On the correlation equalities for the random-bond Ising model

We obtain several equalities of the configurationally averaged spin correlation functions for the random-bond Ising model by means of a gauge transformation. These equalities are shown to be useful to find the exact results for the internal energy, an upper bound of the specific heat, the equality for the zero-field susceptibility and the zero-field spin glass susceptibility, and so on.     

Numerically exact solvable random-bond Ising model

Exact free energies are calculated numerically for aL×L-Ising lattice (L800) with constant nearest neighbour coupling between adjacent columns and random n.n. coupling between adjacent rows. For the latter a gaussian and a double-peaked -distribution are investigated. The result should be useful as a check of the controversially discussed replica trick [1]. In agreement with the numerical treatment a mean field approximation shows a transition to a spinglass phase.     

Bethe-Peierls approximation for the disordered Ising model

We study the Ising model for an alloy with an arbitrary number of components. We develop an approximation which reduces to that of Bethe and Peierls when the concentration of one of the components is unity. We investigate within this approximation the dependence of the various thermodynamic quantities, in particularT _c, on the composition of the alloy and the magnetic properties of its constituents. Comparison with the only exact calculation available, that of F. T. Leeet al., for a linear chain, shows extremely satisfactory agreement.Research supported by ARO (D). It has also benefited from the general support of Materials Science at the University of Chicago by the NSF.     

Electronic structure calculations of Fe-rich ordered and disordered Fe-Al alloys

Tight Binding Linear Muffin-Tin Orbital (TB-LMTO) electronic calculations are presented for the magnetic and structural properties of ordered and disordered FeAl alloys. The total energy, bulk modulus, lattice parameter and magnetic moments of B2, D03 and B32 ordered structures and A2 disordered structure were calculated for different compositions. The different structures are obtained by varying the position of Fe and Al atoms in a BCC superstructure. In this way, we examine the order-disorder transition that takes place in these alloys. Disordered alloys present both larger Fe magnetic moment and lattice parameter than ordered ones. In this work comparison of the calculated quantities with available experimental results is provided and it can be concluded that the results are in quantitative agreement with the experimental trends. Received 7 May 2002 / Received in final form 20 September 2002 Published online 4 February 2003 RID="a" ID="a"e-mail: eaf@we.lc.ehu.es     

Diluted transverse Ising model within the mean field renormalization group

Clusters of different size and symmetry are exploited in the study of the diluted transverse Ising model on several lattices within the mean field renormalization group approach. It is noticed that the critical exponents depend both on the size of clusters as well as on the cluster symmetry. Harris' conjecture is verified for all lattices studied.     

Ising model for the alloys according to the effective field theory

FeAl alloys in their disordered structural phase have been investigated through an Ising model where besides exchange interactions between nearest-neighbors Fe atoms, a superexchange interaction mediated by Al atoms is also taken into account. The model has been approximately treated according to the effective field theory. Although the phase diagram, as a function of Al concentration, is similar to the one previously obtained from Bogoliubov variational approach for the free energy, a different behavior for the superexchange interaction is achieved, which can also be physically accepted for this system.     

Universality aspects of the 2d random-bond Ising and 3d Blume-Capel models

We report on large-scale Wang-Landau Monte Carlo simulations of the critical behavior of two spin models in two- (2d) and three-dimensions (3d), namely the 2d random-bond Ising model and the pure 3d Blume-Capel model at zero crystal-field coupling. The numerical data we obtain and the relevant finite-size scaling analysis provide clear answers regarding the universality aspects of both models. In particular, for the random-bond case of the 2d Ising model the theoretically predicted strong universality’s hypothesis is verified, whereas for the second-order regime of the Blume-Capel model, the expected d = 3 Ising universality is verified. Our study is facilitated by the combined use of the Wang-Landau algorithm and the critical energy subspace scheme, indicating that the proposed scheme is able to provide accurate results on the critical behavior of complex spin systems.     

On the extremality of the disordered state for the Ising model on the Bethe lattice

We give a simple proof that the limit Ising Gibbs measure with free boundary conditions on the Bethe lattice with the forward branching ratio k2 is extremal if and only if is less or equal to the spin glass transition value, given by tanh( _c ^SG = 1/k.The work was partially supported by the NSF grant DMS 9504513.     

Correlation length bounds for disordered Ising ferromagnets

Thed-dimensional, nearest-neighbor disordered Ising ferromagnet: $$H = - \sum {J_{ij} \sigma _i \sigma _j }$$ is studied as a function of both temperature,T, and a disorder parameter,λ, which measures the size of fluctuations of couplingsJ _ij ≧0. A finite-size scaling correlation length,ζ _f (T, λ), is defined in terms of the magnetic response of finite samples. This correlation length is shown to be equivalent, in the scaling sense, to the quenched average correlation lengthζ(T, λ), defined as the asymptotic decay rate of the quenched average two-point function. Furthermore, the magnetic response criterion which definesζ _f is shown to have a scale-invariant property at the critical point. The above results enable us to prove that the quenched correlation length satisfies: $$C\left| {\log \xi (T)} \right|\xi (T) \geqq \left| {T - T_c } \right|^{ - {2 \mathord{\left/ {\vphantom {2 d}} \right. \kern-\nulldelimiterspace} d}}$$ which implies the boundv≧2/d for the quenched correlation length exponent.     

Extremity of the disordered phase in the Ising model on the Bethe lattice

We prove that the disordered Gibbs distribution in the ferromagnetic Ising model on the Bethe lattice is extreme forTT _c ^SG , whereT _c ^SG is the critical temperature of the spin glass model on the Bethe lattice, and it is not extreme forT _c ^SG .     

Correlation functions of the Ising model of solid ortho-hydrogen in the disordered phase

In a generalized molecular field approximation the time independent correlation functions as well as the time dependent ones are calculated for the Ising model of o-H₂ in the orientationally disordered phase. In our treatment the correlations fulfill their characteristic sum rule automatically. The order parameter correlation shows the characteristic singularity at the stability limit fork=0. The time dependent correlations are calculated within the Glauber stochastic model and show critical behaviour at the stability limit.     

Scale invariance in disordered systems: the example of the random-field Ising model

We show by numerical simulations that the correlation function of the random-field Ising model (RFIM) in the critical region in three dimensions has very strong fluctuations and that in a finite volume the correlation length is not self-averaging. This is due to the formation of a bound state in the underlying field theory. We argue that this nonperturbative phenomenon is not particular to the RFIM in 3D. It is generic for disordered systems in two dimensions and may also happen in other three-dimensional disordered systems.     

Two-Dimensional Saddle Point Equation of Ginzburg-Landau Hamiltonian for the Diluted Ising Model

The saddle point equation of Ginzburg-Landau Hamiltonian for the diluted Ising model is developed. The ground state is solved numerically in two dimensions. The result is partly explained by the coarse-grained approximation.     

Large-q asymptotics of the random-bond potts model

We numerically examine the large-q asymptotics of the q-state random bond Potts model. Special attention is paid to the parametrization of the critical line, which is determined by combining the loop representation of the transfer matrix with Zamolodchikov's c-theorem. Asymptotically the central charge seems to behave like c(q)=1 / 2 log(2)(q)+O(1). Very accurate values of the bulk magnetic exponent x(1) are then extracted by performing Monte Carlo simulations directly at the critical point. As q-->infinity, these seem to tend to a nontrivial limit, x(1)-->0.192+/-0.002.