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

Tijl Jappens

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.

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

Abstract

We consider SPT-phases with on-site (where 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 -valued index constructed in \cite{ogata2021h3gmathbb}, an additional -valued index emerges. We also show that if we impose translation invariance in two directions, on top of the expected valued index, an additional -valued index emerges.
Paper Structure (8 sections, 22 equations, 1 figure)

This paper contains 8 sections, 22 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setup and definitions
  3. Quasi local -algebra
  4. States on
  5. Interactions and locally generated automorphisms
  6. The stable equivalence class
  7. Results
  8. Statement of the result in terms of Q-automorphisms

Theorems & Definitions (8)

  • Conjecture 1.1
  • Conjecture 1.2
  • Conjecture 1.3
  • Remark 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1