Interplay of Disorder and Interaction in Majorana Quantum Wires

We study the interplay between disorder and interaction in one-dimensional topological superconductors which carry localized Majorana zero-energy states. Using Abelian bosonization and the perturbative renormalization group approach, we obtain the renormalization group flow and the associated scaling dimensions of the parameters and identify the critical points of the low-energy theory. We predict a quantum phase transition from a topological superconducting phase to a nontopological localized phase, and obtain the phase boundary between these two phases as a function of the electron-electron interaction and the disorder strength in the nanowire. Based on an instanton analysis which incorporates the effect of disorder, we also identify a large regime of stability of the Majorana-carrying topological phase in the parameter space of the model.

Figure 1

(a) Parametric dependence of $y_b(\ell )$ vs $y_{\Delta }(\ell )$, as obtained from the numerical solution of the RG-flow equations (5, 6, 7, 8), for fixed initial parameter $K_0=0.65$ (log-log scale). The thick dashed curve is the critical line, separating the topological SC phase (shaded area) from the nontopological disordered phase, and the thin dotted line is our analytical estimate $y_b\sim y_{\Delta }^{\nu }$, valid in the limit $\{y_b(\ell ),y_{\Delta }(\ell )\}\to 0$. (b) Phase diagram in $y_{\Delta 0}$, $y_{b0}$ space obtained for $y_{s0}=0$ and different values of $K_0$. The curves correspond to the critical lines $y_{b0}$ vs $y_{\Delta 0}$, satisfying the condition $y_{\Delta }({\ell }^{*})=y_b({\ell }^{*})=1$. The area below each curve represents the regime for which the topological SC is expected to dominate over disorder.