Connection between Free-Fermion and Interacting Crystalline Symmetry-Protected Topological Phases

TL;DR

This paper develops a framework connecting free-fermion and interacting crystalline SPT phases using spectral sequences, enabling systematic classification across various dimensions and symmetries.

Contribution

It introduces a novel approach employing the Atiyah-Hirzebruch spectral sequence to relate free-fermion and interacting crystalline SPT phases systematically.

Findings

Established a connection between free-fermion and interacting SPT phases.

Applied the framework to 1d, 2d, and 3d systems with various symmetries.

Demonstrated the method with explicit examples across different dimensions.

Abstract

We present a framework for investigating the effects of interactions on crystalline symmetry-protected topological (SPT) phases. Within this framework, one can establish a direct connection between the equivalence classes of free-fermion systems and their corresponding interacting classes. A central component of this framework is the Atiyah-Hirzebruch spectral sequence, which provides a systematic way to represent crystalline SPT phases as SPT phases with internal symmetries on subspaces. We demonstrate the application of this approach through examples in various dimensions: 1d systems with U(1) and reflection symmetry, 2d systems with U(1) and C_{n} rotation symmetry, and 3d systems with U(1) and inversion symmetry.