超导线阵中的$\mathbb{Z}_3$量子双
$\mathbb{Z}_3$ quantum double in a superconducting wire array
摘要：
我们证明了一个$\mathbb{Z}_3$量子双可以实现在一个阵列的超导导线耦合通过约瑟夫森结。通过适当选择穿过系统的磁通量，线间约瑟夫森耦合采取复杂的阿达玛矩阵的形式，其具有组合规范对称性--涉及超导相的$\pm 2π/3$的置换和移位的局部$\mathbb{Z}_3$对称性。在这个物理实现中，由约瑟夫森能量产生的恒星势的符号被反转，导致非零通量扇区中的大规模简并。编码在阵列电容中的二聚化模式提升了这些简并性，导致$\mathbb{Z}_3$拓扑有序状态。此外，这种二聚化模式导致一个更大的有效视子间隙相比，通常的（uninverted）星条款的典型情况。我们进一步证明，我们的模型映射到一个量子三态Potts模型下的对偶变换。我们认为，使用玻色子化和平均场理论的组合，改变二聚模式的电容导致从$\mathbb{Z}_3$拓扑相到量子XY有序相的过渡。我们的工作突出了组合规范对称性可以作为一个设计原则，以建立量子双模型使用系统与现实的相互作用。
英文摘要：
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 $\pm 2π/3$ 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 $\mathbb{Z}_3$ 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.
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