Physical review. B, Condensed matter

Finite-size effects at the first-order phase transition in the three-dimensional Heisenberg model

Kun Chen, D. P. Landau

Published Jul 1, 1992• 6 citations

Abstract

Finite-size effects on the first-order phase transition in a three-dimensional classical Heisenberg model with nearest-neighbor ferromagnetic interactions and periodic boundary conditions are studied theoretically and numerically in this paper. We find that the current Fisher-Privman theory fails at temperatures close to criticality and/or for small systems. We rederive explicit scaling functions for the longitudinal magnetization and susceptibility under another phenomenological assumption. The results are in very good agreement with Monte Carlo simulations.

DOI: 10.1103/physrevb.46.937

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Physical review. B, Condensed matter

Finite-size effects at the first-order phase transition in the three-dimensional Heisenberg model

Kun Chen, D. P. Landau

Published Jul 1, 1992• 6 citations

Publisher

Abstract

DOI: 10.1103/physrevb.46.937

Related papers

Asymptotic freedom and finite-size scaling of the two-dimensional classical Heisenberg model

Dingyun Yao, Chao Zhang, Z. Y. Xie +2 more

The classical Heisenberg model is one of the most fundamental models in statistical and condensed matter physics. Extensive theoretical and numerical studies suggest that, in two dimensions, this model does not exhibit a finite-temperature phase transition but instead manifests asymptotic freedom. However, some research has also proposed the possibility of a Berezinskii-Kosterlitz-Thouless (BKT) phase transition over the years. In this study, we revisit the classical two-dimensional (2D) Heisenberg model through large-scale simulations with linear system sizes up to $L=16384$ . Our Monte Carlo data, without any extrapolation, clearly reveal an exponential divergence of the correlation length $\ensuremath{\xi}$ as a function of the inverse temperature $\ensuremath{\beta}$, a hallmark of asymptotic freedom. Moreover, extrapolating $\ensuremath{\xi}$ to the thermodynamic limit in the low-temperature regime achieves close agreement with the three-loop perturbative calculations. We further propose a finite-size scaling (FSS) ansatz for $\ensuremath{\xi}$, demonstrating that the pseudocritical point ${\ensuremath{\beta}}_{L}$ diverges logarithmically with $L$ . The thermodynamic and finite-size scaling behaviors of the magnetic susceptibility $\ensuremath{\chi}$ are also investigated and corroborate the predictions of asymptotic freedom. Our paper provides solid evidence for asymptotic freedom in the 2D Heisenberg model and advances understanding of finite-size scaling in such systems.The classical Heisenberg model is one of the most fundamental models in statistical and condensed matter physics. Extensive theoretical and numerical studies suggest that, in two dimensions, this model does not exhibit a finite-temperature phase transition but instead manifests asymptotic freedom. However, some research has also proposed the possibility of a Berezinskii-Kosterlitz-Thouless (BKT) phase transition over the years. In this study, we revisit the classical two-dimensional (2D) Heisenberg model through large-scale simulations with linear system sizes up to $L=16384$. Our Monte Carlo data, without any extrapolation, clearly reveal an exponential divergence of the correlation length $\ensuremath{\xi}$ as a function of the inverse temperature $\ensuremath{\beta}$, a hallmark of asymptotic freedom. Moreover, extrapolating $\ensuremath{\xi}$ to the thermodynamic limit in the low-temperature regime achieves close agreement with the three-loop perturbative calculations. We further propose a finite-size scaling (FSS) ansatz for $\ensuremath{\xi}$, demonstrating that the pseudocritical point ${\ensuremath{\beta}}_{L}$ diverges logarithmically with $L$. The thermodynamic and finite-size scaling behaviors of the magnetic susceptibility $\ensuremath{\chi}$ are also investigated and corroborate the predictions of asymptotic freedom. Our paper provides solid evidence for asymptotic freedom in the 2D Heisenberg model and advances understanding of finite-size scaling in such systems.

