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arXiv:cond-mat/0105326 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Correlation functions, free energies, and magnetizations in the two-dimensional random-field Ising model

S. L. A. de Queiroz, Robin B. Stinchcombe

Published Aug 28, 2001• 5 citations
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Abstract

Transfer-matrix methods are used to calculate spin-spin correlation functions (G), Helmholtz free energies (f) and magnetizations (m) in the two-dimensional random-field Ising model close to the zero-field bulk critical temperature T(c 0), on long strips of width L=3-18 sites, for binary field distributions. Analysis of the probability distributions of G for varying spin-spin distances R shows that describing the decay of their averaged values by effective correlation lengths is a valid procedure only for not very large R. Connections between field and correlation function distributions at high temperatures are established, yielding approximate analytical expressions for the latter, which are used for computation of the corresponding structure factor. It is shown that, for fixed R/L, the fractional widths of correlation-function distributions saturate asymptotically with L-2.2. Considering an added uniform applied field h, a connection between f(h), m(h), the Gibbs free energy g(m) and the distribution function for the uniform magnetization in a zero uniform field, P0(m), is derived and first illustrated for pure systems, and then applied for nonzero random field. From finite-size scaling and crossover arguments, coupled with numerical data, it is found that the width of P0(m) varies against (nonvanishing, but small) random-field intensity H0 as H(-3/7)(0).

DOI
10.1103/physreve.64.036117

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Transfer-matrix methods are used to calculate spin-spin correlation functions (G), Helmholtz free energies (f) and magnetizations (m) in the two-dimensional random-field Ising model close to the zero-field bulk critical temperature T(c 0), on long strips of width L=3-18 sites, for binary field distributions. Analysis of the probability distributions of G for varying spin-spin distances R shows that describing the decay of their averaged values by effective correlation lengths is a valid procedure only for not very large R. Connections between field and correlation function distributions at high temperatures are established, yielding approximate analytical expressions for the latter, which are used for computation of the corresponding structure factor. It is shown that, for fixed R/L, the fractional widths of correlation-function distributions saturate asymptotically with L-2.2. Considering an added uniform applied field h, a connection between f(h), m(h), the Gibbs free energy g(m) and the distribution function for the uniform magnetization in a zero uniform field, P0(m), is derived and first illustrated for pure systems, and then applied for nonzero random field. From finite-size scaling and crossover arguments, coupled with numerical data, it is found that the width of P0(m) varies against (nonvanishing, but small) random-field intensity H0 as H(-3/7)(0).
Transfer-matrix methods are used to study the probability distributions of spin-spin correlation functions G in the two-dimensional random-field Ising model, on long strips of width L=3-15 sites, for binary field distributions at generic distance R, temperature T, and field intensity h(0). For moderately high T, and h(0) of the order of magnitude used in most experiments, the distributions are singly peaked, though rather asymmetric. For low temperatures the single-peaked shape deteriorates, crossing over towards a double-delta ground-state structure. A connection is obtained between the probability distribution for correlation functions and the underlying distribution of accumulated field fluctuations. Analytical expressions are in good agreement with numerical results for R/L > or approximately 1, low T, h(0) not too small, and near G=1. From a finite-size ansatz at T=T(c)(h(0)=0), h(0)-->0, averaged correlation functions are predicted to scale with L(y)h(0), y=7/8. From numerical data we estimate y=0.875+/-0.025, in excellent agreement with theory. In the same region, the rms relative width W of the probability distributions varies for fixed R/L=1 as W approximately h(kappa)(0) f(L h(u)(0)) with kappa approximately 0.45, u approximately 0.8; f(x) appears to saturate when x-->infinity, thus implying W approximately h(kappa)(0) in d=2.
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 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.
The equilibrium properties of the two-dimensional Ising model with nearest-neighbour interactions uniformly distributed between -J and J have been determined using a numerical transfer matrix method. The calculations are performed on a long strip of width L(3<or=L<or=11) with periodic boundary conditions and finite-size scaling arguments are used to extrapolate to L= infinity . The spin-glass correlation length diverges as T - nu at low temperature with nu =2.96+or-0.22. The correlation length scaling function differs from that for the random Ising model with interactions of magnitude J and random sign showing that the two models with different distribution functions belong to different universality classes. The heat capacity is linear at low temperature and can be interpreted using a continuous distribution of two-level states which is finite at zero energy and increases approximately linearly with energy. At zero temperature the entropy is zero and the internal energy extrapolated to L= infinity is -(0.7943+or-0.0010)J.
The two-dimensional random-bond Ising model is numerically studied on long strips by transfer-matrix methods. It is shown that the rate of decay of correlations at criticality, as derived from averages of the two largest Lyapunov exponents plus conformal invariance arguments, differs from that obtained through direct evaluation of correlation functions. The latter is found to be, within error bars, the same as in pure systems. Our results confirm field-theoretical predictions. The conformal anomaly c is calculated from the leading finite-width correction to the averaged free energy on the strips. Estimates thus obtained are consistent with c=1/2, the same as for the pure Ising model.
The statistical mechanics of finite $L\ifmmode\times\else\texttimes\fi{}L$ Ising square lattices in a field $\ifmmode\pm\else\textpm\fi{}h$ of random sign is investigated numerically by a modified recursive transfer matrix method for 6< L< 166<~L<~166< L< 16. Our results are consistent with the absence of a spontaneous magnetization for $h\ensuremath{\ne}0$ even in the ground state. The singularities occurring at T=0T=0T=0 in the range 0< h< 4J0<~h<~4J0< h< 4J, JJJ being the exchange constant, are discussed in terms of a cluster expansion. For nonzero TTT, less than the critical temperature of the pure two-dimensional Ising model, and hhh sufficiently small, the system exhibits a nonzero spin-glass order parameter of the Mattis type although there is no magnetization. The ferromagnetic correlation function becomes long ranged for $h\ensuremath{\rightarrow}0$ and is calculated from the domain-wall density.