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Journal of Applied Physics

The antiferromagnetic Ising model in a magnetic field: Linked-cluster expansion analysis

Christopher D. Wentworth, Y. L. Wang

Published May 15, 1993• 5 citations
Publisher

Abstract

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.

DOI
10.1063/1.353674

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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.
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.
Motived by the $H\text{\ensuremath{-}}T$ phase diagram in the bcc Ising antiferromagnetic with nearest-neighbor interactions obtained by Monte Carlo simulation [Landau, Phys. Rev. B 16, 4164 (1977)] that shows a reentrant behavior at low temperature, with two critical temperatures in magnetic field about 2% greater than the critical value Hc=8J{H}_{c}=8JHc​=8J, we apply the effective field renormalization group (EFRG) approach in this model on three-dimensional lattices (simple cubic-sc and body centered cubic-bcc). We find that the critical curve TN(H){T}_{N}(H)TN​(H) exhibits a maximum point around of $H\ensuremath{\simeq}{H}_{c}$ only in the bcc lattice case. We also discuss the critical behavior by the effective field theory in clusters with one (EFT-1) and two (EFT-2) spins, and a reentrant behavior is observed for the sc and bcc lattices. We have compared our results of EFRG in the bcc lattice with Monte Carlo and series expansion, and we observe a good accordance between the methods.
We study the two-dimensional antiferromagnetic Ising model with a purely imaginary magnetic field, which can be thought of as a toy model for the usual θ physics. Our motivation is to have a benchmark calculation in a system which suffers from a strong sign problem, so that our results can be used to test Monte Carlo methods developed to tackle such problems. We analyze here this model by means of analytical techniques, computing exactly the first eight cumulants of the expansion of the effective Hamiltonian in powers of the inverse temperature, and calculating physical observables for a large number of degrees of freedom with the help of standard multiprecision algorithms. We report accurate results for the free energy density, internal energy, standard and staggered magnetization, and the position and nature of the critical line, which confirm the mean-field qualitative picture, and which should be quantitatively reliable, at least in the high-temperature regime, including the entire critical line.
Abstract The effect of an external magnetic field on the phase transitions and the thermodynamic and magnetic properties of the three-dimensional antiferromagnetic Ising model on a body-centered cubic lattice has been studied using a replica algorithm of the Monte Carlo method, taking into account the interactions of next nearest neighbors. A phase diagram of the dependence of the critical temperature on an external magnetic field has been built. It is shown that a second-order phase transition is observed in the magnetic field range under consideration.
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.
We have calculated the low-temperature series expansions of the spontaneous magnetization and the zero-field susceptibility of the square-lattice ferromagnetic Ising model with first-neighbour interaction J1 and second-neighbour interaction J2 to the 30th and 26th order respectively by computer. Our results extend the previous calculations by Lee and Lin to six more orders. We use the Padé approximants to estimate the critical exponents and the critical temperature for different ratios of R = J2/J1. The estimated critical temperature as a function of R agrees with the estimation by Oitmaa from high-temperature series expansions.