Thermal conductance as a probe of the nonlocal order parameter for a topological superconductor with gauge fluctuations

We investigate the effect of quantum phase slips on a helical quantum wire coupled to a superconductor by proximity. The effective low-energy description of the wire is that of a Majorana chain minimally coupled to a dynamical ${\mathbb{Z}}_2$ gauge field. Hence the wire emulates a matter-coupled gauge theory, with fermion parity playing the role of the gauged global symmetry. Quantum phase slips lift the ground-state degeneracy associated with unpaired Majorana edge modes at the ends of the chain, a change that can be understood as a transition between the confined and the Higgs-mechanism regimes of the gauge theory. We identify the quantization of thermal conductance at the transition as a robust experimental feature separating the two regimes. We explain this result by establishing a relation between thermal conductance and the Fredenhagen-Marcu string order parameter for confinement in gauge theories. Our work indicates that thermal transport could serve as a measure of nonlocal order parameters for emergent or simulated topological quantum order.

Figure 1

Panel (a): An $s$-wave superconductor (gray) is deposited on top of a helical quantum wire (green), which can be, for example, a semiconducting nanowire or the edge of a quantum spin Hall insulator. We consider the effect of quantum phase slips in the superconductor (black arrows). Once a moderate magnetic field is applied to break time reversal invariance, Majorana modes (orange dots) appear at the ends of the wire and at weak links when the phase slips happen. Panel (b): We show an equivalent circuit describing the system [see Eq. (1)]. Here, as usual, a box with a cross denotes a Josephson junction and its capacitance. On the other hand, a box with only half of a cross denotes the $4\pi$-periodic Majorana junction. Arrows represent coupling to external leads to the Majorana modes at the end with strength ${\Gamma }_L$ and ${\Gamma }_R$.

Figure 2

Numerical results: Andreev conductance (left, in units of $G_0=2e^2/h$), Fredenhagen-Marcu order parameter $M_{\mathrm{MF}}$ (center), and thermal conductance (right, in units of $K_0={\pi }^2{k}_B^{2}T/6h$) plotted as a function of $h$ and $\kappa$. The results are obtained for a chain of $N=11$ islands with coupling constant $\Gamma =0.01E_M$ to the leads, in the limit of vanishing temperature $T$ and small applied voltage $V$. The Andreev conductance $G$ is averaged over a small voltage interval ${10}^{-4}{E}_M$ to account for the finite-size energy splitting of the Majorana modes, as for a finite-size wire we trivially have $G=0$ [17, 41]. The Andreev conductance only shows a signal along the axis. On the other hand, the string-order parameter allows us to distinguish a confined regime—corresponding to the topological regime with Majorana end modes—from the trivial Higgs regime. The separation between these two regimes can be clearly identified by the peak in the thermal conductance. The inset in the right panel shows line cuts of the thermal conductance at $\kappa /E_M=0,0.01,0.02$, going from right to left as shown by the arrow. Due to finite-size effects, the transition at $\kappa =0$ is shifted from $h/E_M=1$ to $h/E_M\simeq {(\Gamma /E_M)}^{1/11}\simeq 0.7$, as expected. Increasing $\kappa$, the topological regime shrinks and only the Higgs regime survives.