Experimentally accessible topological quality factor for wires with zero energy modes

Spin-orbit coupled semiconducting nanowires with proximity-induced superconductivity are expected to host Majorana zero modes at their endpoints when a sufficiently strong magnetic field is applied. The resulting phase would be a one-dimensional topological superconductor. However, while a variety of experiments have been performed observing a zero-bias conductance peak (suggestive of Majorana zero modes), the topological nature of these physical systems is still a subject of debate. Here we suggest a quantitative test of the degree to which a system displaying a zero bias peak may be considered topological. The experiment is similar to previous measurements of conductance but is performed with the aid of a quantum dot at the wire's end. We arrive at the surprising result that the nonlocal behavior of a topological system may be tested through a local measurement.

Figure 1

Schematic diagram of the system under consideration. Conductance spectroscopy is performed on a wire containing a single low-lying fermionic level $\widehat{c}=\frac{1}{2}({\gamma }_1+i{\gamma }_2)e^{-i\varphi }$ $(\varphi$ an arbitrary phase) by coupling to a lead at voltage $V_{\mathrm{bias}}$ relative to ground through a quantum dot with a single fermionic level with annihilation operator $\widehat{d}$. The energy difference ${\epsilon }_d$ between the occupied and unoccupied states of that dot is set by a side-gate voltage and varied to perform the experiment suggested here. The dot-wire system is characterized by three additional parameters: the couplings $\eta$ and ${\eta }^{\prime }$ from the dot to the Majorana modes ${\gamma }_1$ and ${\gamma }_2$, respectively, and the Majorana hybridization ${\epsilon }_w$ that sets the energy difference between occupied and unoccupied states of the fermionic mode in the wire. If the wire is in the topological superconducting phase, both ${\eta }^{\prime }$ and ${\epsilon }_w$ should be $\sim 0$. A similar differential conductance experiment without the dot present can only access the value of ${\epsilon }_w$. With the dot, all three parameters may be measured independently. In particular, one can make measure of $q=1-|{\eta }^{\prime }|/|\eta |$, a ‘topological quality factor’ quantifying the extent to which one may expect non-Abelian behavior from an ${\epsilon }_w=0$ mode within the wire.

Figure 2

Expected low-energy spectrum near the dot resonance for various values of the topological quality factor $q$ and in the absence of hybridization between the two Majorana modes. The distance between the outer two mode peaks at resonance is twice the ‘topological’ coupling $\eta$. The distance between the two inner mode peaks at resonance is twice the ‘nontopological’ coupling ${\eta }^{\prime }$. For a perfectly topological system $\left(q=1,{\epsilon }_w=0\right)$, the zero-bias conductance peak does not split as it passes through the dot resonance.

Figure 3

Comparison of differential conductance color plots from Deng et al. [18] (left) with transition plots from our simple model (right). Here we plot the transitions for a quality factor of $q=0.5$ and two values of ${\epsilon }_w/\eta$ that lead to qualitatively similar traces. The circled point in the Deng et al. data (top left) indicates an anticrossing between dot and wire states that is also present within our simple model (top right). Note that $q$ here is chosen only for qualitative similarity to the experimental data and that a wide range of $q$ values may share these qualitative features (see Fig. 2). However, the behavior is sufficiently alike that more exact values for the model parameters might be extracted from similar data in a systematic study.