Quantized Conductance at the Majorana Phase Transition in a Disordered Superconducting Wire

Superconducting wires without time-reversal and spin-rotation symmetries can be driven into a topological phase that supports Majorana bound states. Direct detection of these zero-energy states is complicated by the proliferation of low-lying excitations in a disordered multimode wire. We show that the phase transition itself is signaled by a quantized thermal conductance and electrical shot noise power, irrespective of the degree of disorder. In a ring geometry, the phase transition is signaled by a period doubling of the magnetoconductance oscillations. These signatures directly follow from the identification of the sign of the determinant of the reflection matrix as a topological quantum number.

Figure 1

Thermal conductance and determinant of reflection matrix of a disordered multimode superconducting wire as a function of Fermi energy. The curves are calculated numerically from the Hamiltonian (7, 8, 9) on a square lattice (lattice constant $a=l_{\mathrm{so}}/20$), for parameter values $W=l_{\mathrm{so}}$, $L=10l_{\mathrm{so}}$, $\Delta =10E_{\mathrm{so}}$, $g_{\mathrm{eff}}{\mu }_{B}B=21E_{\mathrm{so}}$, and three different disorder strengths $U_0$. The arrows indicate the expected position of the topological phase transition in an infinite clean wire ($U_0=0$, $L\to \infty$), calculated from Eq. (10). Disorder reduces the topologically nontrivial interval (where $\mathrm{Det}r<0$), and may even remove it completely, but the conductance quantization remains unaffected as long as the phase transition persists.

Figure 2

Fourier amplitude with flux periodicity $h/e$ of the magnetoconductance oscillations, calculated numerically from the Hamiltonian (7, 8, 9) for a single disorder strength $U_0=50E_{\mathrm{so}}$ and seven different temperatures ${\tau }_0$. The inset shows the Aharonov-Bohm ring geometry. The parameters of the superconducting segment of the ring ($S$) are the same as in Fig. 1, with $N=1$ in this range of Fermi energies. The normal part of the ring has $N=8$ propagating modes to avoid localization by the disorder (which has the same strength throughout the ring).