Effect of disorder on the decreasing the critical value of magnetic field in proximity induced topological superconductors

Here we investigate how adding disorder changes the critical value of magnetic field $B_c$ required to observe the topological phase transition in proximity induced topological superconductors .We consider disordered topological superconductors in $D=1,2$ spatial dimensions and use numerical analysis to directly calculate the topological invariant.Qualitatively different behavior are found depending on the dimensionality of the system. In contrast to $D=2$ for $D=1$ we show that adding disorder gradually decreases $B_c$ from the clean the case limit of $B_c=\sqrt{\mu^2+\Delta^2}$ to $B_c=\Delta$. A discussion of why these results are expected is provided. These findings, specially in $D=1$ dimension are of experimental interest since they show that the topological phase transition might be observable at values of magnetic field $B_c$ much smaller than previously expected.