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Journal of Physics C Solid State Physics

Criticality and crossover in the bond-diluted random Ising model

Eytan Domany

Published Apr 28, 1978• 52 citations
Publisher

Abstract

By using the replica trick and a duality transformation, the bond-diluted random Ising model is mapped onto a new Hamiltonian. This demonstrates the higher order critical nature of the percolation point and identifies the appropriate crossover scaling variables. Taking the n to 0 limit of the replica method near the percolation point is shown to be equivalent to the q to 1 limit of the Potts model. The critical line near p c is calculated, yielding for a square lattice exp(-2K c )=2ln2(p- 1 / 2 )+0(p- 1 / 2 ) 2 .

DOI
10.1088/0022-3719/11/8/008

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By using the replica trick and a duality transformation, the bond-diluted random Ising model is mapped onto a new Hamiltonian. This demonstrates the higher order critical nature of the percolation point and identifies the appropriate crossover scaling variables. Taking the n to 0 limit of the replica method near the percolation point is shown to be equivalent to the q to 1 limit of the Potts model. The critical line near p c is calculated, yielding for a square lattice exp(-2K c )=2ln2(p- 1 / 2 )+0(p- 1 / 2 ) 2 .
We introduce a new approach to disordered two-dimensional Ising models based on the extension of the combinatorial solution to randomized supercells. Applying it to the site-diluted Ising model on the square lattice, we resolve the full phase boundary Tc(p)T_c(p)Tc​(p) from the pure-Ising point to the percolation limit Tc(pc)=0T_c(p_c)=0Tc​(pc​)=0 with, in principle, arbitrary precision. The critical eigenvalue governing the transition is found to follow a remarkably accurate linear interpolation between the Ising and percolation endpoints, whose small but systematic deviations reveal the nontrivial fine structure of the phase boundary. Near the percolation threshold, we confirm the crossover exponent ϕRSIM=1ϕ_{\rm RSIM}=1ϕRSIM​=1 and extract the nonuniversal amplitude αRSIM≃1.616{α_{\rm RSIM}\simeq 1.616}αRSIM​≃1.616.
We study by Monte Carlo simulations the influence of bond dilution on the three-dimensional Ising model. This paradigmatic model in its pure version displays a second-order phase transition with a positive specific heat critical exponent α\alphaα . According to the Harris criterion disorder should hence lead to a new fixed point characterized by new critical exponents. We have determined the phase diagram of the diluted model, starting from the pure model limit down to the neighbourhood of the percolation threshold. For the estimation of critical exponents, we have first performed a finite-size scaling study, where we concentrated on three different dilutions to check the stability of the disorder fixed point. We emphasize in this work the great influence of the cross-over phenomena between the pure, disorder and percolation fixed points which lead to effective critical exponents dependent on the concentration. In a second set of simulations, the temperature behaviour of physical quantities has been studied in order to characterize the disorder fixed point more accurately. In particular this allowed us to estimate ratios of some critical amplitudes. In accord with previous observations for other models this provides stronger evidence for the existence of the disorder fixed point since the amplitude ratios are more sensitive to the universality class than the critical exponents. Moreover, the question of non-self-averaging at the disorder fixed point is investigated and compared with recent results for the bond-diluted q = 4 Potts model. Overall our numerical results provide evidence that, as expected on theoretical grounds, the critical behaviour of the bond-diluted model is indeed governed by the same universality class as the site-diluted model.
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 .
The randomly bond-dilute two-dimensional nearest-neighbor Ising model on the square lattice is studied by renormalization-group methods based on the Migdal-Kadanoff approximate recursion relations. Calculations give both thermal and magnetic exponents associated with the percolative fixed point. Differential recursion relations yield a phase diagram which is in quantitative agreement with all known results. Curves for the specific heat, percolation probability, and magnetization are displayed. The critical region of the specific heat becomes unobservably narrow well above the percolation threshold pc{p}_{c}pc​. This provides a possible explanation for the apparent specific-heat rounding in certain experiments.
The techniques developed by Edwards and Anderson (1975) for the spin-glass problem are here applied to the bond-diluted quenched Ising model (S= 1 / 2 ). The n to 0 trick is used to express exactly the free energy in terms of a generalized Hamiltonian representing n different but coupled replicas. The only approximation of the theory consists of decoupling these replicas in a way that reduces the free energy of the diluted system to that of a pure system with renormalized, concentration-dependent parameters. To first order in the concentration of missing bonds the author obtains all the results previously derived by Harris (1974) using a cumulant expansion and which are exact in this limit. More generally, the theory allows a discussion of the concentration dependence of the critical temperature; in particular, the values deduced for the critical concentration agree rather well with those obtained exactly or through series expansions. The behaviour of the system near the transition is also studied and the critical exponents are found to be Fisher-renormalized.