Chiral topologically ordered insulating phases in arrays of interacting integer quantum Hall islands

We study networks of Coulomb-blockaded integer quantum Hall islands with even fillings $\nu =2k$ ($k$ being an integer), including cases with $2k$ layers each of $\nu =1$ fillings. Allowing only spin-current interactions between the islands (i.e., without any charge transfer), we obtain solvable models leading to a rich set of insulating $SU{\left(2\right)}_k$ topologically ordered phases. The case with $k=1$ is dual to the Kalmeyer-Laughlin phase, $k=2$ to Kitaev's chiral spin liquid and the Moore-Read state, and $k=3$ contains a Fibonacci anyon that may be utilized for universal topological quantum computation. Additionally, we show how the $SU{\left(2\right)}_k$ topological phases may be obtained also in an array of islands with $\nu =2k$ integer quantum Hall states and critical spin chains in a checkerboard pattern. The array and checkerboard constructions gap out the charge mode and additional “flavor” modes by virtue of their geometry. Furthermore, we find that a fine tuning of the system parameter is not needed in the checkerboard configuration and the $\nu =2$ case. We also discuss their bulk excitations, and show that their thermal Hall conductance is universal, reflecting the central charge $c=3k/(k+2)$ of the chiral edge modes.