Equivalence of Topological Insulators and Superconductors

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 Kekulé-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.

18 pages, 4 figures

References in corpus (6)

• The electronic properties of graphene
• Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms
• Autocorrelations and Thermal Fragility of Anyonic Loops in Topologically Quantum Ordered Systems
• Many-body characterization of topological superconductivity: The Richardson-Gaudin-Kitaev chain
• Statistical translation invariance protects a topological insulator from interactions
• Bimodal conductance distribution of Kitaev edge modes in topological superconductors

Cited by in corpus (13)

• Relation between $\mathcal{P}\mathcal{T}$-symmetry breaking and topologically nontrivial phases in the SSH and Kitaev models
• Exact solution to an interacting dimerized Kitaev model at symmetric point
• Characterization of topological phases of modified dimerized Kitaev chain via edge correlation functions
• Squaring the fermion: The threefold way and the fate of zero modes
• Generalization of Bloch's theorem for arbitrary boundary conditions: Interfaces and topological surface band structure
• What is a particle-conserving Topological Superfluid?
• Restoring number conservation in quadratic bosonic Hamiltonians with dualities
• Majorana zero modes and their bosonization
• Majorana orthogonal transformation and Majorana zero modes in free fermionic systems
• Duality and topology
• Topological phase transition based on the attractive Hubbard model
• Bloch and Bethe ansatze for the Harper model: A butterfly with a boundary
• Many-body wave function and edge magnetization of an open $p+is$ superconducting chain