| Abstract: |
| Superfluidity in superfluid helium and ultracold atoms is characterized by broken continuous $U(1)$ symmetry. In contrast to these clean systems, superconductors are highly influenced by discrete symmetry originating from crystal structure and multi-band features. To investigate the influence of the discrete symmetry, we consider the two-dimensional Gross-Pitaevskii model having the discrete $\mathbb{Z}_n$ clock symmetry and the characteristics of the superfluid phase transition. The superfluid phase transition is characterized by the number $n$ and the strength $g$ of the discrete symmetry $\mathbb{Z}_n$. When $g$ is large, the superfluid phase transition is first order. When $g$ is small, on the other hand, the phase transition is 2nd order for $n \le 4$, and BKT-type for $n \ge 5$. Our numerical results are consistent with the variational renormalization group analysis. |
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