SPT indices emerging from translation invariance in two dimensional quantum spin systems

TL;DR

This paper explores how translation invariance in two-dimensional quantum spin systems with finite group symmetry leads to new symmetry-protected topological (SPT) indices, expanding the classification framework.

Contribution

It introduces additional SPT indices arising from translation invariance, extending the known classification of 2D SPT phases with finite group symmetry.

Findings

Emergence of an $H^2(G,\mathbb{T})$-valued index with one-direction translation invariance.

Additional $H^1(G,\mathbb{T})$-valued index when translation invariance is in two directions.

Extension of the SPT classification beyond the previously known $H^3(G,\mathbb{T})$ index.

Abstract

We consider SPT-phases with on-site $G$ (where $G$ is any finite group) symmetry for two-dimensional quantum spin systems. We then impose translation invariance in one direction and observe that on top of the $H^3(G,\mathbb{T})$-valued index constructed in \cite{ogata2021h3gmathbb}, an additional $H^2(G,\mathbb{T})$-valued index emerges. We also show that if we impose translation invariance in two directions, on top of the expected $H^3(G,\mathbb{T})\oplus H^2(G,\mathbb{T})\oplus H^2(G,\mathbb{T})$ valued index, an additional $H^1(G,\mathbb{T})$-valued index emerges.