${\mathbb{Z}}_4$ parafermions in an interacting quantum spin Hall Josephson junction coupled to an impurity spin

${\mathbb{Z}}_4$ parafermions can be realized in a strongly interacting quantum spin Hall Josephson junction or in a spin Hall Josephson junction strongly coupled to an impurity spin. In this paper, we study a system that has both features, but with weak (repulsive) interactions and a weakly coupled spin. We show that for a strongly anisotropic exchange interaction, at low temperatures the system enters a strong coupling limit in which it hosts two ${\mathbb{Z}}_4$ parafermions, characterizing a fourfold degeneracy of the ground state. We construct the parafermion operators explicitly and show that they facilitate fractional $e/2$ charge tunneling across the junction. The dependence of the effective low-energy spectrum on the superconducting phase difference reveals an $8\pi$ periodicity of the supercurrent.

Figure 1

Schematic description of the setup. Two superconductors (SC) are connected via a quantum spin Hall insulator (QSHI), with counter propagating chiral electronic modes along its edge. The edge of the QSHI is further coupled to an quantum spin. The channel on the opposite edge was omitted for clarity.

Figure 2

Schematic description of characteristic renormalization group flow of the backscattering strength for interacting (solid line) and noninteracting (dashed) edge. The strength of the backscattering is plotted on a logarithmic scale.

Figure 3

The three different regimes discussed in the main text. (a) The weak exchange regime, where the exchange couplings are much smaller than the superconducting gap, and the interactions with the impurity spin are treated perturbatively. (b) The Kondo regime, where the spin impurity is almost completely screened, and the anisotropic nature of the exchange coupling gives rise to residual interactions between the edge electrons. (c) The strong backscattering regime, where the junction is effectively cut into two separate parts, weakly interacting with each other.

Figure 4

The many-body low-energy spectrum as a function of the phase bias $\mathrm{\Phi }$ across the junction, for (a) the strong backscattering regime and (b) the Kondo regime with weak residual interactions. In panel (a), the hopping between levels within the subspace was taken as $t_{pi/2}=4t_{\pi }$ (see Sec. 6.) In panel (b), the energy is given in units of the superconducting gap, and as a gap opens the four lowest lying states are separated from it. The black circle points to the protected Kramers doublet due to the time-reversal symmetry. Here, the length of the junction was taken as $L/\left(\pi \mathrm{\Delta }{v}_F\right)=1$, in order to ensure that enough Andreev bound states exist and interact in the junction simultaneously. The parameters for the residual exchange coupling and potential scattering are given in Appendix pp2. For both plots, the $8\pi$ periodicity does not depend on the specific choice of parameters.

Figure 5

The renormalization group flow of the energy scale $J_B/\left(\pi a\right)$ given in Eq. (19) for two different sets of bare exchange couplings and for different strengths of electron-electron interactions in the junction. The continuous lines correspond to bare value of $J_B/\left(\pi aD\right)\simeq 0.27$ while the dashed lines correspond to $J_B/\left(\pi aD\right)\simeq 0.16$. The values of all nine exchange couplings for each case are given in Appendix pp1. It should be stressed here that the flow cannot be derived only from the bare value of $J_B$ and $g$, but a full knowledge of all of the exchange couplings is needed.

Figure 6

(a) Schematic depiction of the magnitude of the lowest energy of ${\mathcal{H}}_B$ as a function of the field at $\theta \left(0\right)$. The values for which the local Hamiltonian is at the ground state are denoted by ${\theta }_0+m\pi /2$ and ${\mathcal{H}}_B$ pins the field to these minima. The orientation of the impurity spin (black arrows) alternates between adjacent ground states. (b) The instanton process of the tunneling of $\theta \left(0\right)$ through a potential barrier of height $\sim J_B/\left(\pi a\right)$ that weakly couples two adjacent states within the ground state manifold. The tunneling process is accompanied by the flipping of the impurity spin, which here is schematically depicted as following the effective magnetic field of Eq. (41).

Figure 7

Numerical calculation of the classical instanton action $S_0$, for different randomly chosen values of ${\mathbf{J}}_x$ and ${\mathbf{J}}_y$. In these calculations, the full dynamics of the impurity spin were considered, and the adiabatic limit was not taken explicitly. The red line shows a perfect square root behavior, while the blue dots are the results of the numerical calculations.