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Progress of Theoretical Physics

Monte Carlo Simulation and Static and Dynamic Critical Behavior of the Plane Rotator Model

Seiji Miyashita, Hidetoshi Nishimori, A. Kuroda, Masuo Suzuki

Published Dec 1, 1978• 145 citations
Publisher

Abstract

The properties of phase transition and the lower-temperature phase of the plane rotator model are studied in the square lattice with Monte Carlo method. Physical results thus obtained for this model are quite similar to those of the two-dimensional quantum XY-model,1) that is, the susceptibility diverges and the specific heat does not diverge at the critical point. Furthermore the size-dependence of physical quantities and scaling relations among some temperature-dependent indices are also investigated.

DOI
10.1143/ptp.60.1669

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The properties of phase transition and the lower-temperature phase of the plane rotator model are studied in the square lattice with Monte Carlo method. Physical results thus obtained for this model are quite similar to those of the two-dimensional quantum XY-model,1) that is, the susceptibility diverges and the specific heat does not diverge at the critical point. Furthermore the size-dependence of physical quantities and scaling relations among some temperature-dependent indices are also investigated.
Results of Monte Carlo simulations of the ferromagnetic classical XXZ model on a square lattice are reported for interactions of the out of plane spin components \ensuremath{\lambda} varying between 0 and 0.99. For all the value of \ensuremath{\lambda} the signature of a Berezinskii-Kosterlitz-Thouless (BKT) phase transition was observed, and the corresponding critical temperature was located both by BKT fits and finite-size scaling methods. Data for in-plane and out of plane susceptibility, correlation length, specific heat, and internal energy are given in a broad range of temperature and for lattice sizes ranging from 32\ifmmode\times\else\texttimes\fi{}32 to 256\ifmmode\times\else\texttimes\fi{}256.
The critical behavior of 2D magnetic easy-plane systems has mainly been studied by the classical XY model. However, the z components of the spins have to be considered in order to describe real systems, and their fluctuations cannot positively be neglected when quantum effects are to be included, quantum spins being intrinsically three-component objects. Therefore, Monte Carlo simulations are performed for the Heisenberg ferromagnet with easy-plane anisotropy (XXZ model) on a two-dimensional square lattice with a twofold aim: first, to obtain accurate quantitative results about the critical behavior of the classical model, showing the relevant role played by the out-of-plane fluctuations; second, to open the way for approaching the quantum thermodynamics by means of the effective Hamiltonian method that reduces the quantum thermodynamics of the XXZ ferromagnet to the investigation of an effective classical model with temperature-dependent renormalized interaction parameters. Specific heat, magnetic susceptibility, and correlation length are calculated in the critical region for lattice sizes up to 128×128. These quantities preserve the Kosterlitz–Thouless behavior of the XY model. For the transition temperature of the classical XX0 model we obtain the estimate kBTc/(JS̃2)=0.69±0.01.
We study the classical XY (plane rotator) model at the Kosterlitz–Thouless phase transition. We simulate the model using the single-cluster algorithm on square lattices of a linear size up to L = 2048. We derive the finite-size behaviour of the second moment correlation length over the lattice size ξ 2nd / L at the transition temperature. This new prediction and the analogous one for the helicity modulus ϒ are confronted with our Monte Carlo data. This way β KT = 1.1199 is confirmed as inverse transition temperature. Finally, we address the puzzle of logarithmic corrections of the magnetic susceptibility χ at the transition temperature.
We have performed Monte Carlo calculations of the thermodynamic properties of the bond-dilute Ising model on the square lattice. The results are obtained for the magnetization, internal energy, specific heat, and susceptibility for several concentrations of missing bonds. The dependence of the Curie temperature on dilution is in good agreement with the results obtained by other methods. The phase transition is found to remain sharp with the same value of the susceptibility exponent as for the pure Ising model.
In this work we have used extensive Monte Carlo simulations and finite size scaling theory to study the phase transition in the dipolar planar rotator model (dPRM), also known as dipolar XY model. The true long-range character of the dipolar interactions was taken into account by using the Ewald summation technique. Our results for the critical exponents do not fit those from known universality classes. We observed that the specific heat is apparently non-divergent and the critical exponents are ν = 1.277(2), β = 0.2065(4) and γ = 2.218(5). The critical temperature was found to be T(c) = 1.201(1). Our results are clearly distinct from those of a recent renormalization group study from Maier and Schwabl (2004 Phys. Rev. B 70 134430) and agrees with the results from a previous study of the anisotropic Heisenberg model with dipolar interactions in a bilayer system using a cut-off in the dipolar interactions (Mól and Costa 2009 Phys. Rev. B 79 054404).