Fermion-Parity Anomaly of the Critical Supercurrent in the Quantum Spin-Hall Effect

The helical edge state of a quantum spin-Hall insulator can carry a supercurrent in equilibrium between two superconducting electrodes (separation $L$, coherence length $\xi$). We calculate the maximum (critical) current $I_c$ that can flow without dissipation along a single edge, going beyond the short-junction restriction $L\ll \xi$ of earlier work, and find a dependence on the fermion parity of the ground state when $L$ becomes larger than $\xi$. Fermion-parity conservation doubles the critical current in the low-temperature, long-junction limit, while for a short junction $I_c$ is the same with or without parity constraints. This provides a phase-insensitive, dc signature of the $4\pi$-periodic Josephson effect.

Figure 1

Phase-dependent excitation spectrum of a Josephson junction along a quantum spin-Hall (QSH) edge (left panels) and corresponding zero-temperature supercurrent (right panels). The supercurrent $I_{4\pi }$ is $4\pi$-periodic, with two branches $I_{+}$ (blue solid), $I_{-}$ (red solid) distinguished by the ground-state fermion parity and with a parity switch at $\varphi =\pm \pi$. The top row shows the short-junction limit of Ref. 7, the bottom row the long-junction limit calculated here. (The jump in $I_{-}$ at $\varphi =0$ occurs because of the change in slope indicated by the green arrows in the magnified central part of the spectrum.) The $2\pi$-periodic supercurrent $I_{2\pi }$ without parity constraints is also shown (green dashed). The critical current is the same for $I_{4\pi }$ and $I_{2\pi }$ in the short junction, but different by a factor of two in the long junction.

Figure 2

Phase dependence of the parity-constrained supercurrent $I_{4\pi }$ (solid curves, in units of $e{E}_{\mathrm{T}}/\hslash \propto 1/L$), calculated by a numerical evaluation of the Matsubara sums. The left panel shows the crossover from the short-junction to the long-junction regime in the zero-temperature limit (full interval $-2\pi <\varphi <2\pi$). The right panel shows the temperature dependence in the long-junction limit (reduced interval $0<\varphi <2\pi$). The left panel also shows the supercurrent $I_{2\pi }$ without parity constraints (dashed curves). The insets in the right panel show current-biased superconducting circuits that measure the $I\mathrm{\text{−}}V$ and $I\mathrm{\text{−}}\Phi$ relationships of a Josephson junction.