$U(1)$ symmetry-enriched toric code

Abstract: We propose and study a generalization of Kitaev's $\mathbb Z_2$ toric code on a square lattice with an additional global $U(1)$ symmetry. Using Quantum Monte Carlo simulation, we find strong evidence for a topologically ordered ground state manifold with indications of UV/IR mixing, i.e., the topological degeneracy of the ground state depends on the microscopic details of the lattice. Specifically, the ground state degeneracy depends on the lattice tilt relative to the directions of the torus cycles. In particular, we observe that while the usual compactification along the vertical/horizontal lines of the square lattice shows a two-fold ground state degeneracy, compactifying the lattice at $45^\circ$ leads to a three-fold degeneracy. In addition to its unusual topological properties, this system also exhibits Hilbert space fragmentation. Finally, we propose a candidate experimental realization of the model in an array of superconducting quantum wires.