Proposal for Measuring the Parity Anomaly in a Topological Superconductor Ring

A topological superconductor ring is uniquely characterized by a switch in the ground state fermion number parity upon insertion of one superconducting flux quantum—a direct consequence of the topological “parity anomaly.” Despite the many other tantalizing signatures and applications of topological superconductors, this fundamental, defining property remains to be observed experimentally. Here we propose definitive detection of the fermion parity switch from the charging energy, temperature, and tunnel barrier dependence of the flux periodicity of two-terminal conductance of a floating superconductor ring. We extend the Ambegaokar-Eckern-Schön formalism for superconductors with a Coulomb charging energy to establish new explicit relationships between thermodynamic and transport properties of such a ring and the topological invariant of the superconductor. Crucially, we show that the topological contribution to the conductance oscillations can be isolated from Aharonov-Bohm oscillations of nontopological origin by their different dependence on the charging energy or barrier transparency.

Figure 1

Schematic of a proposed two-terminal transport experiment: a floating superconductor ring (yellow) is coupled to two normal metallic leads (blue). An infinitesimal bias voltage $V=0^{+}$ is applied across the leads. An insulating junction (gray) is present in the middle of the ring, and the enclosed magnetic flux $\mathrm{\Phi }$ varies continuously along with the energy $\delta (\mathrm{\Phi })$ of an Andreev bound state at the junction.

Figure 2

Large and no charging energy limits. (a),(b) Energy spectra for the bound states at the junction of a topological or trivial SC ring. (c),(d) Evolution of the energy of the $N+1$ charge state as a function of the magnetic flux ($\mathrm{\Phi }$). Each color line corresponds to the color point in the spectra in (a) and (b), and $V_g^{*}$ represents the resonance point at which the $N$ and $N+2$ charge states are degenerate. (e),(f) Conductance for the NSN junction with strong Coulomb blockade. Topological (trivial) SC shows $2{\mathrm{\Phi }}_0$ (${\mathrm{\Phi }}_0$) periodicity. (g),(h) Conductance for the NSN junction with no Coulomb blockade. Here, the short (long) SC ring shows $2{\mathrm{\Phi }}_0$ (${\mathrm{\Phi }}_0$) periodicity in conductance, regardless of SC being topological or trivial.

Figure 3

Conductance difference $G(0)-G({\mathrm{\Phi }}_0)$ as a function of the barrier transparency characterized by $g_0$. For a short SC ring (no matter topologically trivial or nontrivial) with no charging energy, the conductance difference increases monotonically with barrier transparency (blue line). By contrast, for a TSC ring with charging energy, although increasing at low barrier transparency, the conductance difference will eventually decrease with barrier transparency when $g_0$ goes beyond some critical value $\sim 0.5$ (red line).