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

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 we study the stability of the TRITOPS phase against the Zeeman field $J$ and the chemical potential $\ensuremath{\mu}$, 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 $\ensuremath{\langle}{S}_{y}\ensuremath{\rangle}=1/4$ at the ends, found for noninteracting wires at $U=0$, is recovered. In addition, the analytic expression that relates $\ensuremath{\langle}{S}_{y}\ensuremath{\rangle}$ with $J$ for finite noninteracting chains is shown to be universal up to moderate values of $U$.