Physical review. B, Condensed matter

Correlated-effective-field treatment of the anisotropic Ising ferromagnet: Thermodynamical properties

R. Honmura

Published Jul 1, 1984• 42 citations

Abstract

Within a new type of correlated-effective-field theory we calculate the effective-field parameters (including their response to external field), critical temperatures, susceptibility, short-range order parameters, and specific heat associated with an anisotropic, square-lattice, spin-\textonehalf{} Ising ferromagnet. The present formalism, which is shown to recover the Bethe-Peierls thermodynamics for the pure $Z$-coordinated-lattice, spin-\textonehalf{} Ising model, yields satisfactory results; in particular, the crossover from a pure Ising ferromagnet on a square lattice to that on a linear chain is reasonably exhibited in the behavior of the susceptibility and the specific heat.

DOI: 10.1103/physrevb.30.348

Related papers

Anisotropic Ising model in a magnetic field: Effective-field theory analysis

Minos A. Neto, Rosana A. dos Anjos, J. Ricardo de Sousa

Physical Review BJun 23, 200632 citations

Critical behavior of the SU(3) ferromagnetic Heisenberg chain

Young-Ju Jee, Kong-Ju-Bock Lee, P. Schlottmann

Physical review. B, Condensed matterFeb 1, 198916 citations

Contribution to the new type of effective-field theory of the Ising model

R. Honmura, T. Kaneyoshi

Journal of Physics C Solid State PhysicsOct 14, 1979622 citations

Low- and High-Temperature Behaviors of Ferromagnetic Ising Model of Infinite-Spin

Tsuyoshi Horiguchi, Ôjirô Nagai, Seiji Miyashita

Journal of the Physical Society of JapanJan 15, 19929 citations

Green’s function study of one-dimensional spin-1/2 Ising-Heisenberg ferromagnets

W. J. Zhang, J. L. Shen, Jian Xu +1 more

Physical review. B, Condensed matterFeb 1, 19957 citations

Thermodynamic properties of the Ising model with equilibrium impurity distribution

А. К. Аржников, A. Vedyayev

A relatively simple method for computing thermodynamic quantities in the Ising model with impurities in thermodynamic equilibrium is proposed. The specific heat, the susceptibility, and the short-range order of spins localized on bonds of different types are investigated for the plane square lattice.

Soviet Journal of Low Temperature PhysicsNov 1, 19821 citation

Physical review. B, Condensed matter

Correlated-effective-field treatment of the anisotropic Ising ferromagnet: Thermodynamical properties

R. Honmura

Published Jul 1, 1984• 42 citations

Publisher

Abstract

Z

-coordinated-lattice, spin-\textonehalf{} Ising model, yields satisfactory results; in particular, the crossover from a pure Ising ferromagnet on a square lattice to that on a linear chain is reasonably exhibited in the behavior of the susceptibility and the specific heat.

DOI: 10.1103/physrevb.30.348

Related papers

Anisotropic Ising model in a magnetic field: Effective-field theory analysis

Minos A. Neto, Rosana A. dos Anjos, J. Ricardo de Sousa

The anisotropic two-dimensional Ising model in the presence of a magnetic field is studied within two different approaches: the effective-field theory (EFT) with correlation and the Bethe-Peierls (BP) approximation. The model consists of ferromagnetic interaction $({J}_{x})$ in the $x$ direction and antiferromagnetic interaction $({J}_{y})$ in the $y$ direction. The phase diagram in the $T\text{\ensuremath{-}}H$ plane is obtained for the particular case ${J}_{x}={J}_{y}$ . Special focus is given in the low-temperature region of the phase diagram, where a first-order phase transition is observed using the mean-field approximation, which is in disagreement with the linear chain approximation (LCA). Our results indicate a second-order phase transition for all values of $H∕J∊[0,2]$ , with the presence of a reentrant behavior only observed in the BP approximation in accordance with the results of the LCA and exact solution. The null field critical temperature is an increasing function of $r={J}_{y}∕{J}_{x}$ , and in the $r\ensuremath{\rightarrow}0$ limit we have found, using BP and EFT, the approximate form ${k}_{B}{T}_{N}∕{J}_{x}\ensuremath{\simeq}A∕\mathrm{ln}(1∕r)$ in accordance with the exact result of Onsager.The anisotropic two-dimensional Ising model in the presence of a magnetic field is studied within two different approaches: the effective-field theory (EFT) with correlation and the Bethe-Peierls (BP) approximation. The model consists of ferromagnetic interaction $({J}_{x})$ in the $x$ direction and antiferromagnetic interaction $({J}_{y})$ in the $y$ direction. The phase diagram in the $T\text{\ensuremath{-}}H$ plane is obtained for the particular case ${J}_{x}={J}_{y}$. Special focus is given in the low-temperature region of the phase diagram, where a first-order phase transition is observed using the mean-field approximation, which is in disagreement with the linear chain approximation (LCA). Our results indicate a second-order phase transition for all values of $H∕J∊[0,2]$, with the presence of a reentrant behavior only observed in the BP approximation in accordance with the results of the LCA and exact solution. The null field critical temperature is an increasing function of $r={J}_{y}∕{J}_{x}$, and in the $r\ensuremath{\rightarrow}0$ limit we have found, using BP and EFT, the approximate form ${k}_{B}{T}_{N}∕{J}_{x}\ensuremath{\simeq}A∕\mathrm{ln}(1∕r)$ in accordance with the exact result of Onsager.

