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Physical review. B, Condensed matter

Finite-size scaling on random magnetic structures

Fabio D. A. Aarao Reis

Published May 1, 1997• 6 citations
Publisher

Abstract

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}${)}_{\mathrm{L}}$ and the pseudocritical temperature ${\mathrm{T}}^{\mathrm{*}}$(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.

DOI
10.1103/physrevb.55.11084

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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.
We studied the ferromagnetic Ising model on two-dimensional finite systems with non-integer characteristic lengths. First we considered very long strips with a finite number I of complete rows and one partially filled row with probability x , which is the two-dimensional version of a layer-by-layer growth. At fixed temperature, when the characteristic length L=I+x increases, the free energy per spin, the specific heat and the magnetic susceptibility oscillate, attaining relative extremes at integer L . The oscillations in the free energy are interpreted as surface corrections related to the oscillations in the mean coordination number. Finite-size scaling relations are not satisfied with continuous L , but still hold for fixed x and variable I , where the differences of mean widths are integers. For fixed x , the fits of the free energy give the conformal anomaly with very good accuracy. We also studied strips with discretized Gaussian distributions of widths, with non-integer means L and rms deviation . In these structures, the thermodynamic quantities vary monotonically with continuous L , but some methods for calculating critical exponents do not work properly when generalized to continuous L . We also obtain with good accuracy in these systems. We discuss the possible implications of our results to real systems behaviour.
We have calculated finite-size correlation lengths for strips of the square-lattice Ising spin glass using the transfer-matrix method. In order to minimize sampling errors we study strip lengths up to 5\ifmmode\times\else\texttimes\fi{}106{10}^{6}106 lattice spacings. A phenomenological renormalization-group analysis indicates that there are strong corrections to simple power-law scaling near the zero-temperature critical point, as is to be expected near a lower critical dimension. We examine models with Gaussian, exp(-J2{J}^{2}J2/2), and exponential, exp(-\ensuremath{\Vert}J\ensuremath{\Vert}), distributions of couplings; the Gaussian distribution shows stronger finite-size corrections. The correlation-length exponent is estimated to be \ensuremath{\nu}=4.2\ifmmode\pm\else\textpm\fi{}0.5, although we do not want to rule out the possibility that \ensuremath{\nu} is significantly larger than this.
We consider long strips of finite width L\ensuremath{\le}13 sites of ferromagnetic Ising spins with random couplings distributed according to the binary distribution P(Jij{\mathit{J}}_{\mathit{ij}}Jij​)=1/2[\ensuremath{\delta}(Jij{\mathit{J}}_{\mathit{ij}}Jij​-J0{\mathit{J}}_{0}J0​)+\ensuremath{\delta}(Jij{\mathit{J}}_{\mathit{ij}}Jij​- rJ0{\mathit{rJ}}_{0}rJ0​)], 0r1. Spin-spin correlation functions 〈${\mathrm{\ensuremath{\sigma}}}_{0}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{R}}$〉 along the ``infinite'' direction are computed by transfer-matrix methods, at the critical temperature of the corresponding two-dimensional system, and their probability distribution is investigated. We show that, although in-sample fluctuations do not die out as strip length is increased, averaged values converge satisfactorily. These latter are very close to the critical correlation functions of the pure Ising model, in agreement with recent Monte Carlo simulations. A scaling approach is formulated, which provides the essential aspects of the R and L dependence of the probability distribution of ln〈${\mathrm{\ensuremath{\sigma}}}_{0}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{R}}$〉, including the result that the appropriate scaling variable is R/L. Predictions from scaling theory are borne out by numerical data, which show the probability distribution of ln〈${\mathrm{\ensuremath{\sigma}}}_{0}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{R}}$〉 to be remarkably skewed at short distances, approaching a Gaussian only as R/L\ensuremath{\gg}1. \textcopyright{} 1996 The American Physical Society.
We address the question of weak versus strong universality scenarios for the random-bond Ising model in two dimensions. A finite-size scaling theory is proposed, which explicitly incorporates lnL\mathrm{ln} LlnL corrections (L(L(L is the linear finite size of the system) to the temperature derivative of the correlation length. The predictions are tested by considering long, finite-width strips of Ising spins with randomly distributed ferromagnetic couplings, along which free energy, spin-spin correlation functions, and specific heats are calculated by transfer-matrix methods. The ratio $\ensuremath{\gamma}/\ensuremath{\nu}$ is calculated and has the same value as in the pure case; consequently conformal invariance predictions remain valid for this type of disorder. Semilogarithmic plots of correlation functions against distance yield average correlation lengths ${\ensuremath{\xi}}^{\mathrm{av}}$, whose size dependence agrees very well with the proposed theory. We also examine the size dependence of the specific heat, which clearly suggests a divergency in the thermodynamic limit. Thus our data consistently favor the Dotsenko-Shalaev picture of logarithmic corrections (enhancements) to pure system singularities, as opposed to the weak universality scenario.
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.
We consider long, finite-width strips of Ising spins with randomly distributed couplings. Frustration is introduced by allowing both ferromagnetic and antiferromagnetic interactions. Free energy and spin-spin correlation functions are calculated by transfer-matrix methods. Numerical derivatives and finite-size scaling concepts allow estimates of the usual critical exponents $\ensuremath{\gamma}/\ensuremath{\nu},$ $\ensuremath{\alpha}/\ensuremath{\nu},$ and $\ensuremath{\nu}$ to be obtained, whenever a second-order transition is present. Low-temperature ordering persists for suitably small concentrations of frustrated bonds, with a transition governed by pure-Ising exponents. Contrary to the unfrustrated case, subdominant terms do not fit a simple, logarithmic-enhancement form. Our analysis also suggests a vertical critical line at and below the Nishimori point. Approaching this point along either the temperature axis or the Nishimori line, one finds nondiverging specific heats. A percolationlike ratio $\ensuremath{\gamma}/\ensuremath{\nu}$ is found upon analysis of the uniform susceptibility at the Nishimori point. Our data are also consistent with frustration inducing a breakdown of the relationship between correlation-length amplitude and critical exponents, predicted by conformal invariance for pure systems.