Fractional Spin and Josephson Effect in Time-Reversal-Invariant Topological Superconductors

Time-reversal-invariant topological superconducting (TRITOPS) wires are known to host a fractional spin $\hslash /4$ at their ends. We investigate how this fractional spin affects the Josephson current in a TRITOPS-quantum dot-TRITOPS Josephson junction, describing the wire in a model that can be tuned between a topological and a nontopological phase. We compute the equilibrium Josephson current of the full model by continuous-time Monte Carlo simulations and interpret the results within an effective low-energy theory. We show that in the topological phase, the 0-to-$\pi$ transition is quenched via formation of a spin singlet from the quantum-dot spin and the fractional spins associated with the two adjacent topological superconductors.

Figure 1

Top: Josephson current vs $\varphi$ for a quantum dot at $T=0$ with $U=0$, $t^{\prime }=t$, $\lambda =t/2$, $\mathrm{\Delta }=t/5$, and values of $\mu$ in the topological ($\mu <t$) as well as the nontopological ($\mu >t$) phase. The wires have $N=500$ sites. Inset: Josephson current at finite temperature. Red, blue, and green lines correspond to $\beta =400,200$, and 100, respectively. The $T=0$ case is plotted in black for reference. Bottom: Spectrum of ${\widehat{\mathcal{H}}}_{\text{BdG}}$ for $\mu ={ϵ}_d=0$ (left), $\mu =0$, ${ϵ}_d=t$ (middle), and $\mu =-{ϵ}_d=1.8$ (right). Other parameters are as they are in the top panel. Energies are measured in units of $t=1$.

Figure 2

Josephson current for an interacting quantum dot at $\beta =400$ with $t^{\prime }=t$, ${ϵ}_d=-U/2$, $\lambda =t/2$, $\mathrm{\Delta }=t/5$. The upper (lower) panel corresponds to the topological (nontopological) phase. The values of $U$ and $\mu$ are indicated in the figure. Energies are expressed in units of $t=1$.

Figure 3

Local density of states (LDOS) at the quantum dot obtained by QMC. The black dashed lines are the predictions for the peaks in the density of states on the basis of $H_{\mathrm{eff}}$ with $J=0.2$. The amplitude of the superconducting gap $\mathrm{\Delta }=0.2$ is indicated in thin lines. Other parameters are $U=4t$, $J=0.2$, $\lambda =0.5t$, $t^{\prime }=t$, $\mu =0$, and $\beta =400$.