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

We consider a model proposed before for a time-reversal-invariant topological superconductor which contains a hopping term $t$, a chemical potential $\mu$, an extended $s$-wave pairing $\mathrm{\Delta }$, and spin-orbit coupling $\lambda$. We show that for $\left|\mathrm{\Delta }\right|=\left|\lambda \right|$, $\mu =t=0$, 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.

Figure 1

Sketch of the construction of the annihilation operators of finite energy, leaving the zero-energy modes $a_{1\sigma }$ and $b_{L\sigma }$ at the ends.

Figure 2

Many-body ground-state manifold. For the four orthonormal many-body ground states, we sketch the $S_z$ value at each chain site $n$. The local parity ($P_{\text{left}}$, $P_{\text{right}}$) value is indicated with a $+$ or $-$ at the corresponding end. The states are accommodated such that left and right panels are related by time-reversal symmetry and top and bottom states are connected by the action of $a_{1\uparrow }^{†}$.