Monte Carlo calculation of the transition temperature of the anisotropic three-dimensional<i>XY</i>model

Costa, B. V.; Pereira, A. R.; Pires, A.S.T.

doi:10.1103/physrevb.54.3019

Cited by 25 publications

(40 citation statements)

References 23 publications

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“…From hereon the present approach means the results obtained by solving the non-linear system of equations (39) and (40) numerically (by using standard iterative procedures) for the variational parameters. As can be seen from this figure the results are quite similar to those from reference [16] in the region close to the isotropic three-dimensional case J z ∼ J. On the other hand, the slope of the phase boundary close to isotropic two-dimensional case J z = 0 is zero according to the present approach while the procedure from reference [16] for the XY model (and also for the planar rotator model [15]) furnishes a positive slope.…”

Section: Numerical Results Parametric Procedures and Simple Assumptionssupporting

confidence: 84%

“…Again, we find the same transition temperature for the two-dimensional model t c = 1.472 and, for the three-dimensional model, t c = 2.190 [16]. The transition temperatures for D = 0 and D → ∞ are given in Table I for the isotropic twoand three-dimensional models together with those coming from other approaches.…”

Section: Gsupporting

confidence: 72%

“…The transition temperature for the two-dimensional model J z = 0, as well as the isotropic three-dimensional model J z = J, are also identical to those from reference [16], respectively, t c = 1.076 and t c = 1.605. However, it is worth to stress here that the present approach not only is a generalization over the model treated by Costa et al [16] to other values of the axial interaction J z , but also includes, as it will be seen below, the important contribution of the crystal field interaction D.…”

Section: Gsupporting

confidence: 61%

“…The above model has been treated according to the self-consistent harmonic approximation (SCHA) in the case D = 0 for the three-dimensional model [16] as well as the planar rotator model [15] (in this case there is no z spin components). The approach in [16], however, only considers the cases J z ≈ J and J z << J, that is, quasi-isotropic case or weak layer coupling and no general treatment has been done for any value of 0 ≤ J z ≤ J.…”

Section: Introductionmentioning

confidence: 99%

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Free-energy variational approach for the classical anisotropicXYmodel in a crystal field

2007

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A variational approach for the free energy is used to study the three-dimensional anisotropic XY model in the presence of a crystal field. The magnetization and the phase diagrams as a function of the parameters of the Hamiltonian are obtained. Some limiting results for isotropic XY and planar rotator models in two and three dimensions are analyzed and compared to previous results obtained from analytical approximations as well as from those obtained from more reliable approaches such as series expansion and Monte Carlo simulations. It is also shown that from this general variational approach some simple assumptions can drastically simplify the self-consistent implicit equations. The validity of the low temperature region of this approach is analyzed and compared to Monte Carlo results as well.

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Section: Numerical Results Parametric Procedures and Simple Assumptionssupporting

confidence: 84%

Section: Gsupporting

confidence: 72%

Section: Gsupporting

confidence: 61%

Section: Introductionmentioning

confidence: 99%

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Free-energy variational approach for the classical anisotropicXYmodel in a crystal field

2007

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“…Several applications to classical systems were found to agree very well with Monte Carlo and experimental results. 10,11 The SCHA was also used in the study of the 1D quantum sine-Gordon problem, where it describes correctly the phase transition of the model. The reason is that it is equivalent to a renormalization-group analysis to one loop.…”

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Phase diagram of the antiferromagneticXYmodel in two dimensions in a magnetic field

2008

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The phase diagram of the quasi-two-dimensional easy-plane antiferromagnetic model, with a magnetic field applied in the easy plane, is studied using the self-consistent harmonic approximation. We found a linear dependence of the transition temperature as a function of the field for large values of the field. Our results are in agreement with experimental data for the spin-1 honeycomb compound BaNi 2 V 2 O 3 . DOI: 10.1103/PhysRevB.78.212408 PACS number͑s͒: 75.10.Hk, 75.30.Kz, 75.40.Mg The XY model in two dimensions provides the best example of a phase transition mediated by topological defects. A variety of analytical and numerical methods have been presented in the literature in an attempt to fully understand the nature of its transition. As it is well known the model has a phase transition at a temperature T BKT called the Berezinskii-Kosterlitz-Thouless ͑BKT͒ temperature. 1,2 This phase transition is associated with the emergence of a topological order, resulting from the pairing of vortices with opposite circulation. The BKT mechanism does not involve any spontaneous symmetry breaking 3 and emergence of a spatially uniform order parameter. The low-temperature phase is associated with a quasi-long-range order for finite T, with the correlation of the order parameter decaying algebraically in space. Above the critical temperature the correlations decay exponentially. This picture is applicable to a wide variety of two-dimensional phenomena. 4 A recent experiment in a trapped atomic gas 5 not only confirms the BKT theory in a new system, but also reveals for the first time the role played by local topological defects or vortices. A very convenient technique to study the XY model is the self-consistent harmonic approximation ͑SHCA͒. The SHCA was originally proposed by Pokrovsky and Uimin 6 to study the twodimensional ͑2D͒ classical planar rotor model. Later, Minnhagen 7 pointed out that the SCHA overestimated the transition temperature because it did not take into account vortex fluctuations and he suggested a way to improve the thermodynamics of the planar rotor by replacing the exchange constant J with a "renormalized" J͑T͒. This procedure leads to a better estimate of T BKT . Menezes et al. 8 extended the SCHA to the classical XY model and Pires 9 applied it to the quantum model. The approximation consists in replacing the Hamiltonian of the system by an effective harmonic Hamiltonian with renormalized parameters. Several applications to classical systems were found to agree very well with Monte Carlo and experimental results. 10,11The SCHA was also used in the study of the 1D quantum sine-Gordon problem, where it describes correctly the phase transition of the model. The reason is that it is equivalent to a renormalization-group analysis to one loop. 12 To test the reliability of the quantum SCHA we will compare the transition temperature calculated theoretically with experimental results, for two S = 1 compounds. 13 In these cases we have an exchange anisotropy of the form A͑S i z S j z ͒. The calculati...

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Spin transport in the two-dimensional quantum disordered anisotropic Heisenberg model

Lima

Pires

Costa

2014

Journal of Magnetism and Magnetic Materials

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Monte Carlo calculation of the transition temperature of the anisotropic three-dimensionalXYmodel

Cited by 25 publications

References 23 publications

Free-energy variational approach for the classical anisotropicXYmodel in a crystal field

Free-energy variational approach for the classical anisotropicXYmodel in a crystal field

Phase diagram of the antiferromagneticXYmodel in two dimensions in a magnetic field

Spin transport in the two-dimensional quantum disordered anisotropic Heisenberg model

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