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arXiv:cond-mat/0001291 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Coexistence of excited states in confined Ising systems

A. Drzewiński

Published Sep 1, 2000• 4 citations
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Abstract

Using the density-matrix renormalization-group method we study the two-dimensional Ising model in an infinite strip geometry with free-boundary conditions. The renormalization scheme enables us to consider systems of width up to 300 (lattice spacings) and study the influence of the bulk magnetic field on correlation function structure for all temperatures. From our numerical results we have determined the crossover line for the correlation length related to the coexistence of the excited states. A detailed scaling study of this line is performed. Our numerical results support and further specify previous conclusions reached by Abraham, Parry, and Upton based on the bubble model of correlations.

DOI
10.1103/physreve.62.4378

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Using the density-matrix renormalization-group method we study the two-dimensional Ising model in an infinite strip geometry with free-boundary conditions. The renormalization scheme enables us to consider systems of width up to 300 (lattice spacings) and study the influence of the bulk magnetic field on correlation function structure for all temperatures. From our numerical results we have determined the crossover line for the correlation length related to the coexistence of the excited states. A detailed scaling study of this line is performed. Our numerical results support and further specify previous conclusions reached by Abraham, Parry, and Upton based on the bubble model of correlations.
Two-dimensional Ising strips subject to identical surface fields h(1)=h(2) > or =0 are studied for temperatures above and below the bulk critical temperature T(c) and a range of bulk fields h by means of the density-matrix renormalization-group method. In the case of nonvanishing surface fields, the near-critical behavior of the solvation force f(solv), total adsorption Gamma, inverse longitudinal correlation length xi(parallel)( -1) and specific heat C(H) is strongly influenced by the (pseudo) capillary condensation that occurs below T(c). We obtain scaling functions of f(solv), Gamma, and xi(parallel)(-1). C(H) exhibits a weakly rounded singularity on crossing the pseudocoexistence line. We contrast these results with those for the case of free boundaries where, for temperatures slightly below T(c), f(solv) and C(H) exhibit a sharp extremum away from h=0. Our results have direct repercussions for the properties of near-critical Ising films in three dimensions and we argue that the long-ranged solvation (Casimir) force in confined fluids should be more attractive in the neighborhood of the capillary critical point than exactly at the bulk critical point.
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).
The results of high-accuracy density-matrix renormalization-group calculations for infinite Ising strips of finite widths $40\ensuremath{\leqslant}L\ensuremath{\leqslant}400$ with open boundary conditions are presented. It is shown that in the presence of a small external magnetic field the infinite-system critical power laws can be observed for LLL of the order of hundreds of nanometers. The single power law describes the field dependence of the magnetization or the longitudinal correlation length only on the infinite-system critical isotherm independently of the value of LLL. The approximate power law which describes how the magnetization varies with distance from the infinite-system critical point for several directions in the plane (temperature, external field) is also studied.
We investigate two-dimensional Ising films at the critical temperature T(c) and nonzero bulk magnetic field h using the density-matrix renormalization-group method. The crossover between ordinary (h(1)=0) and normal (h(1)=infinity) transitions corresponding to finite values of the surface fields h(1)=h(2), is studied. The structure and the solvation force f(solv) as a function of h, crucially depend on the value of h(1). Scaling functions for f(solv) and the longitudinal correlation length are given and discussed.
We consider the two-dimensional (2D) random Ising model on a diagonal strip of the square lattice, where the bonds take two values, J1>J2, with equal probability. Using an iterative method, based on a successive application of the star-triangle transformation, we have determined at the bulk critical temperature the correlation length along the strip xi(L) for different widths of the strip L</=21. The ratio of the two lengths xi(L)/L=A is found to approach the universal value A=2/pi for large L, independent of the dilution parameter J(1)/J(2). With our method we have demonstrated with high numerical precision, that the surface correlation function of the 2D dilute Ising model is self-averaging, in the critical point conformally covariant and the corresponding decay exponent is eta( ||)=1.
The excess adsorption Gamma in two-dimensional Ising strips (infinityxL), subject to identical boundary fields at both one-dimensional surfaces decaying in the orthogonal direction j as -h1j(-p), is studied for various values of p and along various thermodynamic paths below the bulk critical point by means of the density-matrix renormalization-group method. The crossover behavior between the complete-wetting and critical-adsorption regimes, occurring in semi-infinite systems, is strongly influenced by confinement effects. Along isotherms T=const the asymptotic power-law dependences on the external bulk field, which characterize these two regimes, are pre-empted by capillary condensation. Along the pseudo-first-order phase-coexistence line of the strips, which varies with temperature, we find a broad crossover regime in which both the thickness of the wetting film and Gamma increase as functions of the reduced temperature tau but do not follow any power law. Above the wetting temperature the order-parameter profiles are not slablike but exhibit wide interfacial variations and pronounced tails.