Geometric effects on critical behaviours of the Ising model

Shima, Hiroyuki; Sakaniwa, Yasunori

doi:10.1088/0305-4470/39/18/010

Cited by 60 publications

(84 citation statements)

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“…More importantly, in the latter model, the surface curvature effects manifest themselves in scaling behavior in a different way from that in the donut-shaped Ising model. In what follows, we will give a brief review of our previous study on the Ising model on the pseudosphere [37,38]. By comparing the results obtained for the two distinct curved surfaces, we gain a deeper understanding of the effect of geometry on the Ising model.…”

Section: Ising Models On Negatively Curved Surfacesmentioning

confidence: 99%

Novel scaling behavior of the Ising model on curved surfaces

2007

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We demonstrate the nontrivial scaling behavior of Ising models defined on (i) a donut-shaped surface and (ii) a curved surface with a constant negative curvature. By performing Monte Carlo simulations, we find that the former model has two distinct critical temperatures at which both the specific heat C(T ) and magnetic susceptibility χ(T ) show sharp peaks. The critical exponents associated with the two critical temperatures are evaluated by the finite-size scaling analysis; the result reveals that the values of these exponents vary depending on the temperature range under consideration. In the case of the latter model, it is found that static and dynamic critical exponents deviate from those of the Ising model on a flat plane; this is a direct consequence of the constant negative curvature of the underlying surface.

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Section: Ising Models On Negatively Curved Surfacesmentioning

confidence: 99%

Novel scaling behavior of the Ising model on curved surfaces

2007

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“…This structure has been verified having a nontrivial impact on the critical behavior of many models of statistical physics. For example, in the context of the Ising model, significant shifts in static and dynamic critical exponents toward the mean-field values were noticed [1,2]; small-sized ferromagnetic domains were observed to exist at temperatures far greater than the critical temperature [4]; An apparent zero-temperature orientational glass transition in the XY spin model on a negatively curved surface was recently demonstrated [6].…”

Section: Introductionmentioning

confidence: 99%

Majority-vote model on hyperbolic lattices

Holme

2010

Phys. Rev. E

114

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We study the critical properties of a non-equilibrium statistical model, the majority-vote model, on heptagonal and dual heptagonal lattices. Such lattices have the special feature that they only can be embedded in negatively curved surfaces. We find, by using Monte Carlo simulations and finite-size analysis, that the critical exponents 1/ν, β/ν and γ/ν are different from those of the majority-vote model on regular lattices with periodic boundary condition, which belongs to the same universality class as the equilibrium Ising model. The exponents are also from those of the Ising model on a hyperbolic lattice. We argue that the disagreement is caused by the effective dimensionality of the hyperbolic lattices. By comparative studies, we find that the critical exponents of the majority-vote model on hyperbolic lattices satisfy the hyperscaling relation 2β/ν + γ/ν = D eff , where D eff is an effective dimension of the lattice. We also investigate the effect of boundary nodes on the ordering process of the model.

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“…This argument is based on the fact that a negatively curved surface contains a huge amount of boundary points: that is, for a negatively curved surface, the ratio of surface area to perimeter (which is the * Corresponding author, E-mail: beomjun@skku.edu two-dimensional example of the so-called surface-volume ratio in general dimension) remain nonvanishing even in the large-system limit. 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.…”

Section: Introductionmentioning

confidence: 99%

Phase transition ofq-state clock models on heptagonal lattices

et al. 2009

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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.

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Geometric effects on critical behaviours of the Ising model

Cited by 60 publications

References 50 publications

Novel scaling behavior of the Ising model on curved surfaces

Novel scaling behavior of the Ising model on curved surfaces

Majority-vote model on hyperbolic lattices

Phase transition ofq-state clock models on heptagonal lattices

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