Phase transition ofq-state clock models on heptagonal lattices

Abstract

We study the q-state clock models on heptagonal lattices assigned on a negatively curved surface. We show that the system exhibits three classes of equilibrium phases; in between ordered and disordered phases, an intermediate phase characterized by a diverging susceptibility with no magnetic order is observed at every q ≥ 2. The persistence of the third phase for all q is in contrast with the disappearance of the counterpart phase in a planar system for small q, which indicates the significance of nonvanishing surface-volume ratio that is peculiar in the heptagonal lattice. Analytic arguments based on Ginzburg-Landau theory and generalized Cayley trees make clear that the two-stage transition in the present system is attributed to an energy gap of spin-wave excitations and strong boundary-spin contributions. We further demonstrate that boundary effects breaks the mean-field character in the bulk region, which establishes the consistency with results of clock models on boundary-free hyperbolic lattices.

Majority-vote model on hyperbolic lattices

“…The critical exponents 1/ν, β/ν, and γ/ν are estimated in Figs. 7(a), (b) and (d) to 0.32(9), 0.093(6), and 0.761 (9) respectively. The data collapse of the magnetization in Fig.…”

“…Heptagonal lattice and dual heptagonal lattice Figure 1 shows two examples of the heptagonal lattice and the dual heptagonal lattice. One peculiar property of the heptagonal lattice is that, if we consider the innermost heptagon as the level one, then the number of nodes of a heptagonal lattice with level l can be calculated by using the formulation [4,9],…”

“…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].…”

Geometry-Property Relations in Physics on Curved Surfaces

“…On the issue, Dr. Seung Ki Baek studied low-temperature properties of the XY- 28) and q-state 29) spin models on a negatively curved surface. Geometric curvature of the surface gives rise to frustration in local spin configuration, which results in the formation of high-energy spin clusters scattered over the system.…”

Analysis of quantum spin models on hyperbolic lattices and Bethe lattice

“…A remarkable demand for an appropriate numerical tool persists. Implementation of the Monte Carlo simulations fails due to exponential increase of the number of the lattice sites for models on hyperbolic lattices with respect to the expanding lattice size from the lattice center [11,12]. Our desire is to propose a novel and sufficiently accurate numerical algorithm, which originates from the solid state physics and inherits the typical features coming from widely accepted renormalization group approaches, especially based on the Density Matrix Renormalization Group [13,14,15].…”