Anderson topological superconductor

In this paper we study the phase diagram of a disordered, spin-orbit coupled superconductor with $s$-wave or $d+id$-wave pairing symmetry in symmetry class $D$. We analyze the topological phase transitions by applying three different methods, which include a disorder averaged entanglement entropy, a disorder averaged real-space Chern number, and an evaluation of the momentum space Chern number in a disorder averaged effective model. We find evidence for a disorder-induced topological state. While in the clean limit there is a single phase transition from a trivial phase with a Chern number $C=4$ to a topological phase with $C=1$, in the disordered system there is an intermediate phase with $C=3$. The phase transition from the trivial $C=4$ phase into the intermediate phase with $C=3$ is seen in the real-space calculation of the Chern number. In spite of this, this phase transition is not detectable in the entanglement entropy. A second phase transition from the disorder-induced $C=3$ into the $C=1$ phase is seen in all three quantities.

Figure 1

(a) Chern number $C$ for $d+id$-wave coupling ${\mathrm{\Delta }}_1=0.8t,{\mathrm{\Delta }}_2=0.4t,M=0.8t,\mu =0$, and $A=0.25t$. (b) Chern number $C$ for $s$-wave coupling ${\mathrm{\Delta }}_s=1t,B=0,\mu =-4t$, and $A=0.25t$. Derivative of the entanglement entropy for (c) $d+id$-wave coupling and (d) $s$-wave coupling. The insets show the entanglement entropy.

Figure 2

(a) Renormalization of the chemical potential (solid line) and Zeeman coupling (dashed line) for varying disorder strengths for $d+id$ wave and (b) Chern number $C_{\mathrm{ren}}$ (solid line) calculated from the renormalized parameters and through a real-space formula $C_{RS}$ (dashed line).