Equivariant Parameter Families of Spin Chains: A Discrete MPS Formulation

TL;DR

This paper develops a framework for analyzing topological phase transitions in 1D quantum spin systems by incorporating symmetry actions into a discretized parameter space, leading to new invariants and insights into phase transition points.

Contribution

It introduces a $G$-compatible discretization method for parameter spaces, enabling systematic construction of equivariant topological invariants and a fixed-point formula for higher Berry curvature.

Findings

Phase transition point acts as a monopole-like defect.

Higher Berry curvature emanates from phase transition points.

Hierarchical topological defect structures are governed by symmetry reductions.

Abstract

We analyze topological phase transitions and higher Berry curvature in one-dimensional quantum spin systems, using a framework that explicitly incorporates the symmetry group action on the parameter space. Based on a $G$-compatible discretization of the parameter space, we incorporate both group cochains and parameter-space differentials, enabling the systematic construction of equivariant topological invariants. We derive a fixed-point formula for the higher Berry invariant in the case where the symmetry action has isolated fixed points. This reveals that the phase transition point between Haldane and trivial phases acts as a monopole-like defect where higher Berry curvature emanates. We further discuss hierarchical structures of topological defects in the parameter space, governed by symmetry reductions and compatibility with subgroup structures.