A $\mathbb{Z}_3$ quantum double in a superconducting wire array
Abstract
We show that a $\mathbb{Z}_3$ quantum double can be realized in an array of superconducting wires coupled via Josephson junctions. With a suitably chosen magnetic flux threading the system, the interwire Josephson couplings take the form of a complex Hadamard matrix, which possesses combinatorial gauge symmetry  a local $\mathbb{Z}_3$ symmetry involving permutations and shifts by $\pm 2\pi/3$ of the superconducting phases. The sign of the star potential resulting from the Josephson energy is inverted in this physical realization, leading to a massive degeneracy in the nonzero flux sectors. A dimerization pattern encoded in the capacitances of the array lifts up these degeneracies, resulting in a $\mathbb{Z}_3$ topologically ordered state. Moreover, this dimerization pattern leads to a larger effective vison gap as compared to the canonical case with the usual (uninverted) star term. We further show that our model maps to a quantum threestate Potts model under a duality transformation. We argue, using a combination of bosonization and mean field theory, that altering the dimerization pattern of the capacitances leads to a transition from the $\mathbb{Z}_3$ topological phase into a quantum XYordered phase. Our work highlights that combinatorial gauge symmetry can serve as a design principle to build quantum double models using systems with realistic interactions.
 Publication:

arXiv eprints
 Pub Date:
 January 2021
 arXiv:
 arXiv:2101.01720
 Bibcode:
 2021arXiv210101720Y
 Keywords:

 Condensed Matter  Strongly Correlated Electrons;
 Condensed Matter  Mesoscale and Nanoscale Physics;
 Quantum Physics
 EPrint:
 Added an appendix on the construction of the canonical $\mathbb{Z}_3$ quantum double, updated references and fixed typos. 16 pages, 14 figures