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

Two-dimensional ferromagnetic $N$-state clock models are studied on a hyperbolic lattice represented by tessellation of pentagons. The lattice lies on the hyperbolic plane with a constant negative scalar curvature. We observe the spontaneous magnetization, the internal energy, and the specific heat at the center of sufficiently large systems, where fixed boundary conditions are imposed, for the cases $N\ge 3$ up to $N=30$. The model with $N=3$, which is equivalent to the three-state Potts model on the hyperbolic lattice, exhibits a first-order phase transition. A mean-field-like phase transition of second order is observed for the cases $N\ge 4$. When $N\ge 5$ we observe a Schottky-type specific heat below the transition temperature, where its peak height at low temperatures scales as $N^{-2}$. From these facts we conclude that the phase transition of the classical $XY$ model deep inside hyperbolic lattices is not of the Berezinskii-Kosterlitz-Thouless type.

Figure 1

Pentagonal lattice drawn on the Poincaré disk. The open circles represent the $N$-state spin variables ${\theta }_i$. Two geodesics drawn by thick arcs divide the system into four equivalent quadrants.

Figure 2

Temperature dependence of the spontaneous magnetization ${\mathcal{M}}^{\left(N\right)}$ for $3\le N\le 30$. The open circle denotes the discontinuity in ${\mathcal{M}}^{\left(3\right)}$.

Figure 3

Square of ${\mathcal{M}}^{\left(N\right)}$ with respect to $\left(T_0^{\left(N\right)}-T\right)/T_0^{\left(N\right)}$.

Figure 4

Absolute value of the internal energy $\left|{\mathcal{E}}^{\left(N\right)}\right|$. The open circles denote the jump in the case $N=3$.

Figure 5

Rescaled specific heat $C^{\left(N\right)}/C_{\text{max}}^{\left(N\right)}$ versus the rescaled temperature $T/T_0^{\left(N\right)}$. The inset shows a typical example for the case $N=10$ without rescaling.

Figure 6

Schottky peak position $T_{\text{Sch}}^{\left(N\right)}$ versus $1/N^2$.