Topological phases of strongly-interacting time-reversal invariant topological superconducting chains under a magnetic field

TL;DR

This study uses density-matrix renormalization group methods to explore the topological phases and edge states of a strongly-interacting, time-reversal invariant topological superconductor chain under a magnetic field, revealing phase stability and spin properties.

Contribution

It provides a detailed analysis of how magnetic fields and interactions affect topological phases and Majorana modes in TRITOPS chains, including universal relations for spin projections.

Findings

Majorana zero modes remain degenerate until a critical magnetic field.

Fractional spin projection at edges is recovered for certain field orientations.

Universal relation between spin projection and magnetic field up to moderate interactions.

Abstract

Using the density-matrix renormalization group, we determine the different topological phases and low-energy excitations of a time-reversal invariant topological superconducting (TRITOPS) wire with extended s-wave superconductivity, Rashba spin-orbit coupling (SOC) and on-site repulsion $U$, under an externally applied Zeeman field $J$. For the case in which $J$ is perpendicular to the SOC, the model describes a chain of Shiba impurities on top of a superconductor with extended superconductor pairing. We identify the different topological phases of the model at temperature $T=0$, and in particular study the stability of the TRITOPS phase against the Zeeman field $J$ and the chemical potential $\mu$, for different values of $U$. In the case where the magnetic field $J$ is perpendicular to the SOC axis, the pair of Kramers-degenerate Majorana zero modes at the edges of the system that exist for $J=0$, remain degenerate until a critical value of the magnetic field is reached. For $J$ parallel to the SOC and up to moderate values of $U$, the fractional spin projection $\langle S_y \rangle=1/4$ at the ends, found for non-interacting wires at $U=0$, is recovered. In addition, the analytic expression that relates $\langle S_y \rangle$ with $J$ for finite non-interacting chains is shown to be universal up to moderate values of $U$.