Novel scaling behavior of the Ising model on curved surfaces

Hasegawa, Isaku; Sakaniwa, Yasunori; Shima, Hiroyuki

doi:10.1016/j.susc.2007.04.207

Cited by 15 publications

(19 citation statements)

References 51 publications

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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 . If defining a ratio of the boundary sites to the total number of sites, the ratio converges to zero in the Euclidean case, whereas it goes to non-zero values on the hyperbolic lattices in the thermodynamic limit.…”

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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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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“…Contrary to the case considered in [15], the crystal structure induces anisotropy and the actual geometry is shown in Fig. 1.…”

mentioning

confidence: 95%

Axial hypertoroidal moment in a ferroelectric nanotorus: A way to switch local polarization

et al. 2014

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International audienceGrowth of ferroelectric nanotori is reported and first-principles-based effective Hamiltonian simulations were performed on these new objects. They could reproduce the nonpolar (phase I) and homogeneously toroidized (phase II) states of an isolated nanotorus. Computation of an axial hypertoroidal moment leads to numerical observation of two new phases: (i) a homogeneously hypertoroidized one (phase III) that can be switched by a homogeneous electric field and (ii) another one with striking newpolarization patterns (phase IV) due to azimuthal variations of the hypertoroidal moment. In both phases, hypertoroidization coexists with a homogeneous axial toroidal moment

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“…The universality class of the ferromagnetic-paramagnetic phase transition of this model has been so far considered to be mean-field like. Recent numerical studies have supported this conjecture [6,7,8,9].…”

Section: Introductionmentioning

confidence: 80%

Tricritical point of theJ1−J2Ising model on a hyperbolic lattice

et al. 2008

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A ferromagnetic-paramagnetic phase transition of the two-dimensional frustrated Ising model on a hyperbolic lattice is investigated by use of the corner transfer matrix renormalization group method. The model contains ferromagnetic nearest-neighbor interaction J 1 and the competing antiferromagnetic interaction J 2 . A mean-field like second-order phase transition is observed when the ratio κ = J 2 /J 1 is less than 0.203. In the region 0.203 < κ < 1/4, the spontaneous magnetization is discontinuous at the transition temperature. Such tricritical behavior suggests that the phase transitions on hyperbolic lattices need not always be mean-field like.

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Novel scaling behavior of the Ising model on curved surfaces

Cited by 15 publications

References 51 publications

Free-energy analysis of spin models on hyperbolic lattice geometries

Free-energy analysis of spin models on hyperbolic lattice geometries

Axial hypertoroidal moment in a ferroelectric nanotorus: A way to switch local polarization

Tricritical point of theJ1−J2Ising model on a hyperbolic lattice

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