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Europhysics Letters (EPL)

Disorder-Induced Phase in the Random Bond Ising Model on the Square Lattice

K. Ziegler

Published Mar 1, 1991• 16 citations
Publisher

Abstract

The random bond Ising model in two dimensions is considered in terms of a random matrix representation. A generalization of this approach enables us to study an N → ∞ limit and the corresponding 1/N expansion. With g as the strength of the random bond fluctuations, the critical temperature T c of the pure system is split into two critical temperatures T ± = T c (1 ± c 0 exp[− c 1 / g ]) with a new phase for T − < T < T + . The specific heat, for instance, grows logarithmically (or double logarithmically in a crossover region near T ± ) with|1− T / T c | up to a bound of the order 1/ g .

DOI
10.1209/0295-5075/14/5/005

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The random bond Ising model in two dimensions is considered in terms of a random matrix representation. A generalization of this approach enables us to study an N → ∞ limit and the corresponding 1/N expansion. With g as the strength of the random bond fluctuations, the critical temperature T c of the pure system is split into two critical temperatures T ± = T c (1 ± c 0 exp[− c 1 / g ]) with a new phase for T − < T < T + . The specific heat, for instance, grows logarithmically (or double logarithmically in a crossover region near T ± ) with|1− T / T c | up to a bound of the order 1/ g .
A 2D Ising model in which the bonds K fluctuate randomly about Kc{\mathrm{K}}_{\mathrm{c}}Kc​, the critical value of the pure system, is considered. The ensemble average of the square of the two-point function, s0{\mathrm{s}}_{0}s0​sR{\mathrm{s}}_{\mathrm{R}}sR​〉2{\mathrm{〉}}^{2}〉2〉Av{\mathrm{〉}}_{\mathrm{Av}}〉Av​, is shown to decay as (lnR)1/4{)}^{1/4})1/4${\mathrm{R}}^{\mathrm{\ensuremath{-}}1/2}$ aat the critical point. This implies that s0{\mathrm{s}}_{0}s0​sR{\mathrm{s}}_{\mathrm{R}}sR​Av{}_{\mathrm{Av}}Av​ is bounded above by (lnR)1/8{)}^{1/8})1/8${\mathrm{R}}^{\mathrm{\ensuremath{-}}1/4}$ in disagreement with the exp [-(ln lnR)2{)}^{2})2] decay law found by Dotsenko and Dotsenko by a different method. On the other hand, the present calculation reproduces their specific-heat singularity C\ensuremath{\sim}ln\ensuremath{\Vert}ln\ensuremath{\tau}\ensuremath{\Vert} (\ensuremath{\tau}=K-Kc{\mathrm{K}}_{\mathrm{c}}Kc​).
The two-dimensional random bond Ising model with two distinct ferromagnetic nearest-neighbour couplings, J and J ', distributed randomly over the lattice with concentration 1/2 is studied for various ratios J '/ J at criticality. The critical temperature is exactly known from self-duality. The size-dependent specific heat shows a crossover from the perfect Ising to the randomness-dominated critical behaviour of doubly logarithmic form. In the latter region, our estimates for the critical exponents of the size-dependent magnetization and susceptibility are in very good agreement with analytic results suggesting β/ν = 1/8 and γ/ν = 7/4.
The free energy and the specific heat of the two-dimensional Gaussian random bond Ising model on a square lattice are found with high accuracy using graph expansion and analysis of continuation of the high-temperature series. At low temperatures the specific heat follows the power law C∝T3/2C\propto T^{3/2}C∝T3/2, the result confirmed independently by counting many-body states in finite size samples. The interpretation of the free energy in terms of independent droplet excitations gives the density of the two-level states, that follows a novel power law ρ(ϵ)∝ϵ\rho(\epsilon)\propto \sqrt{\epsilon}ρ(ϵ)∝ϵ​ at low energies.
The problem of the effect of inhomogeneous impurities in a lattice structure on the critical behaviour of systems which undergo a second-order phase transition is studied on a model system, the 2D Ising Model with impurity bonds randomly distributed over the lattice. It is known that in the critical region the 2D Ising Model is equivalent to the model of free fermions. We show that the effect of impurities is to add a four-fermion interaction with the corresponding charge proportional to the concentration of impurities. The resulting fermion model is simple enough and can be studied exactly by renormalization group methods. We show that for any small concentration of impurity bonds a new critical regime is established if we go sufficiently close to the phase-transition point. In particular we find that the specific heat singularity changes from C ˜ ln 1/|τ|(τ=(Τ − Τc/Τc to C ˜ ln ln 1/|τ| for τ≪τi ˜ exp(−const/c i), c i being the concentration of impurities, while the spin-spin correlation function in the critical point changes more seriously: from <σ R σΟ> ˜ 1/R1/4 to <σ R σΟ> ˜ exp { − (const/c i)(ln ln R2 )} for R≫R i˜ exp (const/c i), which corresponds to a change of the critical exponent η from 1/4 to 0.
The effects of bond randomness on the phase diagram and critical behavior of the square lattice ferromagnetic Blume-Capel model are discussed. The system is studied in both the pure and disordered versions by the same efficient two-stage Wang-Landau method for many values of the crystal field, restricted here in the second-order phase-transition regime of the pure model. For the random-bond version several disorder strengths are considered. We present phase diagram points of both pure and random versions and for a particular disorder strength we locate the emergence of the enhancement of ferromagnetic order observed in an earlier study in the ex-first-order regime. The critical properties of the pure model are contrasted and compared to those of the random model. Accepting, for the weak random version, the assumption of the double-logarithmic scenario for the specific heat we attempt to estimate the range of universality between the pure and random-bond models. The behavior of the strong disorder regime is also discussed and a rather complex and yet not fully understood behavior is observed. It is pointed out that this complexity is related to the ground-state structure of the random-bond version.
We investigate the effects of quenched bond randomness on the critical properties of the two-dimensional ferromagnetic Ising model embedded in a triangular lattice. The system is studied in both the pure and disordered versions by the same efficient two-stage Wang-Landau method. In the first part of our study, we present the finite-size scaling behavior of the pure model, for which we calculate the critical amplitude of the specific heat's logarithmic expansion. For the disordered system, the numerical data and the relevant detailed finite-size scaling analysis along the lines of the two well-known scenarios-logarithmic corrections versus weak universality--strongly support the field-theoretically predicted scenario of logarithmic corrections. A particular interest is paid to the sample-to-sample fluctuations of the random model and their scaling behavior that are used as a successful alternative approach to criticality.