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Physical Review B

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

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

Published Jun 23, 2006• 32 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.73.214439

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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 (Jx)({J}_{x})(Jx​) in the xxx direction and antiferromagnetic interaction (Jy)({J}_{y})(Jy​) in the yyy direction. The phase diagram in the T\ensuremath-HT\text{\ensuremath{-}}HT\ensuremath-H plane is obtained for the particular case Jx=Jy{J}_{x}={J}_{y}Jx​=Jy​. 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]H∕J∊[0,2]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=Jy∕Jxr={J}_{y}∕{J}_{x}r=Jy​∕Jx​, and in the r\ensuremath→0r\ensuremath{\rightarrow}0r\ensuremath→0 limit we have found, using BP and EFT, the approximate form kBTN∕Jx\ensuremath≃A∕ln(1∕r){k}_{B}{T}_{N}∕{J}_{x}\ensuremath{\simeq}A∕\mathrm{ln}(1∕r)kB​TN​∕Jx​\ensuremath≃A∕ln(1∕r) in accordance with the exact result of Onsager.
The mean-field approximation of the critical behavior of the antiferromagnetic Ising model in a magnetic field shows that the critical field increases first at low temperature. However, the exact solution of a decorated square lattice proves that the critical field rapidly decreases with increasing temperature for the two-dimensional lattice. Three-dimensional lattices of this model have been studied as well using the high temperature expansion, low temperature expansion, and Monte Carlo simulation. The results are different. In this article we have investigated the critical field of the body centered cubic lattice of this model using the linked-cluster expansion technique. An eighth-order perturbation series for the free energy has been calculated. The magnetization and staggered magnetic susceptibility series have been calculated from the free energy. These series have been analyzed using Padé approximants to obtain the phase diagram in the field-temperature plane. A comparison is made with Monte Carlo simulation calculations.
The critical behavior at the phase transition of the ferromagnetic two-dimensional anisotropic Ising model with next-nearest neighbor (NNN) couplings in the presence of the field is determined using mainly Monte Carlo (MC) method. This method is used to investigate the phase diagram of the model and to verify the existence of a divergence at null temperature which often appears in two-dimensional systems. We analyze also the influence of the report of the NNN interactions [Formula: see text] and the magnetic field [Formula: see text] on the critical temperature of the system, and we show that the critical temperature depends on the magnetic field for positive values of the interaction. Finally, we have investigated other thermodynamical qualities such as the magnetic susceptibility [Formula: see text]. It has been shown that their thermal behavior depends qualitatively and quantitatively on the strength of NNN interactions and the magnetic field.
We have studied the anisotropic three-dimensional nearest-neighbor Ising model with competitive interactions in an uniform longitudinal magnetic field HHH. The model consists of ferromagnetic interaction Jx(Jz)J_{x}(J_{z})Jx​(Jz​) in the x(z)x(z)x(z) direction and antiferromagnetic interaction JyJ_{y}Jy​ in the yyy direction. We have compared our calculations within a effective-field theory in clusters with four spins (EFT-4) in the simple cubic (sc) lattice with traditional Monte Carlo (MC) simulations. The phase diagrams in the h−kBT/Jxh-k_{B}T/J_{x}h−kB​T/Jx​ plane (h=H/Jxh=H/J_{x}h=H/Jx​) were obtained for the particular case λ1=Jy/Jx(λ2=Jz/Jx)=1λ_{1}=J_{y}/J_{x} (λ_{2}=J_{z}/J_{x})=1λ1​=Jy​/Jx​(λ2​=Jz​/Jx​)=1 (anisotropic sc). Our results indicate second-order frontiers for all values of HHH for the particular case λ2=0λ_{2}=0λ2​=0 (square lattice), while in case λ1=λ2=1λ_{1}=λ_{2}=1λ1​=λ2​=1, we observe first- and second-order phase transitions in the low and high temperature limits, respectively, with presence of a tricritical point. Using EFT-4, a reentrant behavior at low temperature was observed in contrast with results of MC.
The interface method is used to derive the order-disorder critical temperature of the two-dimensional Ising model on a rectangular lattice with ferromagnetic interactions in the x direction and antiferromagnetic interactions in the y direction. Employing this approximation in two different ways gives different critical temperatures Tc{T}_{c}Tc​ for nonzero uniform magnetic fields H. However, both of the phase boundaries Tc{T}_{c}Tc​(H) reduce to the exact results available at H=0 and T=0. .AE
The two-dimensional Ising model is the simplest model of statistical mechanics exhibiting a second-order phase transition. While in absence of magnetic field it is known to be solvable on the lattice since Onsager's work of the 1940s, exact results for the magnetic case have been missing until the late 1980s, when A Zamolodchikov solved the model in a field at the critical temperature, directly in the scaling limit, within the framework of integrable quantum field theory. In this paper, we review this field theoretical approach to the Ising universality class, with particular attention to the results obtained starting from Zamolodchikov's scattering solution and to their comparison with the numerical estimates on the lattice. The topics discussed include scattering theory, form factors, correlation functions, universal amplitude ratios and perturbations around integrable directions. Although we restrict our discussion to the Ising model, the emphasis is on the general methods of integrable quantum field theory which can be used in the study of all universality classes of critical behaviour in two dimensions.
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 ZZZ-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.