We show that a 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 Z 3 symmetry involving permutations and shifts by ±2π/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 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 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.
We show that a
Z3 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
Z3 symmetry involving permutations and shifts by
±2π/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
Z3 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
Z3 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.
We propose a superconducting quantum circuit based on a general symmetry principle-combinatorial gauge symmetry-designed to emulate topologically ordered quantum liquids and serve as a foundation for the construction of topological qubits.The proposed circuit exhibits rich features: in the classical limit of large capacitances its ground state consists of two superimposed loop structures; one is a crystal of small loops containing disordered U(1) degrees of freedom, and the other is a gas of loops of all sizes associated with Z 2 topological order.We show that these classical results carry over to the quantum case, where phase fluctuations arise from the presence of finite capacitances, yielding Z 2 quantum topological order.A key feature of the exact gauge symmetry is that amplitudes connecting different Z 2 loop states arise from paths having zero classical energy cost.As a result, these amplitudes are controlled by dimensional confinement rather than tunneling through energy barriers.We argue that this effect may lead to larger energy gaps than previous proposals that are limited by such barriers, potentially making it more likely for a topological phase to be experimentally observable.Finally, we discuss how our superconducting circuit realization of combinatorial gauge symmetry can be implemented in practice.
Electric-magnetic self-duality of the
Z2 gauge theory, realized microscopically as a half-lattice-translation exchanging electric charge and magnetic flux, has been an influential example of a duality symmetry with an exact lattice realization. We construct the first non-Abelian generalization of this construction: a lattice model of the
S3 quantum double
D(S3) on a tensor product Hilbert space in which the
Z2em anyon-permutation symmetry, exchanging the non-Abelian chargeon
C and fluxon
F, is realized via lattice translation. Consequently we find that the zigzag boundary termination of the model realizes, without fine-tuning, a gapless critical edge state described by the tetracritical Ising CFT. The bulk admits three independent
Z2em-preserving bosonic perturbations, driving
D(S3) into distinct gapped phases. We analyze these transitions by three independent methods: category-theoretic anyon condensation, microscopic lattice Hamiltonians, and Chern-Simons-Higgs theory, which all agree, yielding a unified picture. These examples motivate a minimal-condensation principle: proliferating a bosonic anyon generically drives condensation of a minimal condensable algebra containing it, with symmetry-related condensates appearing as degenerate vacua that spontaneously break the anyon-permutation symmetry. Our model construction extends to an infinite family of self-dual dihedral quantum doubles
D(D2n). Notably, each model is sign-problem-free, opening the door to large-scale numerical exploration of the phases of non-Abelian Chern-Simons-Higgs theories.
Evidence is reported that topological effects in graph-shaped arrays of superconducting islands can condition superconducting energy gap and transition temperature. The carriers giving rise to the new phase are couples of electrons (Cooper pairs) which, in the superconducting state, behave as predicted for bosons in our structures. The presented results have been obtained both on star and double comb-shaped arrays and the coupling between the islands is provided by Josephson junctions whose potential can be tuned by external magnetic field or temperature. Our peculiar technique for probing distribution on the islands is such that the hopping of bosons between the different islands occurs because their thermal energy is of the same order of the Josephson coupling energy between the islands. Both for star and double comb graph topologies the results are in qualitative and quantitative agreement with theoretical predictions.
Circuit quantum electrodynamics embeds Josephson junction qubits within superconducting cavities, and has emerged as a leading approach to quantum computing and quantum simulation. Despite the many permutations of circuit geometry that have been explored, Josephson connectivities have so far been planar, making them effectively low-dimensional. Here we show that a non-planar qubit -- a
3×3 crossbar Josephson array -- gives rise to flux-tunable
Z3 combinatorial gauge symmetry (CGS), potentially enabling spin-liquid behavior when networked into a lattice. The observed excitation spectrum shows excellent agreement with predictions from a neural network trained to generate variational quantum states, demonstrating that we have predictive power over our high-dimensional quantum system. Fine-structure splittings near the CGS point are compatible with weak tunneling or symmetry breaking due to experimental imperfections. We additionally use the superconducting cavity to externally induce symmetry breaking, observing a restoration of symmetry at the CGS point where ground states differ only by a
Z3 phase. This work initiates a general program exploring lattice gauge theories using the toolbox of circuit quantum electrodynamics. More broadly, introducing non-planar Josephson connectivities opens a vast space for experimental and theoretical exploration of structures in almost any imaginable dimensionality and geometry.
Superconductivity provides a canonical example of a quantum phase of matter. When superconducting islands are connected by Josephson junctions in a lattice, the low-temperature state of the system can map to the celebrated XY model and its associated universality classes. This has been used to experimentally implement realizations of Mott insulator and Berezinskii-Kosterlitz-Thouless transitions to vortex dynamics analogous to those in type-II superconductors. When an external magnetic field is added, the effective spins of the XY model become frustrated, leading to the formation of topological defects (vortices). Here, we observe the many-body dynamics of such an array, including frustration, via its coupling to a superconducting microwave cavity. We take the design of the transmon qubit, but replace the single junction between two antenna pads with the complete array. This allows us to probe the system at 10 mK with minimal self-heating by using weak coherent states at the single (microwave) photon level to probe the resonance frequency of the cavity. We observe signatures of an ordered vortex lattice at rational flux fillings of the array.