantum number and $\times$ represents negative $\mathbb{Z}_2$ quantum number. Total boson number $l$ and winding number $m$ are labeled as $(l,m)$ for each primary field, represented by the shaded $+$ or $\times$. States in the same conformal tower have the same $l$ and $m$.
  • Figure 2: Reduce combination of $T(g_2)$ and $T(g_1)$ into $T(g_1g_2)$.
  • Figure 3: Different ways to reduce combination of $T(g_3)$, $T(g_2)$ and $T(g_1)$ into $T(g_1g_2g_3)$. Only the right projection operators are shown. Their combined actions differ by a phase factor $\phi(g_1,g_2,g_3)$.