Duality and topology

TL;DR

This paper explores how dualities and mappings between different quantum models, especially involving Majorana fermions, can alter topological invariants, highlighting the impact of unitary transformations on topological properties.

Contribution

It systematically analyzes the effect of dualities on topological invariants using Majorana fermion representations in spin and fermionic systems.

Findings

Unitary transformations can change topological invariants.

Mappings between models can alter topological properties.

Majorana fermion representation aids in analyzing dualities.

Abstract

Mappings between models may be obtained by unitary transformations with preservation of the spectra but in general a change in the states. Non- canonical transformations in general also change the statistics of the operators involved. In these cases one may expect a change of topological properties as a consequence of the mapping. Here we consider some dualities resulting from mappings, by systematically using a Majorana fermion representation of spin and fermionic problems. We focus on the change of topological invariants that results from unitary transformations taking as examples the mapping between a spin system and a topological superconductor, and between different fermionic systems.