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Progress of Theoretical Physics

Gaussian Random Bond Problem for Annealed System in 1/n Expansion

M. Oku, R. Abe

Published Jan 1, 1982• 2 citations
Publisher

Abstract

Critical phenomena for the for the annealed n-vector model are studied in 1/n expansion by assuming that the exchange interactions between spins are fluctuating variables obeying the Gaussian distribution. It is shown that for an overall fluctuation of the interaction between spins the susceptibility never diverges at any finite temperature except for n→∞ (spherical limit) and n=1 (Ising) cases. On the other hand, if the fluctuation is restricted to only nearest-neighbor pairs, the susceptibility is proved to diverge at an appropriate transition temperature in the limit n→∞. The critical exponent γ in this case is shown to be the same as the spherical model value. Also the dependence of transition temperature on the strength of fluctuation is discussed.

DOI
10.1143/ptp.67.118

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Critical phenomena for the for the annealed n-vector model are studied in 1/n expansion by assuming that the exchange interactions between spins are fluctuating variables obeying the Gaussian distribution. It is shown that for an overall fluctuation of the interaction between spins the susceptibility never diverges at any finite temperature except for n→∞ (spherical limit) and n=1 (Ising) cases. On the other hand, if the fluctuation is restricted to only nearest-neighbor pairs, the susceptibility is proved to diverge at an appropriate transition temperature in the limit n→∞. The critical exponent γ in this case is shown to be the same as the spherical model value. Also the dependence of transition temperature on the strength of fluctuation is discussed.
Using the high-temperature series expansion method, a system of randomly mixed ferromagnetic and antiferromagnetic Ising bonds is considered in quenched and annealed systems for an arbitrary value of spin and a BCC lattice. The dependence of the critical temperature on the concentration p of ferromagnetic bonds is determined. It is found that the resulting phase diagrams depend on the system (whether quenched or annealed), the spin value and the strength of the coupling constants. Also the dependence of the critical exponent gamma on p and the crossover exponent phi are estimated.
The critical behavior of the two-dimensional Ising model (square lattice, exchange constant J) in an uniform field, and in an annealed random field is considered. The random field is generated by decorating the horizontal and vertical bonds of the lattice, and it satisfies an arbitrary distribution which is imposed by introducing a pseudo-chemical potential. By decimating the decorating variables the model can be mapped onto a homogeneous Ising model with effective exchange constant J′ and effective external field h′, dependent on the temperature. These parameters, which satisfy a set of coupled equations, depend on the spin average and nearest-neighbor two-spin correlation, and are obtained numerically. For the symmetric field distribution [Formula: see text] the mapping of the critical frontier on the (K′=βJ′,H′=βh′) plane onto the (K=β J,H=βh) plane is determined and, as in the model introduced by Essam and Place, there is a region on the (K, H) plane which cannot be reached from any real values of (K′, H′). The critical exponents are determined numerically, and it is shown that they do not satisfy renormalization relations obtained for their model.
We study the equilibrium properties of an Ising model on a disordered random network where the disorder can be quenched or annealed. The network consists of fourfold coordinated sites connected via variable length one-dimensional chains. Our emphasis is on nonuniversal properties and we consider the transition temperature and other equilibrium thermodynamic properties, including those associated with one-dimensional fluctuations arising from the chains. We use analytic methods in the annealed case, and a Monte Carlo simulation for the quenched disorder. Our objective is to study the difference between quenched and annealed results with a broad random distribution of interaction parameters. The former represents a situation where the time scale associated with the randomness is very long and the corresponding degrees of freedom can be viewed as frozen, while the annealed case models the situation where this is not so. We find that the transition temperature and the entropy associated with one-dimensional fluctuations are always higher for quenched disorder than in the annealed case. These differences increase with the strength of the disorder up to a saturating value. We discuss our results in connection to physical systems where a broad distribution of interaction strengths is present.
The two-dimensional Ising model with two kinds of bonds –ferro and antiferromagnetic– is investigated in both quenched and annealed approximations. High temperature series expansion technique is used to get the respective free energies and susceptibilities. Form the analysis of the ratio method results, the phase diagrams are drawn and critical concentrations are estimated. It is shown that in both approximations the system behaves similarly, although in the annealed case the magnetically ordered phases are more stable against increasing impurity concentration. It is also proved that for Bethe lattices initial susceptibilities for quenced and annealed systems coincide.
This paper investigates the role of statistical fluctuations about the equilibrium value of the order parameter in the vicinity of a phase-transition temperature. The systems under consideration are assumed to be of finite size, but the behavior in the thermodynamic limit is also described. Our objective is to examine the simplest exactly solvable model of non-Gaussian static fluctuations for samples of finite size. To this end a Landau form is employed for the model Hamiltonian and various expansions are analyzed in order to describe first- and second-order phase transitions, field-induced transitions, and liquid-vapor critical phenomena. The approach differs from previous treatments primarily in the application of a quartic non-Gaussian model for statistical fluctuations which redresses some of the shortcomings of the standard Gaussian approximation. In particular, it is found that for finite values of V (volume) no singularities exist even at the critical temperature. The treatment is generalized by examining a class of non-Gaussian distribution functions, and particular emphasis is placed on strict adherence to the uniform convergence criterion so that general conclusions can be drawn.
A cumulant expansion is used to calculate the transition temperature of Ising models with random-bond defects. For a concentration, x, of missing interactions in the simple-square Ising model the author finds -T c -1 dT c /dx mod x=0 =1.329 compared with the mean-field value of one. If the interactions are independent random variable with a width delta J/J identical to epsilon , the result is -T c -1 dT c /d epsilon 2 mod epsilon =0 =0.312 compared with the mean-field results of zero. An approximation yields the specific heat in the critical regime as C approximately C 0 /(1+x gamma 2 C 0 ), where gamma is a constant and C 0 is the unperturbed specific heat at a renormalized temperature. Thus, the specific heat divergence is broadened over a temperature interval Delta T, with Delta T/T c approximately x (1 alpha )/, where alpha is the critical exponent for the specific heat, and a maximum value of order x -1 is attained. Heuristic arguments show that this smoothing effect occurs if alpha >0.