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

## TL;DR

This paper provides an exact analytical solution for a specific time-reversal-invariant topological superconductor model, revealing zero-energy modes and their entanglement properties, and discusses implications for the time-reversal anomaly.

## Contribution

The authors derive an exact solution for a special parameter case of the TRITOPS model, clarifying zero-energy modes and correcting misconceptions about the time-reversal anomaly.

## Key findings

- Exact solution at specific parameters with zero-energy modes
- Degeneracy and entanglement of zero-energy states
- Effects of small hopping and magnetic field on degeneracy

## Abstract

We consider a model proposed before for a time-reversal-invariant topological superconductor (TRITOPS) which contains a hopping term $t$, a chemical potential $\mu$, an extended $s$-wave pairing $\Delta$ and spin-orbit coupling $\lambda$. We show that for $|\Delta|=|\lambda|$, $\mu=t=0$, the model can be solved exactly defining new fermion operators involving nearest-neighbor sites. The many-body ground state is four-fold degenerate due to the existence of two zero-energy modes localized exactly at the first and the last site 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 correct some statements related with the so called "time-reversal anomaly". Addition of a small hopping term 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 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 with numerical results %for a small chain obtaining good agreement.