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.
Highlights
The identification of possible experimental realizations of topologically ordered states of matter remains a central problem in condensed-matter physics
The fractional quantum Hall (FQH) effects [1,2] are the quintessential and best-characterized topological states. Both the fractional charge [3,4] and, more recently, the fractional statistics [5,6] of the quasiparticle excitations of Abelian FQH states have been experimentally measured. In addition to their fundamental importance, topological phases such as those associated with non-Abelian FQH states have potential application to topological quantum computation
Having established the local combinatorial gauge symmetry of Hamiltonian (1), we look at the minima of the classical potential energy HJ on a single waffle
Summary
The identification of possible experimental realizations of topologically ordered states of matter remains a central problem in condensed-matter physics. We provide the first example outside of the family of the simplest type of Z2 topological order, and construct a quantum double for the group Z3 using superconducting wire arrays. The particular construction discussed in this paper has the following interesting feature: the star potential that usually constrains states to lie in the zero flux sector is inverted, i.e., the states with nonzero flux have the lowest energy This inverted potential by itself would lead to an extensive degeneracy, but the degeneracy can be lifted by a dimerization pattern encoded in the capacitances of the array. In the appendices we present details of the calculations, including the estimates of the fields that enter in the clock model as function of the microscopic parameters of the superconducting wire array
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