One-dimensional symmetric phases protected by frieze symmetries

TL;DR

This paper systematically classifies one-dimensional symmetry-protected topological phases in quantum spin chains with frieze symmetries, using matrix product states and group cohomology, identifying 17 distinct phases and constructing explicit fixed-point wave functions.

Contribution

It provides a comprehensive classification of 1D SPT phases with frieze symmetries, introduces canonical forms, and links topological indices with group cohomology calculations.

Findings

Identified 17 distinct non-trivial phases.

Compared topological indices with group cohomology predictions.

Constructed explicit RG fixed-point wave functions.

Abstract

We make a systematic study of symmetry-protected topological gapped phases of quantum spin chains in the presence of the frieze space groups in one dimension using matrix product states. Here, the spatial symmetries of the one-dimensional lattice are considered together with an additional 'vertical reflection', which we take to be an on-site $\mathbb{Z}_2$ symmetry. We identify seventeen distinct non-trivial phases, define canonical forms, and compare the topological indices obtained from the MPS analysis with the group cohomological predictions. We furthermore construct explicit renormalization group fixed-point wave functions for symmetry-protected topological phases with global on-site symmetries, possibly combined with time-reversal and parity symmetry. En route, we demonstrate how group cohomology can be computed using the Smith normal form.