Monte Carlo study of the planar spin model

Tobochnik, Jan; Chester, G. V.

doi:10.1103/physrevb.20.3761

Cited by 449 publications

(211 citation statements)

References 21 publications

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“…Many Monte Carlo simulations of this model have been done in order to check the predictions of the theory. Among them, we quote those of Tobochnik and Chester [22] and of Gupta and Baillie [23] which, on the basis of accurate numerical analysis, confirmed the predictions of the theory and fixed the critical temperature at T c = 0.89 (J = 1).…”

Section: D Xy Modelmentioning

confidence: 85%

Hamiltonian dynamics and geometry of phase transitions in classicalXYmodels

2000

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The Hamiltonian dynamics associated to classical, planar, Heisenberg XY models is investigated for two-and three-dimensional lattices. Besides the conventional signatures of phase transitions, here obtained through time averages of thermodynamical observables in place of ensemble averages, qualitatively new information is derived from the temperature dependence of Lyapunov exponents. A Riemannian geometrization of newtonian dynamics suggests to consider other observables of geometric meaning tightly related with the largest Lyapunov exponent. The numerical computation of these observables -unusual in the study of phase transitions -sheds a new light on the microscopic dynamical counterpart of thermodynamics also pointing to the existence of some major change in the geometry of the mechanical manifolds at the thermodynamical transition. Through the microcanonical definition of the entropy, a relationship between thermodynamics and the extrinsic geometry of the constant energy surfaces ΣE of phase space can be naturally established. In this framework, an approximate formula is worked out, determining a highly non-trivial relationship between temperature and topology of the ΣE. Whence it can be understood that the appearance of a phase transition must be tightly related to a suitable major topology change of the ΣE. This contributes to the understanding of the origin of phase transitions in the microcanonical ensemble.

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Section: D Xy Modelmentioning

confidence: 85%

Hamiltonian dynamics and geometry of phase transitions in classicalXYmodels

2000

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“…Computer simulations have been performed on other nonseparable and separable potentials. Whilst quasi-LRO has been observed for the separable models studied, 15,16 contrasting evidence is found for nonseparable potentials. 9,[16][17][18] Indeed, for ellipses, the nature of the transition appears to depend on the aspect ratio, and the phase diagram possesses a tricritical point at which the transition changes from first order to continuous.…”

Section: D Nematic Phasesmentioning

confidence: 96%

Phase behavior of two-dimensional hard rod fluids

Bates¹,

Frenkel²

2000

The Journal of Chemical Physics

241

254

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Monte Carlo simulations are used to study two-dimensional hard rod fluids consisting of spherocylinders confined to lie in a plane. The phase behavior is mapped out as a function of the aspect ratio (L/D) of the particles, from the hard disc limit at one extreme (L/Dϭ0) to the thin hard needle limit at the other (L/Dϭϱ). For long rods, a 2D nematic phase is observed at high density in which the orientational correlation functions decay algebraically, indicating that the phase does not possess true long range orientational order. The simulation data indicate that the transition from this phase to the low density isotropic phase is continuous, via a Kosterlitz-Thouless disclination unbinding type mechanism, rather than first order. For short rods the nematic phase disappears so that, on expansion, the solid phase undergoes a first order transition directly to an isotropic phase. Although the latter phase is globally isotropic, we find evidence for strong local positional and orientational correlations between the particles. Where possible, the simulation results are compared and contrasted to experimental, simulation and theoretical data for other two-dimensional liquid crystalline systems.

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“…7 of Ref. [9]). The problem one encounters with direct visualization is that the spins move too fast to allow the observation of coherent structures.…”

Section: The Phase Transitionmentioning

confidence: 99%

“…However, there is a number of observations in the literature that seems to complicate this simple picture of the phase transition mechanism. Among the different results those worth mentioning are: the visualization of the vortex distribution [9], the presence of domains delimited by topological defects [10], the precise determination of the transition temperature T KT ≈ 0.89 [11], much lower than the transition temperature of the Villain model. Results are also found on the interaction between vortices [10], or the vortex interaction energy [12].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Hamiltonian dynamics and the phase transition of theXYmodel

Leoncini

Verga²,

Ruffo³

1998

Phys. Rev. E

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A Hamiltonian dynamics is defined for the XY model by adding a kinetic energy term. Thermodynamical properties (total energy, magnetization, vorticity) derived from microcanonical simulations of this model are found to be in agreement with canonical Monte-Carlo results in the explored temperature region. The behavior of the magnetization and the energy as functions of the temperature are thoroughly investigated, taking into account finite size effects. By representing the spin field as a superposition of random phased waves, we derive a nonlinear dispersion relation whose solutions allow the computation of thermodynamical quantities, which agree quantitatively with those obtained in numerical experiments, up to temperatures close to the transition. At low temperatures the propagation of phonons is the dominant phenomenon, while above the phase transition the system splits into ordered domains separated by interfaces populated by topological defects. In the high temperature phase, spins rotate, and an analogy with an Ising-like system can be established, leading to a theoretical prediction of the critical temperature T KT ≈ 0.855.

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Monte Carlo study of the planar spin model

Cited by 449 publications

References 21 publications

Hamiltonian dynamics and geometry of phase transitions in classicalXYmodels

Hamiltonian dynamics and geometry of phase transitions in classicalXYmodels

Phase behavior of two-dimensional hard rod fluids

Hamiltonian dynamics and the phase transition of theXYmodel

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