Equivalence of Topological Insulators and Superconductors

TL;DR

This paper demonstrates that topological insulators and superconductors can be mapped onto each other through Gaussian dualities, revealing deep equivalences and symmetry transmutations that preserve topological features and the bulk-boundary correspondence.

Contribution

It introduces a broad class of Gaussian dualities that establish equivalences between insulators and superconductors, including their edge modes and symmetries, extending to interacting systems.

Findings

Any insulator is dual to a superconductor.

Fermionic edge modes are dual to Majorana edge modes.

Graphene can be dual to a p-wave superconductor.

Abstract

Systems of free fermions are classified by symmetry, space dimensionality, and topological properties described by K-homology. Those systems belonging to different classes are inequivalent. In contrast, we show that by taking a many-body/Fock space viewpoint it becomes possible to establish equivalences of topological insulators and superconductors in terms of duality transformations. These mappings connect topologically inequivalent systems of fermions, jumping across entries in existent classification tables, because of the phenomenon of symmetry transmutation by which a symmetry and its dual partner have identical algebraic properties but very different physical interpretations. To constrain our study to established classification tables, we define and characterize mathematically Gaussian dualities as dualities mapping free fermions to free fermions (and interacting to interacting). By introducing a large, flexible class of Gaussian dualities we show that any insulator is dual to a superconductor, and that fermionic edge modes are dual to Majorana edge modes, that is, the Gaussian dualities of this paper preserve the bulk-boundary correspondence. Transmutation of relevant symmetries, particle number, translation, and time reversal is also investigated in detail. As illustrative examples, we show the duality equivalence of the dimerized Peierls chain and the Majorana chain of Kitaev, and a two-dimensional Kekul\'e-type topological insulator, including graphene as a special instance in coupling space, dual to a p-wave superconductor. Since our analysis extends to interacting fermion systems we also briefly discuss some such applications.