Physical Review BJun 23, 200632 citations

Critical behavior of the SU(3) ferromagnetic Heisenberg chain

Young-Ju Jee, Kong-Ju-Bock Lee, P. Schlottmann

We have extracted the leading low-temperature contributions to the specific heat and the magnetic susceptibility from the thermodynamic Bethe-Ansatz equations of the SU(3) invariant ferromagnetic Heisenberg chain of spin 1. The critical exponents for the specific heat, the susceptibility, and the correlation length are $\ensuremath{\alpha}=\ensuremath{-}\frac{1}{2}$, $\ensuremath{\gamma}=2$, and $\ensuremath{\nu}=1$, respectively. The susceptibility and the correlation length exhibit logarithmic corrections, which are quenched by relatively small magnetic fields. In large fields, the energy required to flip a spin gives rise to an exponential activation.We have extracted the leading low-temperature contributions to the specific heat and the magnetic susceptibility from the thermodynamic Bethe-Ansatz equations of the SU(3) invariant ferromagnetic Heisenberg chain of spin 1. The critical exponents for the specific heat, the susceptibility, and the correlation length are $\ensuremath{\alpha}=\ensuremath{-}\frac{1}{2}$, $\ensuremath{\gamma}=2$, and $\ensuremath{\nu}=1$, respectively. The susceptibility and the correlation length exhibit logarithmic corrections, which are quenched by relatively small magnetic fields. In large fields, the energy required to flip a spin gives rise to an exponential activation.

Physical review. B, Condensed matterFeb 1, 198916 citations

Contribution to the new type of effective-field theory of the Ising model

R. Honmura, T. Kaneyoshi

A new type of effective-field theory of the Ising model is presented. The differential-operator method is introduced into the exact spin correlation function identity obtained by Callen. The Curie temperatures are evaluated by using two different types of effective Hamiltonians. It is also shown how the Zernike and the Bethe-Peierls equations can be reproduced within one framework depending on the choice of effective fields in an effective Hamiltonian. The spin correlation function and the specific heat are presented.A new type of effective-field theory of the Ising model is presented. The differential-operator method is introduced into the exact spin correlation function identity obtained by Callen. The Curie temperatures are evaluated by using two different types of effective Hamiltonians. It is also shown how the Zernike and the Bethe-Peierls equations can be reproduced within one framework depending on the choice of effective fields in an effective Hamiltonian. The spin correlation function and the specific heat are presented.

Journal of Physics C Solid State PhysicsOct 14, 1979622 citations

Low- and High-Temperature Behaviors of Ferromagnetic Ising Model of Infinite-Spin

Tsuyoshi Horiguchi, Ôjirô Nagai, Seiji Miyashita

We investigate low- and high-temperature properties of the ferromagnetic Ising model of infinite-spin. By using the cummulant expansion methods, we obtain low-temperature expansions of the free energy, the specific heat, the spontaneous magnetization, the zero-field susceptibility and so on, and high-temperature expansions of the free energy and the zero-field susceptibility. We compare the results with those obtained by Thompson for the system on the one-dimensional lattice, with those by using Monte Carlo simulations for the systems on the square lattice and on the simple cubic lattice and with those with the aid of the mean-field approximation.We investigate low- and high-temperature properties of the ferromagnetic Ising model of infinite-spin. By using the cummulant expansion methods, we obtain low-temperature expansions of the free energy, the specific heat, the spontaneous magnetization, the zero-field susceptibility and so on, and high-temperature expansions of the free energy and the zero-field susceptibility. We compare the results with those obtained by Thompson for the system on the one-dimensional lattice, with those by using Monte Carlo simulations for the systems on the square lattice and on the simple cubic lattice and with those with the aid of the mean-field approximation.

