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

Understand

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 $2$.