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Using a Monte Carlo procedure with a self-consistent-field boundary condition, the magnetization of a simple cubic classical Heisenberg ferromagnet with nearest-neighbor interactions only is calculated. For reduced temperatures and reduced fields in the range $0.02<~|\frac{T}{{T}_{c}}\ensuremath{-}1|$, $\frac{H}{J}<~0.5$ the data of this computer experiment can be described in terms of the "effective" critical exponents $0.31<~\ensuremath{\beta}<~0.35$, $1.33<~\ensuremath{\gamma}<~1.39$, and $4.9<~\ensuremath{\delta}<~5.3$. Although the present data are not very close to the critical point they do obey the homogeneity requirement and determine the scaling function rather precisely. This function agrees very well with the scaling function for the face-centered-cubic classical Heisenberg magnet derived recently by Milo\ifmmode \check{s}\else \v{s}\fi{}evi\ifmmode \acute{c}\else \'{c}\fi{} and Stanley using high-temperature---series-expansion techniques. This agreement supports their hypothesis that neither critical exponents nor the scaling function depend on the lattice structure in the Heisenberg model.Using a Monte Carlo procedure with a self-consistent-field boundary condition, the magnetization of a simple cubic classical Heisenberg ferromagnet with nearest-neighbor interactions only is calculated. For reduced temperatures and reduced fields in the range $0.02<~|\frac{T}{{T}_{c}}\ensuremath{-}1|$, $\frac{H}{J}<~0.5$ the data of this computer experiment can be described in terms of the "effective" critical exponents $0.31<~\ensuremath{\beta}<~0.35$, $1.33<~\ensuremath{\gamma}<~1.39$, and $4.9<~\ensuremath{\delta}<~5.3$. Although the present data are not very close to the critical point they do obey the homogeneity requirement and determine the scaling function rather precisely. This function agrees very well with the scaling function for the face-centered-cubic classical Heisenberg magnet derived recently by Milo\ifmmode \check{s}\else \v{s}\fi{}evi\ifmmode \acute{c}\else \'{c}\fi{} and Stanley using high-temperature---series-expansion techniques. This agreement supports their hypothesis that neither critical exponents nor the scaling function depend on the lattice structure in the Heisenberg model.

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Marcelo D. Grynberg, Horacio Ceva

The magnetic critical exponent of the two-dimensional axial next-nearest-neighbor Ising model is estimated by means of the finite-size scaling of the order parameter. For a finite system without ordering field, this magnitude is computed from the two leading eigenvectors of the transfer matrix. The ferromagnetic-paramagnetic transition temperatures estimated with this method exhibit a small lattice-size dependence. Moreover, they are in good agreement with those obtained using phenomenological renormalization, and also with analytical approximations. The correlation-length critical exponent is calculated by means of phenomenological renormalization. The results found with these numerical techniques indicate that the ferromagnetic-paramagnetic transition of the model seems to be continuous, in contradiction with a previous result obtained using the persistence-length criterion, which indicated that the transition is always first order. We found that the phase transition is first order only when the ratio of the next-nearest-neighbor and nearest-neighbor interactions equals one-half.The magnetic critical exponent of the two-dimensional axial next-nearest-neighbor Ising model is estimated by means of the finite-size scaling of the order parameter. For a finite system without ordering field, this magnitude is computed from the two leading eigenvectors of the transfer matrix. The ferromagnetic-paramagnetic transition temperatures estimated with this method exhibit a small lattice-size dependence. Moreover, they are in good agreement with those obtained using phenomenological renormalization, and also with analytical approximations. The correlation-length critical exponent is calculated by means of phenomenological renormalization. The results found with these numerical techniques indicate that the ferromagnetic-paramagnetic transition of the model seems to be continuous, in contradiction with a previous result obtained using the persistence-length criterion, which indicated that the transition is always first order. We found that the phase transition is first order only when the ratio of the next-nearest-neighbor and nearest-neighbor interactions equals one-half.

Physical review. B, Condensed matterNov 1, 19886 citations

Finite-size scaling at first-order phase transitions

Kurt Binder, D. P. Landau

Using thermodynamic fluctuation theory, we study the finite-size rounding of anomalies occurring at first-order phase transitions of the corresponding infinite system. Explicit expressions for thermodynamic functions are derived both for "symmetric transitions" (such as the jump of the spontaneous magnetization in the Ising model from $+{M}_{\mathrm{sp}}$ to $\ensuremath{-}{M}_{\mathrm{sp}}$ as the field changes from ${0}^{+}$ to ${0}^{\ensuremath{-}}$) as well as for asymmetric cases, but restricting attention to (hyper)cubic system shapes. As an explicit example for the usefulness of these considerations in Monte Carlo simulations where it may be a problem to (i) locate a phase transition and (ii) distinguish first-order from second-order transitions, we present numerical results for the two-dimensional nearest-neighbor Ising ferromagnet in a field, both below the critical temperature ${T}_{c}$ and at ${T}_{c}$ . The numerical results are found to be in very good agreement with the phenomenological theory and it is shown that one may extract the magnitudes of jumps occurring at first-order phase transitions in a well-defined and accurate way.Using thermodynamic fluctuation theory, we study the finite-size rounding of anomalies occurring at first-order phase transitions of the corresponding infinite system. Explicit expressions for thermodynamic functions are derived both for "symmetric transitions" (such as the jump of the spontaneous magnetization in the Ising model from $+{M}_{\mathrm{sp}}$ to $\ensuremath{-}{M}_{\mathrm{sp}}$ as the field changes from ${0}^{+}$ to ${0}^{\ensuremath{-}}$) as well as for asymmetric cases, but restricting attention to (hyper)cubic system shapes. As an explicit example for the usefulness of these considerations in Monte Carlo simulations where it may be a problem to (i) locate a phase transition and (ii) distinguish first-order from second-order transitions, we present numerical results for the two-dimensional nearest-neighbor Ising ferromagnet in a field, both below the critical temperature ${T}_{c}$ and at ${T}_{c}$. The numerical results are found to be in very good agreement with the phenomenological theory and it is shown that one may extract the magnitudes of jumps occurring at first-order phase transitions in a well-defined and accurate way.