Journal of the Physical Society of JapanJan 15, 19929 citations

Green’s function study of one-dimensional spin-1/2 Ising-Heisenberg ferromagnets

W. J. Zhang, J. L. Shen, Jian Xu +1 more

The one-dimensional spin-1/2 Ising-Heisenberg ferromagnets are studied by the Green's function method at finite temperatures. The Green's function is decoupled in terms of the correlation functions of 〈 ${\mathit{S}}_{0}^{\mathit{z}}$ ${\mathit{S}}_{\mathit{n}}^{\mathit{z}}$ 〉 and 〈 ${\mathit{S}}_{0}^{+}$ ${\mathit{S}}_{\mathit{n}}^{\mathrm{\ensuremath{-}}}$〉 under the constraint of 〈 ${\mathit{S}}^{\mathit{z}}$ 〉=0, from which a set of self-consistent equations are derived and solved numerically. The thermodynamic properties such as internal energy, specific heat, and zero-field parallel susceptibility are calculated. Our results predict the correct behavior for the correlation functions and thermodynamic quantities in the entire temperature region, in agreement both with the exact numerical results on the finite chain and the experimental data.The one-dimensional spin-1/2 Ising-Heisenberg ferromagnets are studied by the Green's function method at finite temperatures. The Green's function is decoupled in terms of the correlation functions of 〈${\mathit{S}}_{0}^{\mathit{z}}$${\mathit{S}}_{\mathit{n}}^{\mathit{z}}$〉 and 〈${\mathit{S}}_{0}^{+}$${\mathit{S}}_{\mathit{n}}^{\mathrm{\ensuremath{-}}}$〉 under the constraint of 〈${\mathit{S}}^{\mathit{z}}$〉=0, from which a set of self-consistent equations are derived and solved numerically. The thermodynamic properties such as internal energy, specific heat, and zero-field parallel susceptibility are calculated. Our results predict the correct behavior for the correlation functions and thermodynamic quantities in the entire temperature region, in agreement both with the exact numerical results on the finite chain and the experimental data.

Physical review. B, Condensed matterFeb 1, 19957 citations

Thermodynamic properties of the Ising model with equilibrium impurity distribution

А. К. Аржников, A. Vedyayev

Soviet Journal of Low Temperature PhysicsNov 1, 19821 citation

Z

({J}_{x})

in the

x

direction and antiferromagnetic interaction

({J}_{y})

in the

y

direction. The phase diagram in the $T\text{\ensuremath{-}}H$ plane is obtained for the particular case

{J}_{x}={J}_{y}

. Special focus is given in the low-temperature region of the phase diagram, where a first-order phase transition is observed using the mean-field approximation, which is in disagreement with the linear chain approximation (LCA). Our results indicate a second-order phase transition for all values of

H∕J∊[0,2]

, with the presence of a reentrant behavior only observed in the BP approximation in accordance with the results of the LCA and exact solution. The null field critical temperature is an increasing function of

r={J}_{y}∕{J}_{x}

, and in the $r\ensuremath{\rightarrow}0$ limit we have found, using BP and EFT, the approximate form ${k}_{B}{T}_{N}∕{J}_{x}\ensuremath{\simeq}A∕\mathrm{ln}(1∕r)$ in accordance with the exact result of Onsager.

{\mathit{S}}_{0}^{\mathit{z}}

{\mathit{S}}_{\mathit{n}}^{\mathit{z}}

〉 and 〈

{\mathit{S}}_{0}^{+}

${\mathit{S}}_{\mathit{n}}^{\mathrm{\ensuremath{-}}}$〉 under the constraint of 〈

{\mathit{S}}^{\mathit{z}}

〉=0, from which a set of self-consistent equations are derived and solved numerically. The thermodynamic properties such as internal energy, specific heat, and zero-field parallel susceptibility are calculated. Our results predict the correct behavior for the correlation functions and thermodynamic quantities in the entire temperature region, in agreement both with the exact numerical results on the finite chain and the experimental data.