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 $\gamma \cos (q\phi )$ term to the ordinary O(2) model with angular values restricted to a $2\pi$ interval. In the $\gamma \to \infty$ 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 $2\pi$ 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-$\beta$ peak is associated with a crossover, and the large-$\beta$ 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 $\mathrm{\Delta }q$ and the system is close to the small-$\beta$ Berezinskii-Kosterlitz-Thouless transition, the system has a magnetic susceptibility that scales as $\sim 1/(\mathrm{\Delta }q{)}^{1-1/{\delta }^{\prime }}$ with ${\delta }^{\prime }$ estimates consistent with the magnetic critical exponent $\delta =15$. 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 $(\beta ,q)$ 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.