Clock model interpolation and symmetry breaking in O(2) models

Abstract

Motivated by recent attempts to quantum simulate lattice models with continuous Abelian symmetries using discrete approximations, we define an extended-O(2) model by adding a γcos(qφ) term to the ordinary O(2) model with angular values restricted to a 2π interval. In the γ→∞ limit, the model becomes an extended q-state clock model that reduces to the ordinary q-state clock model when q is an integer and otherwise is a continuation of the clock model for noninteger q. By shifting the 2π integration interval, the number of angles selected can change discontinuously and two cases need to be considered. What we call case 1 has one more angle than what we call case 2. We investigate this class of clock models in two space-time dimensions using Monte Carlo and tensor renormalization group methods. Both the specific heat and the magnetic susceptibility show a double-peak structure for fractional q. In case 1, the small-β peak is associated with a crossover, and the large-β peak is associated with an Ising critical point, while both peaks are crossovers in case 2. When q is close to an integer by an amount Δq and the system is close to the small-β Berezinskii-Kosterlitz-Thouless transition, the system has a magnetic susceptibility that scales as ∼1/(Δq)1-1/δ′ with δ′ estimates consistent with the magnetic critical exponent δ=15. The crossover peak and the Ising critical point move to Berezinskii-Kosterlitz-Thouless transition points with the same power-law scaling. A phase diagram for this model in the (β,q) plane is sketched. These results are possibly relevant for configurable Rydberg-atom arrays where the interpolations among phases with discrete symmetries can be achieved by varying continuously the distances among atoms and the detuning frequency.