$8\pi$-periodic dissipationless ac Josephson effect on a quantum spin Hall edge via a quantum magnetic impurity

Time-reversal invariance places strong constraints on the properties of the quantum spin Hall edge. One such restriction is the inevitability of dissipation in a Josephson junction between two superconductors formed on such an edge without the presence of interaction. Interactions and spin-conservation breaking are key ingredients for the realization of the dissipationless ac Josephson effect on such quantum spin Hall edges. We present a simple quantum impurity model that allows us to create a dissipationless fractional Josephson effect on a quantum spin Hall edge. We then use this model to substantiate a general argument that shows that any such nondissipative Josephson effect must necessarily be $8\pi$ periodic.

Figure 1

(a) The system with a short Josephson junction whose Andreev state is tunnel coupled to a quantum dot. The quantum dot is coupled to another localized spin [Eq. (1)]. (b) Schematic diagram showing the $8\pi$ cycle of states. $|\uparrow ,\downarrow \rangle$ represent the states of the spin and $d_{\uparrow ,\downarrow }^{†}$ represents the electron in the quantum dot. $J,J_A$ represents the spin-symmetric and -asymmetric exchange interactions inside the dot, respectively. In each $2\pi$ cycle of phase shown by the bold arrow a spin-up electron is pumped into or a spin-down hole leaves the edge.

Figure 2

The (many-electron) energy spectrum of a coupled SC junction/QD/spin system [Eqs. (1, 2, 3)] as a function of superconducting phase $\varphi$ with $J=0.2\mathrm{\Delta }$ and ${\varepsilon }_d=0.5\mathrm{\Delta }$ and (a) $J_A=0$ (spin-conserving case) and (b) $J_A=J$ (spin-anisotropic case). The solid lines represent odd total fermion parity (Kramers degenerate) states and the dashed lines represent even total fermion parity states. The fermion parity changes with each $2\pi$ cycle of $\varphi$ and the color represents the curves corresponding to various eigenstates. Following the spectrum in panel (a) we find that the ground state necessarily couples to the bulk states. In the spin-conservation breaking in panel (b), the spectrum is isolated from the bulk states and $8\pi$ periodic.

Figure 3

(a) A short Josephson junction on a one-dimensional TI edge. (b) Equivalent construction where now the phase difference is controlled by threading a flux through the center of the Corbino disk. (c) The single-particle Andreev spectrum for the junction with (blue thin lines) or without (black thick lines) the FI element. (d) The many-body $\mathrm{E}\mathrm{\Phi }\mathrm{R}$ with (blue thin lines) or without (black thick lines) the FI element, where solid (dotted) lines indicate states with even (odd) parity.

Figure 4

(a) $\mathrm{E}\mathrm{\Phi }\mathrm{R}$ for a junction with length $L=\pi v_F/{\mathrm{\Delta }}_0$. The solid (dotted) lines are solutions with eigenvalues $s_z=\pm 1$ (b) Many-body spectrum for the same noninteracting junction. Black (dotted) lines are states with even (odd) parity. The labels indicate which of the ABSs are occupied. (c) Many-body spectrum for a junction with $s_z$-conserving interactions. Detailed calculations are presented in Appendix pp2.

Figure 5

The path of an adiabatically followed even-parity state. Blue (red) and light blue (light red) lines are time-reversed paths of each other. The crucial point is $n\ne 0$, because otherwise ${|0\rangle }_0$ would join to $|\tilde{m}{\rangle }_{2\pi }$ and ${|m\rangle }_{2\pi }$ simultaneously.

Figure 6

For an odd-parity state we define $m$ and $n$ as the states that Kramers pair $\left\{\right|0_0\rangle ,|{\tilde{0}}_0\rangle \}$ traverses as a flux quantum is inserted in either direction. Here again, deep-colored and light-colored lines are time-reversed paths of each other. Use Eq. (7) to shift the light blue and light red lines by $4\pi$, we get the dashed path, completing the full $8\pi$-periodic path that the state follows.