Exact analytical solution of a time-reversal-invariant topological superconducting wire

Abstract

We consider a model proposed before for a time-reversal-invariant topological superconductor which contains a hopping term t, a chemical potential μ, an extended s-wave pairing Δ, and spin-orbit coupling λ. We show that for |Δ|=|λ|, μ=t=0, the model has an exact analytical solution defining new fermion operators involving nearest-neighbor sites. The many-body ground state is fourfold degenerate due to the existence of two zero-energy modes localized exactly at the first and last sites of the chain. These four states show entanglement in the sense that creating or annihilating a zero-energy mode at the first site is proportional to a similar operation at the last site. By continuity, this property should persist for general parameters. Using these results, we discuss some statements related to the so-called time-reversal anomaly. The addition of a small hopping term t for a chain with an even number of sites breaks the degeneracy, and the ground state becomes unique with an even number of particles. We also consider a small magnetic field B applied to one end of the chain. We compare the many-body excitation energies and spin projection along the spin-orbit direction for both ends of the chains obtained treating t and B as small perturbations with numerical results in a short chain, obtaining good agreement.