Fractionalization in Josephson junction arrays hinged by quantum spin Hall edges

We study a superconductor-ferromagnet-superconductor (SC-FM-SC) Josephson array deposited on top of a two-dimensional quantum spin Hall insulator. The Majorana bound state at the interface between SC and FM leads to charge-$e$ tunneling between neighboring superconductor islands, in addition to the usual charge-$2e$ Cooper pair tunneling. Moreover, because Majorana fermions encode the information of charge number parity, an exact $Z_2$ gauge structure naturally emerges and leads to many new phases, including a deconfined phase where electrons fractionalize into charge-$e$ bosons and topological defects. A deconfined SC-insulator transition has also been found.

Figure 1

(Color online) (a) The lattice structure for case 1. The green circles are the Majorana fermions and the dashed brown line is the edge of the underlying QSH insulator. (b) The phase diagram for ordinary SC Josephson array. (c) The phase diagram for Josephson array in our case, where the number of MI lobes doubles compared with (b).

Figure 2

(Color online) (a) The Josephoson lattice structure of case 2. (b) The effective lattice of the Josephson array in (a). The SC islands are denoted by $j$ and the $Z_2$ gauge field ${\sigma }^x$ and ${\sigma }^z$ are defined on links $\left(j,j+\nu \right)$. The dual Ising variables ${\tau }^a$ are defined on the sites $\overline{j}$ of the dual lattice, represented by red dashed lines. (c) Case 3 with compressed $y$ direction, the intraisland tunneling $Ji{\gamma }_{j,3}{\gamma }_{j,4}$ is denoted by the pink dashed lines. (d) The phase diagram of case 3 with integer $\overline{n}$, plotted against $t_2/U$ and $J/U$. Phase A, B, and C represent the SF, the $Z_2$ liquid, and the ${\sigma }^x$ ordered phase, respectively. The green line between phase A and B is a 3D XY transition and the transition between B and C is a first-order transition described by Eq. (18). The direct transition between phase A and C may expand into a stable roton liquid phase.

Figure 3

(Color online) (a) The electron at site $j,1$ [Fig. 2a] can be represented as $\text{exp}\left(i{\varphi }_j/2\right)$ times a string of ${\sigma }^z$, denoted by red circles; by binding two visons (blue circles), the fermion becomes a boson. (b) The Wilson loop operator as a product of ${\sigma }^z$ (red circles) in the dual formalism is a product of ${\tau }^x$ on the dual lattice (blue circles). In the $Z_2$ liquid phase, the perturbation of $J$ term will only change the Wilson loop at the corner, represented by the green dashed square.