Supersymmetric Free Fermions and Bosons: Locality, Symmetry and Topology

TL;DR

This paper explores how supersymmetry relates fermionic and bosonic topological phases, revealing new classifications and resolving paradoxes in free particle systems through a systematic theoretical framework.

Contribution

It introduces a classification of topological insulators and superconductors based on supersymmetry properties and clarifies the encoding of topological information in free bosonic and fermionic systems.

Findings

Topological phases are classified into three supersymmetry-based classes.

Topological information is encoded in the identification map, resolving fermion-boson paradox.

Supersymmetric entanglement duality provides new insights into band topology.

Abstract

Supersymmetry, originally proposed in particle physics, refers to a dual relation that connects fermionic and bosonic degrees of freedom in a system. Recently, there has been considerable interest in applying the idea of supersymmetry to topological phases, motivated by the attempt to gain insights from the fermion side into the boson side and vice versa. We present a systematic study of this construction when applied to band topology in noninteracting systems. First, on top of the conventional ten-fold way, we find that topological insulators and superconductors are divided into three classes depending on whether the supercharge can be local and symmetric, must break a symmetry to preserve locality, or needs to break locality. Second, we resolve the apparent paradox between the nontriviality of free fermions and the triviality of free bosons by noting that the topological information is encoded in the identification map. We also discuss how to understand a recently revealed supersymmetric entanglement duality in this context. These findings are illustrated by prototypical examples. Our work sheds new light on band topology from the perspective of supersymmetry.