How to infer non-Abelian statistics and topological visibility from tunneling conductance properties of realistic Majorana nanowires

We consider a simple conceptual question with respect to Majorana zero modes in semiconductor nanowires: can the measured nonideal values of the zero-bias-conductance-peak in the tunneling experiments be used as a characteristic to predict the underlying topological nature of the proximity induced nanowire superconductivity? In particular, we define and calculate the topological visibility, which is a variation of the topological invariant associated with the scattering matrix of the system as well as the zero-bias-conductance-peak heights in the tunneling measurements, in the presence of dissipative broadening, using precisely the same realistic nanowire parameters to connect the topological invariants with the zero-bias tunneling conductance values. This dissipative broadening is present in both (the existing) tunneling measurements and also (any future) braiding experiments as an inevitable consequence of a finite braiding time. The connection between the topological visibility and the conductance allows us to obtain the visibility of realistic braiding experiments in nanowires, and to conclude that the current experimentally accessible systems with nonideal zero-bias conductance peaks may indeed manifest (with rather low visibility) non-Abelian statistics for the Majorana zero modes. In general, we find that a large (small) superconducting gap (Majorana peak splitting) is essential for the manifestation of the non-Abelian braiding statistics, and in particular, a zero-bias conductance value of around half the ideal quantized Majorana value should be sufficient for the manifestation of non-Abelian statistics in experimental nanowires. Our work also establishes that as a matter of principle the topological transition associated with the emergence of Majorana zero modes in finite nanowires is always a crossover (akin to a quantum phase transition at finite temperature) requiring the presence of dissipative broadening (which must be larger than the Majorana energy splitting in the system) in the system. For braiding, this dissipation is supplied by the finite speed of the braiding process itself, which must be diabatic in any real experiment for braiding to succeed.

Figure 1

A schematic diagram for measuring the tunneling conductance. One end of the Rashba nanowire is shown attached to a normal lead. The lead is connected to the nanowire through a potential barrier. Magnetic field is parallel to the nanowire. Proximate $s$-wave superconductor is responsible for the superconducting order parameter in the nanowire.

Figure 2

LDOS for clean nanowire with $L=1.5\mu \text{m}$ and Zeeman field strengths (a)–(d), $V_Z=4.2$, 4.3, 4.5, and 5.0 K. The corresponding Majorana splitting (a)–(d) are $\delta =0.012$, 0.036, 0.094, and 0.18 K respectively clearly vary strongly with $V_Z$.

Figure 3

Conductance plot corresponding to the LDOS splittings for ${\mathrm{\Gamma }}_L/\mathrm{\Gamma }=10$ (blue solid curve) and $0.25$ (red dashed curve). (a)–(d) The parameter $\delta /\mathrm{\Gamma }=0.27$, 0.71, 1.74, and 3.27, respectively. The TV $\left(Q\right)$ values (a)–(d) are $\left(-0.80,-0.75,-0.46,0.12\right)$ and $\left(0.44,0.58,0.82,0.93\right)$ for blue solid and red dashed curves, respectively. The conductance peaks split for large $\delta /\mathrm{\Gamma }$ and the conductance decreases for small ${\mathrm{\Gamma }}_L/\mathrm{\Gamma }$.

Figure 4

TV for $\delta /\mathrm{\Gamma }$ values corresponding to Fig. 3 for coupling parameter ${\mathrm{\Gamma }}_L/\mathrm{\Gamma }=10$ (blue dots) and $0.25$ (red plus). The TV is an increasing function of $\delta /\mathrm{\Gamma }$, i.e., the system tends to become non-topological as $\delta /\mathrm{\Gamma }$ increases.

Figure 5

(Top) Plot of the TV as a function of ZBCP for $\delta /\mathrm{\Gamma }=0.16$ (blue dot) and $2.43$ (red plus). Conductance is varied by varying ${\mathrm{\Gamma }}_L/\mathrm{\Gamma }$. (Bottom) TV vs ZBCP for ${\mathrm{\Gamma }}_L/\mathrm{\Gamma }=10$ (blue dot) and $0.25$ (red plus). Conductance is varied by varying $\delta /\mathrm{\Gamma }$. The TV is a decreasing function of the ZBCP.

Figure 6

Plot of the lowest Andreev bound state energy (top) and bulk quasiparticle energy gap (bottom) as a function of Zeeman field strength for different physical lengths of Majorana nanowire. The bulk TPT is at $V_Z=3\mathrm{K}$. In the topological phase, $\left(V_Z>3\mathrm{K}\right)$, lowest Andreev bound state energy is the Majorana splitting.

Figure 7

Conductance plot for ${\mathrm{\Gamma }}_L/\mathrm{\Delta }=0.20$ (blue solid curve) and 0.005 (red dashed curve). Broadening $\mathrm{\Gamma }$ is chosen so that $\delta /\mathrm{\Gamma }=0.16$ is held fixed for all panels with $\mathrm{\Delta }/\mathrm{\Gamma }$ (a)–(d) being 19.33, 0.90, 0.28, $-12.1$, respectively. The TV $\left(Q\right)$ values (a)–(d) are $(-0.75,0.34,0.56,0.98)$ and $\left(0.51,0.95,0.97,1.0\right)$ for blue solid curve and red dashed curve, respectively.

Figure 8

TV for $\mathrm{\Delta }/\mathrm{\Gamma }$ values corresponding to Fig. 7 for coupling parameters near TPT with ${\mathrm{\Gamma }}_L/\mathrm{\Delta }=0.20$ (blue dots) and 0.005 (red plus), respectively. $\delta /\mathrm{\Gamma }=0.16$ is held fixed. The TV is an decreasing function of $\mathrm{\Delta }/\mathrm{\Gamma }$ with $Q⟶1$ as $\mathrm{\Delta }/\mathrm{\Gamma }⟶0$, i.e., the system tends to become non-topological (topological) as the system tends to small (large) topological gap and the TV tends to +1 as the gap completely closes (system approaches TPT).

Figure 9

A schematic diagram for braiding a pair of Majoranas using a trijunction. The system is initialized (a) such that ${\gamma }_1^{\prime },{\gamma }_1,{\gamma }_2^{\prime }$, and ${\gamma }_2$ are four localized Majorana modes and $\left({\gamma }_3^{\prime },{\gamma }_3\right)$ Majorana pair is paired into a Dirac fermion. Paired Majorana modes are depicted as green discs paired by enhanced wave-function overlap over the region depicted by pink oval. At every move an unpaired Majorana mode is moved from one position to another. The movement of the Majorana resulting from the move resulting in each configuration (b)–(d) is shown by a dashed arrow.