Phase-locked magnetoconductance oscillations as a probe of Majorana edge states

We calculate the Andreev conductance of a superconducting ring interrupted by a flux-biased Josephson junction, searching for electrical signatures of circulating edge states. Two-dimensional pair potentials of spin-singlet $d$-wave and spin-triplet $p$-wave symmetry support, respectively, (chiral) Dirac modes and (chiral or helical) Majorana modes. These produce $h/e$-periodic magnetoconductance oscillations of amplitude $\simeq (e^2/h)N^{-1/2}$, measured via an $N$-mode point contact at the inner or outer perimeter of the grounded ring. For Dirac modes the oscillations in the two contacts are independent, while for an unpaired Majorana mode they are phase locked by a topological phase transition at the Josephson junction.

Figure 1

Left panel: Superconducting ring (S) containing a weak link (J), forming a flux-biased Josephson junction. A current can be injected into the grounded superconductor from a voltage-biased normal-metal (N) contact at the inner or outer perimeter. The Andreev conductance $G_n=I/V_n$ is the ratio of the current-to-ground $I$ and the applied voltage $V_n$ to contact $n=1,2$ (with $V=0$ for the other contact). Right panel: Scattering processes at the outer perimeter, involving the coupling of a chiral edge state to $2N$ incoming and $2N$ outgoing electron-hole modes at the normal metal. This coupling introduces a dependence of the conductance on the phase difference $\varphi =(2e/\hslash )\Phi$ across the Josephson junction and on its topological quantum number $\sigma$.

Figure 2

Probability distribution of the change $\delta G$ in the conductance of the superconducting ring when the Josephson junction switches from topologically trivial to nontrivial. These are results from random-matrix theory in the circular real ensemble (CRE, symmetry class D), for an unpaired chiral Majorana mode circulating along the ring and $N$ modes in the contact to the normal metal. The solid lines for $N=1,2$ are from Eq. (3.6), the histograms are obtained numerically by averaging the scattering matrix $S$ uniformly over $\mathrm{SO}(2N+1)$, and the dashed curves are the large-$N$ Gaussian limit (3.9).

Figure 3

Probability distribution of the conductance change upon a topological phase transition in symmetry class DIII, involving a pair of helical Majorana modes circulating along the ring. The contact to the normal metal has $M$ orbital modes, each with a twofold Kramers degeneracy. The results are from the time-reversally invariant circular real ensemble (T-CRE) of random-matrix theory. The solid line for $M=1$ is from Eq. (3.16), the histograms are obtained numerically, and the dashed curves are the large-$N$ Gaussian limit (3.17).

Figure 4

Excitation spectrum of an infinitely long Josephson junction along the $y$ axis, for different values of the phase difference $\varphi$, calculated numerically from the discretized Bogoliubov–De Gennes Hamiltonian (4.1) (Ref. 28). Panel (a) is for the class-D chiral $p$-wave pair potential (4.4) and panel (b) for the class-DIII helical $p$-wave pair potential (4.11). The closing of the excitation gap at $\varphi =\pi$ is topologically protected.

Figure 5

Same as Fig. 4, for a class-C chiral $d$-wave pair potential. The two curves are for $\varphi =\pi$, with and without $\pm x$ symmetry of the Josephson junction along the $y$ axis. The gap closing now has no topological protection, but requires a spatial symmetry.

Figure 6

Electrical conductance (upper row) and thermal conductance (lower row) in the ring geometry of Fig. 1, as a function of the enclosed flux $\Phi$, for the three pair potentials (4.4), (4.11), and (4.5). The red and blue curves are calculated numerically for two different realizations of the disorder potential (Ref. 28). The thermal conductance is measured between the inner and outer perimeters of the ring. The electrical conductance is measured either between the inner perimeter and ground ($G_1$, solid curves) or between the outer perimeter and ground ($G_2$, dashed curves).