Kitaev spin models from topological nanowire networks

We show that networks of superconducting topological nanowires can realize the physics of exactly solvable Kitaev spin models on trivalent lattices. This connection arises from the low-energy theory of both systems being described by a tight-binding model of Majorana modes. In Kitaev spin models the Majorana description provides a convenient representation to solve the model, whereas in an array of Josephson junctions of topological nanowires it arises from localized physical Majorana modes tunneling between the wire ends. We explicitly show that an array of junctions of three wires—a setup relevant to topological quantum computing with nanowires—can realize the Yao-Kivelson model, a variant of Kitaev spin models on a decorated honeycomb lattice. Employing properties of the latter, we show that the network can be constructed to give rise to two-dimensional collective topological states characterized by Chern numbers $\nu =0,\pm 1$, and $\pm 2$, and that defects in the array can be associated with vortex-like quasiparticle excitations. In addition we show that the collective states are stable in the presence of disorder and superconducting phase fluctuations. When the network is operated as a quantum information processor, the connection to Kitaev spin models implies that decoherence mechanisms can in general be understood in terms of proliferation of the vortex-like quasiparticles.

Figure 1

(a) The 3-junction nanowire array. Majorana states ${\gamma }_{b\left(w\right)}$ are denoted with black (white) circles. (b) The Yao-Kivelson variant of Kitaev spin models on the decorated honeycomb lattice. The couplings $J_{\alpha }$ span the triangular plaquettes, while the couplings $J_{\alpha }^{\prime }$ connect them.

Figure 2

The fermion gap in the vortex-free sector (squares) and vortex gap (circles) calculated from the effective Majorana model (9) with derived couplings (10) (solid line) and from the full microscopic array model (dashed lines) presented in Appendix pp1. By vortex gap we mean the ground state energy difference between the vortex-free sector and the a sector with two neighboring vortices. The full microscopic model consists of 48 wires each of length $L=20$ (19 sites with lattice spacing $a=1$). The used parameters are $t=1,\mu =-1$, and $\Delta =0.15$ that correspond to a $L/\xi =1.5$ and $L/{\lambda }_F\approx 3$. The interwire tunnel couplings that model the Josephson couplings span $\tau \in [0,0.6]$ and the superconducting phases are taken equal, i.e., corresponding to $\beta =1/3$.

Figure 3

Phase diagram of the 3-junction wire array as a function of the Josephson couplings (here $S=J/sin\delta \varphi$) and the array deformation parameter $\beta$ [which gives the relative superconducting phases ${\varphi }_n=2\pi (n-1)\beta$]. This parametrization enables us to consider the pure tunneling effects separately from the array geometry that, under our simplifying assumptions, gives the superconducting phases. The $\nu =0$ phases in the $S/J^{\prime }\ll 1$ regime correspond to parameter ranges where the wire end Majoranas do no hybridize, whereas in the $\nu =\pm 1$ phases they form an extended collective state. The $\nu =0$ phases inside these phases emerge when $R>2J^{\prime }$ is satisfied with the Josephson couplings $J_{\alpha }$ being unequal, as studied in Ref. ([57]). On the right we illustrate the uniform Y-K and brick wall array geometries that are obtained for $\beta =1/3$ and $1/4$, respectively.

Figure 4

The vortex lattices and higher Chern number phases in the Y-K model. (a) By alternating the lengths of the wires (or equivalently staggering the chemical potential), the tunneling couplings can become sign staggered such that each thin black link has $J$ while every thick red link has $-J$. This corresponds to having a $\pi$-flux vortex ($W^{\left(12\right)}=-1$) on every dodecagonal plaquette. (b) In the presence of such staggering, we find that the gap closure moves to a larger value of $J/J^{\prime }$ and the collective state is now characterized by $\nu =2$. Both features are consistent with the studies in the honeycomb model [22].

Figure 5

Illustrations for (a) an array with alternating 6- and 3-junctions and (b) a 4-junction array. Majorana states ${\gamma }_{b\left(w\right)}$ are denoted with black (white) circles.

Figure 6

The phase diagram for the 4-junction array with $S^{\left(1\right)}=S^{\left(2\right)}=S$.

Figure 7

The phase diagram for an array of alternating 3- and 6-junctions for (a) $S^{\left(1\right)}=S^{\left(2\right)}=S^{\left(3\right)}=S$ and (b) $S^{\left(1\right)}=S$,$S^{\left(2\right)}=0.9S$,$S^{\left(3\right)}=0.7S$. The white regions are gapless.

Figure 8

A basis rotation allows one to write all $J^{\prime }$ links as ${\sigma }^z{\sigma }^z$ and all $J$ links as ${\sigma }^x{\sigma }^y$. The corresponding plaquette symmetries are given in the right hand panel.

Figure 9

The spin representation offers a simple picture of both Aharonov-Bohm and Aharonov-Casher processes at the effective level. Panels (a) and (b) correspond to electron/hole motion in a closed loop. In this case we show the process for the situation where there is only one electron shared between all links. In panel (c) we show how a vortex tunneling through the Josephson links corresponds to the process $-{\sigma }^z{\sigma }^z=-2(c^{†}c-I)$ measuring the parity of the enclosed wire. The branch cuts which connect vortices indicate all of the Josephson couplings connecting to the wire are changed by $-1$, corresponding with a $2\pi$ phase slip of the superconducting phase.