Phase transition of XY model in heptagonal lattice

Baek, Seung Ki; Minnhagen, Petter; Kim, Beom Jun

doi:10.1209/0295-5075/79/26002

Cited by 24 publications

(35 citation statements)

References 19 publications

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“…5(b), we display the N -dependence of the magnetization at q c . From the slope of the dashed line, which corresponds to the best fit to the data points, we estimate the corresponding value of the critical exponent to β/ν = 0.114 (5). Using these values we proceed to plot M N β/ν against (q − q c )N 1/ν .…”

Section: Numerical Results and Finite Size Scaling Analysismentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Numerical Results and Finite Size Scaling Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Majority-vote model on hyperbolic lattices

Holme

2010

Phys. Rev. E

114

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“…Strong boundary effects on the hyperbolic lattices prevent the Monte Carlo (MC) simulations from the accurate analysis of phase transition phenomena on the hyperbolic lattices [20][21][22][23] . The necessity to subtract a couple of boundary site layers were performed to detect the correct bulk properties 24 .…”

Section: A Absence Of Phase Transition On Non-euclidean Latticesmentioning

confidence: 99%

Free-energy analysis of spin models on hyperbolic lattice geometries

Serina¹,

Genzor²,

Lee³

et al. 2016

Phys. Rev. E

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We investigate relations between spatial properties of the free energy and the radius of Gaussian curvature of the underlying curved lattice geometries. For this purpose we derive recurrence relations for the analysis of the free energy normalized per lattice site of various multistate spin models in the thermal equilibrium on distinct non-Euclidean surface lattices of the infinite sizes. Whereas the free energy is calculated numerically by means of the corner transfer matrix renormalization group algorithm, the radius of curvature has an analytic expression. Two tasks are considered in this work. First, we search for such a lattice geometry, which minimizes the free energy per site. We conjecture that the only Euclidean flat geometry results in the minimal free energy per site regardless of the spin model. Second, the relations among the free energy, the radius of curvature, and the phase transition temperatures are analyzed. We found out that both the free energy and the phase transition temperature inherit the structure of the lattice geometry and asymptotically approach the profile of the Gaussian radius of curvature. This achievement opens new perspectives in the AdS-CFT correspondence theories.

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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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Phase transition of XY model in heptagonal lattice

Cited by 24 publications

References 19 publications

Majority-vote model on hyperbolic lattices

Majority-vote model on hyperbolic lattices

Free-energy analysis of spin models on hyperbolic lattice geometries

Phase transition ofq-state clock models on heptagonal lattices

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