$\mathbb{Z}_3$ quantum double in a superconducting wire array

TL;DR

This paper proposes a way to realize a $ ext{Z}_3$ quantum double in a superconducting wire array with specific flux and coupling configurations, leading to topological order and phase transitions, guided by combinatorial gauge symmetry.

Contribution

It introduces a novel physical implementation of a $ ext{Z}_3$ quantum double using superconducting wires, leveraging combinatorial gauge symmetry and dimerization patterns.

Findings

Realization of $ ext{Z}_3$ quantum double in superconducting wire arrays.

Identification of a dimerization pattern that induces topological order.

Prediction of a phase transition to XY order via capacitance pattern modification.

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 inter-wire 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 non-zero 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 three-state 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 XY-ordered phase. Our work highlights that combinatorial gauge symmetry can serve as a design principle to build quantum double models using systems with realistic interactions.