Density of states at disorder-induced phase transitions in a multichannel Majorana wire

An $N$-channel spinless $p$-wave superconducting wire is known to go through a series of $N$ topological phase transitions upon increasing the disorder strength. Here, we show that at each of those transitions the density of states shows a Dyson singularity $\nu \left(\varepsilon \right)\propto {\varepsilon }^{-1}{|ln\varepsilon |}^{-3}$, whereas $\nu \left(\varepsilon \right)\propto {\varepsilon }^{\left|\alpha \right|-1}$ has a power-law singularity for small energies $\varepsilon$ away from the critical points. Using the concept of “superuniversality” [Gruzberg et al., Phys. Rev. B 71, 245124 (2005)], we are able to relate the exponent $\alpha$ to the wire's transport properties at zero energy and, hence, to the mean free path $l$ and the superconducting coherence length $\xi$.

Figure 1

Dimensionless distance to the critical point as a function of the ratio $\xi /l$ of superconducting coherence length and mean free path. The red dots show the exponent $\alpha$ obtained by fitting the numerically computed integrated density of states to the functional form in Eq. (11) for a single-channel wire (top) and a three-channel wire (bottom). The dashed curve is the exponent expected from the mapping onto a single channel hopping model [see Eq. (13)]. Inset: Example of a fit of the integrated density of states normalized by the wire length as a function of energy. The squares show the numerically obtained data and the continuous curve is the analytical result, Eq. (11), using the value $\alpha =0.0906$ obtained from the fitting procedure. The value of $\overline{l}$ can be obtained directly from the model parameters and need not be fitted [see Eqs. (12) and (15)].

Figure 2

Dimensionless distance $\alpha$ to the critical point for the model (2) with ${\Delta }_x^{\prime }={\Delta }_y^{\prime }$ and $N=2$ transverse channels. The dashed line is the analytical prediction (13). The continuous line is the analytical prediction, Eq. (12), corrected for the slight shift of the critical disorder strength at the second phase transition by inserting the value of $l_{\mathrm{crit}}^{\left(n\right)}$ observed in the numerics.