Physical review. B, Condensed matterAug 1, 1984634 citations

Finite-size scaling of O(n) models with singular behaviour

Scott Allen, Raj K. Pathria

The finite-size scaling hypothesis of Privman and Fisher (Phys. Rev. B, 30, 322 (1984)) is applied to systems with O(n) symmetry [Formula: see text], confined to geometry Ld − d′ × ∞d′ (where d and d′ are continuous variables such that 2 < d′ < d < 4) and subjected to periodic boundary conditions. Predictions, involving amplitudes as well as exponents, are made on the singular part of the specific heat c(s), spontaneous magnetization m0, magnetic susceptibility χ0, and correlation length ξ0 in the region of the second-order phase transition [Formula: see text]. Analytical verification of these predictions is carried out in the case of the spherical model of ferromagnetism (n = ∞), which includes evaluation of the shift in the critical temperature of the system.The finite-size scaling hypothesis of Privman and Fisher (Phys. Rev. B, 30, 322 (1984)) is applied to systems with O(n) symmetry [Formula: see text], confined to geometry Ld − d′ × ∞d′ (where d and d′ are continuous variables such that 2 < d′ < d < 4) and subjected to periodic boundary conditions. Predictions, involving amplitudes as well as exponents, are made on the singular part of the specific heat c(s), spontaneous magnetization m0, magnetic susceptibility χ0, and correlation length ξ0 in the region of the second-order phase transition [Formula: see text]. Analytical verification of these predictions is carried out in the case of the spherical model of ferromagnetism (n = ∞), which includes evaluation of the shift in the critical temperature of the system.

Canadian Journal of PhysicsOct 1, 198913 citations

L=16384

. Our Monte Carlo data, without any extrapolation, clearly reveal an exponential divergence of the correlation length $\ensuremath{\xi}$ as a function of the inverse temperature $\ensuremath{\beta}$, a hallmark of asymptotic freedom. Moreover, extrapolating $\ensuremath{\xi}$ to the thermodynamic limit in the low-temperature regime achieves close agreement with the three-loop perturbative calculations. We further propose a finite-size scaling (FSS) ansatz for $\ensuremath{\xi}$, demonstrating that the pseudocritical point ${\ensuremath{\beta}}_{L}$ diverges logarithmically with

L

. The thermodynamic and finite-size scaling behaviors of the magnetic susceptibility $\ensuremath{\chi}$ are also investigated and corroborate the predictions of asymptotic freedom. Our paper provides solid evidence for asymptotic freedom in the 2D Heisenberg model and advances understanding of finite-size scaling in such systems.

\frac{H}{J}<~0.5

the data of this computer experiment can be described in terms of the "effective" critical exponents $0.31<~\ensuremath{\beta}<~0.35$, $1.33<~\ensuremath{\gamma}<~1.39$, and $4.9<~\ensuremath{\delta}<~5.3$. Although the present data are not very close to the critical point they do obey the homogeneity requirement and determine the scaling function rather precisely. This function agrees very well with the scaling function for the face-centered-cubic classical Heisenberg magnet derived recently by Milo\ifmmode \check{s}\else \v{s}\fi{}evi\ifmmode \acute{c}\else \'{c}\fi{} and Stanley using high-temperature---series-expansion techniques. This agreement supports their hypothesis that neither critical exponents nor the scaling function depend on the lattice structure in the Heisenberg model.

+{M}_{\mathrm{sp}}

to $\ensuremath{-}{M}_{\mathrm{sp}}$ as the field changes from

{0}^{+}

to ${0}^{\ensuremath{-}}$) as well as for asymmetric cases, but restricting attention to (hyper)cubic system shapes. As an explicit example for the usefulness of these considerations in Monte Carlo simulations where it may be a problem to (i) locate a phase transition and (ii) distinguish first-order from second-order transitions, we present numerical results for the two-dimensional nearest-neighbor Ising ferromagnet in a field, both below the critical temperature

{T}_{c}

and at

{T}_{c}

. The numerical results are found to be in very good agreement with the phenomenological theory and it is shown that one may extract the magnitudes of jumps occurring at first-order phase transitions in a well-defined and accurate way.