  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 $\gamma \cos(q\varphi)$ term to the ordinary
O(2) model with angular values restricted to a $2\pi$ 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 $2\pi$ 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 $\Delta q$
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.
