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Physical review. B, Condensed matter

Interface energy in random systems

Bernard Derrida, J. Vannimenus

Published Apr 1, 1983• 35 citations
Publisher

Abstract

We study the interface energy $\ensuremath{\sigma}$ as a function of disorder in two-dimensional Ising-type systems at $T=0$ for the percolation and frustration models. Our approach consists in calculating this energy for long strips of varying widths by a random sampling method, then extrapolating the results. In the weak-disorder limit, very wide strips may be studied and accurate values are obtained for the first-order correction to the interface energy in both models. Only moderate widths (up to $N=9\ensuremath{-}10$) can be studied in the general situation and we use finite-size scaling to analyze the data in the region of the threshold, where $\ensuremath{\sigma}$ vanishes with a critical exponent $v$. For percolation, we obtain $\frac{v}{\ensuremath{\nu}}=0.98\ifmmode\pm\else\textpm\fi{}0.05$, where $\ensuremath{\nu}$ is the correlation-length exponent, in agreement with Deutscher and Rappaport's proposal that $v=\ensuremath{\nu}$ exactly. The analysis of the results is more difficult in the case of frustration, because size effects are important and the scaling region is not reached for $N=9$. Our data show that Monte Carlo results for $\ensuremath{\sigma}$ are unreliable and that much care is necessary to reach firm conclusions on the frustration threshold ${x}_{c}$ or the exponents $\ensuremath{\nu}$ and $v$.

DOI
10.1103/physrevb.27.4401

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We address the issue of universality in two-dimensional disordered Ising systems, by considering long, finite-width strips of ferromagnetic Ising spins with randomly distributed couplings. We calculate the free energy and spin-spin correlation functions (from which averaged correlation lengths, ${\ensuremath{\xi}}^{\mathrm{av}}$, are computed) by transfer-matrix methods. An ansatz for the size dependence of logarithmic corrections to ${\ensuremath{\xi}}^{\mathrm{av}}$ is proposed. Data for both random-bond and site-diluted systems show that pure system behavior (with $\ensuremath{\nu}=1$) is recovered if these corrections are incorporated, discarding the weak-universality scenario.
Systems with quenched disorder possess complex energy landscapes that are challenging to explore under the conventional Monte Carlo method. In this work, we implement an efficient entropy sampling scheme for accurate computation of the entropy function in low-energy regions. The method is applied to the two-dimensional ±J\pm J±J random-bond Ising model, where frustration is controlled by the fraction ppp of ferromagnetic bonds. We investigate the low-temperature paramagnetic--ferromagnetic phase boundary below the multicritical point at TN=0.9530(4)T_N = 0.9530(4)TN​=0.9530(4), PN=0.89078(8)P_N = 0.89078(8)PN​=0.89078(8), as well as the zero-temperature ferromagnetic--spin-glass transition. Finite-size scaling analysis reveals that the phase boundary for T<TNT < T_NT<TN​ exhibits reentrant behavior. By analyzing the evolution of the magnetization-resolved density of states g(E,M)g(E, M)g(E,M) and ground-state spin configurations against increasing frustration, we provide strong evidence that the zero-temperature transition is quasi-first order. Finite-size scaling conducted on the spin-glass side supports the validity of β=0β= 0β=0, with a correlation length exponent ν=1.50(8)ν= 1.50(8)ν=1.50(8). Our results provide new insights into the nature of the ferromagnetic-to-spin-glass phase transition in an extensively degenerate ground state.
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We study the ferromagnetic Ising model on very long strips of random widths using transfer matrix techniques. The width distribution is Gaussian with mean L and rms deviation \ensuremath{\Delta}L, which may be constant for any L (type I) or proportional to L (type II). We calculate the initial susceptibility for strips with 4\ensuremath{\leqslant}L\ensuremath{\leqslant}12, obtaining estimates of the ratio of exponents (\ensuremath{\gamma}/\ensuremath{\nu})L{)}_{\mathrm{L}})L​ and the pseudocritical temperature T∗{\mathrm{T}}^{\mathrm{*}}T∗(L,L-1). In strips of type I, those estimates satisfy finite-size scaling, with finite-size corrections increasing with \ensuremath{\Delta}L, and in strips of type II it is satisfied only for very large lengths L. We discuss the influence of nonuniform thicknesses on the magnetism of low-dimensional systems, comparing the finite-size corrections of this problem to those which fit the experimental data of magnetic thin films.