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

Abstract

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>or=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>or=4 . When N>or=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.

Majority-vote model on hyperbolic lattices

“…The susceptibility values in the critical region reach their maximal values [ Fig. 4(b)], and the reduced fourth-order cumulants cross at the critical point, giving q c = 0.034 (8).…”

“…Recently, there has been a growing interest the critical behavior of statistical-physics models on curved surfaces-ranging from spin models, such as the ferromagnetic Ising model [1,2,3,4], the XY model [5,6], the Heisenberg model [7], the q-state clock models [8,9], to other traditional models, such as percolation [10], diffusion [11], etc. One reason for this interest is that many newly discovered soft materials (e.g., carbon nanotubes) show a negatively curved structure in the nanoscale [12].…”

Phase transition ofq-state clock models on heptagonal lattices

“…Hence, the present system involving the boundary effects will exhibit distinct properties from the meanfield character observed in Ref. [14] wherein the boundary effects are artificially excluded. We also note that our discussion in the previous paragraph supports the validity of the mean-field description in the boundary-free system since the correlation function decays so fast at T c [27].…”

“…We also comment that the spin-wave excitation is observable in the hyperbolic lattice without boundary since it is the basic excitation mode. In the latter system, however, the excitation is not sufficient to destroy the ordered phase but arises as a separate peak in specific heat at q > 4 [14].…”

“…Since it was pointed out that a system may have a novel behavior due to the presence of a nonvanishing boundary [7], there have been ongoing studies to clarify this issue [4,6,[8][9][10][11]. While the boundary effects can be sometimes excluded, for example, by using a periodic boundary condition [12] or by mathematical abstractions [5,[13][14][15], it is often crucial to understand how a boundary affects the physical properties since it may give the most important contribution to an observed behavior as will be explained in this work.…”