Electrical detection of topological quantum phase transitions in disordered Majorana nanowires

We study a disordered superconducting nanowire, with broken time-reversal and spin-rotational symmetry, which can be driven into a topological phase with end Majorana bound states by an externally applied magnetic field. As a function of disorder strength, it is known that the Majorana nanowire has a delocalization quantum phase transition from a topologically nontrivial phase, which supports Majorana bound states, to a nontopological insulating phase without them. On both sides of the transition, the system is localized at zero energy albeit with very different topological properties. We exploit this deep connection between topology and localization properties to propose an electrical transport measurement to detect the localization-delocalization transition occurring in the bulk of the nanowire. The basic idea consists of measuring the difference of conductance at one end of the wire obtained at different values of the coupling to the opposite lead. We show that this measurement reveals the nonlocal correlations emergent only at the topological transition. Hence, while the proposed experiment does not directly probe the end Majorana bound states, it can provide direct evidence for the bulk topological quantum phase transition itself.

Figure 1

Schematic representation of the N-S-N system under consideration. The basic idea consists of measuring the conductance at one end of the NW (e.g., left end) for different values of ${\gamma }_R$, the coupling to the opposite (i.e., right) lead. The difference of conductances $\delta G_{LL}$ [see Eq. (6)] contains information about the nonlocal correlations between the end points of the NW, which can be used to detect the TQPT.

Figure 2

(a) Transmission probability $T_{1N}$ between the end points (sites 1 and $N$) of the isolated NW and topological phase diagram as a function of disorder strength $v_0$ and Zeeman field $V_Z$. At the topological critical point, the topological invariant $Q=\text{sgn}\left({\prod }_{n=1}^4\tanh {\lambda }_n\right)$, where ${\lambda }_n$ are the Lyapunov exponents, changes sign and the transmission probability $T_{1N}$ becomes 1 [see also Fig. 3a]. The bold dashed line follows the TQPT. (b) Difference of left conductances $\delta G_{LL}$ [see Eq. (6)]. We have chosen parameters ${\gamma }_L=0.2t$, ${\gamma }_R={10}^{-5}t$, $\delta {\gamma }_R={10}^{-3}t$, $N=300$, $\alpha =0.07t$, $\Delta =0.37t$, and ${\mu }_0=-2t$. In the clean case and at $V_z=0.45t=6.4\alpha$ we obtain the maximal ratio of $L/\xi \sim 10$ where $L$ is the length of the NW.

Figure 3

(a) Topological invariant for an isolated NW. The localization-delocalization transition is evident from the appearance of a quantized peak in the transmission probability $T_{1N}$. (b) Difference in conductance in the left lead for different couplings to the right lead in the regime $\delta {\gamma }_R\ll {\gamma }_R$. The main point is that these two quantities follow each other which can be used to detect the topological phase transition. Here we have chosen fixed disorder amplitude $v_0=3.14\alpha$; see dashed vertical lines in Fig. 2.