Quantum Spin Models with Exponential U(1) Symmetry

We study the quantum breakdown model generalized in 1D spin systems, which has an exponential U(1) symmetry with the charge exponentially decaying versus real space position. We show that for large spin systems, the ground state exhibits a transition from ferromagnetic phase to a quantum breakdown condensate phase, which spontaneously break the exponential U(1) symmetry and has an exponentially large ground state degeneracy resembling a glass. The two phases are related a particle-vertex duality. For small spin systems, symmetry protected topological phases can be realized in this model, which is analogous to the Haldane phase.