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 $gamma cos(qvarphi)$ term to the ordinary O(2) model with angular values restricted to a $2pi$ interval. In the $gamma rightarrow infty$ 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 $2pi$ 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 $Delta q$ and the system is close to the small-$beta$ Berezinskii-Kosterlitz-Thouless transition, the system has a magnetic susceptibility that scales as $sim 1 / (Delta q)^{1 - 1/delta}$ with $delta$ 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.
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