The field of two-dimensional classical spin systems has been studied for many decades using a variety of analytical and numerical methods. This field of study contains many interesting models including the O(2) model, which has an infinite-order Berezinskii-Kosterlitz-Thouless (BKT) transition and the class of q-state clock models which have second-order phase transitions when q = 2, 3, 4 but BKT transitions when q > 5. Motivated by attempts to quantum simulate lattice models with continuous Abelian symmetries using discrete approximations, we study an extended-O(2) model that differs from the ordinary O(2) model by the addition of an explicit symmetry breaking term. Its coupling allows us to smoothly interpolate between the O(2) model (zero coupling) and a q-state clock model (infinite coupling). In the latter case, a q-state clock model can also be defined for non-integer values of q. Thus, such a limit can also be considered as an analytic continuation of an ordinary q-state clock model to non-integer q. In the infinite coupling limit, the extended-O(2) model can be simplified, and so we start by establishing the phase diagram in that case. Using Monte Carlo and tensor methods, we show that for non-integer q, there is a second-order phase transition at low temperature and a crossover at high temperature. Next we establish the phase diagram at finite values of the coupling again using both Monte Carlo and tensor methods. We show that for non-integer q, the second-order phase transition at low temperature and crossover at high temperature persist to finite coupling. For integer q = 2, 3, 4, there is a second-order phase transition at infinite coupling (i.e. the clock models). At intermediate coupling, there are second-order phase transitions, but the critical exponents vary with the coupling. At small coupling, the second-order phase transitions may turn into BKT transitions.
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