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Two-dimensional ferromagnetic N-state clock models are studied on ahyperbolic lattice represented by tessellation of pentagons. The lattice lieson the hyperbolic plane with a constant negative scalar curvature. We observethe spontaneous magnetization, the internal energy, and the specific heat atthe center of sufficiently large systems, where the fixed boundary conditionsare imposed, for the cases N>=3 up to N=30. The model with N=3, which isequivalent to the 3-state Potts model on the hyperbolic lattice, exhibits thefirst order phase transition. A mean-field like phase transition of the secondorder is observed for the cases N>=4. When N>=5 we observe the Schottky typespecific heat below the transition temperature, where its peak hight at lowtemperatures scales as N^{-2}. From these facts we conclude that the phasetransition of classical XY-model deep inside the hyperbolic lattices is not ofthe Berezinskii-Kosterlitz-Thouless type.

Phase transition of clock models on hyperbolic lattice studied by corner transfer matrix renormalization group method