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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Effect of bulk magnetic field on critical Ising films

A. Drzewiński, Anna Maciołek, A. Ciach

Published May 1, 2000• 19 citations
Publisher

Abstract

Two-dimensional Ising films $L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ in a nonvanishing bulk magnetic field H are studied at the bulk critical temperature ${T}_{c}$ for two choices of surface fields (a) ${H}_{1}{=H}_{L}=0$ (ordinary transition), and (b) ${H}_{1}{=H}_{L}=\ensuremath{\infty}$ (normal transition) by the density-matrix renormalization-group method. Universal scaling functions for magnetization profiles, the excess magnetization $\ensuremath{\Gamma},$ the longitudinal correlation length ${\ensuremath{\xi}}_{\ensuremath{\Vert}},$ and for the analog of the solvation force ${f}_{\mathrm{solv}}$ are found and discussed. When ${H}_{1}=0$ the scaling function for ${f}_{\mathrm{solv}}$ has two symmetric minima at $y=\mathrm{sgn}(H)L|H{|}^{\ensuremath{\nu}/\ensuremath{\Delta}}\ensuremath{\approx}\ifmmode\pm\else\textpm\fi{}1$ with an amplitude at the minimum about 3.8 times the value at $H=0,$ the Casimir amplitude. For the normal transition the scaling function for ${f}_{\mathrm{solv}}$ has a single minimum near the continuation of the pseudocoexistence (capillary condensation) line, with an amplitude about 100 times the Casimir amplitude.

DOI
10.1103/physreve.61.5009

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We investigate two dimensional critical Ising films of width L with surface fields H(1)=H(L) in the crossover between ordinary (H(1)=0) and normal (H(1)=infinity) transitions. Using exact transfer-matrix diagonalization and density matrix renormalization-group (DMRG) methods, we calculate magnetization profiles m(z), the excess magnetization Gamma, and the analog of the solvation force f(solv) as functions of H1 for several L. Scaling functions of the above quantities deviate substantially from their asymptotic forms at fixed points for a broad region of the scaling variable LH21 approximately L/l(1), where l(1) is the length induced by the surface field H1. The scaling function for /f(solv)/ has a deep minimum near LH(2)(1)=1, which is about one order of magnitude smaller than its value at both fixed points (the "Casimir" amplitude). For weak H1 (l(1)>L) the magnetization profile has a maximum at the center of the film, and f(solv) decays much faster than L-2. For stronger H1 (1<l(1)<L), the magnetization has two maxima at a distance approximately l(1) from the walls, and the solvation force decays much slower than L-2. For L>>l(1) the solvation force decays according to the universal power law f(solv) approximately L(-2). The results of the approximate DMRG method show remarkable agreement with the exact ones.
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.
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.
Using the quasi-exact density-matrix renormalization-group method we calculate the solvation forces in two-dimensional Ising films of thickness L subject to identical algebraically decaying boundary fields with various decay exponents p. At the bulk critical point the solvation force acquires a universal contribution which is long-ranged in L due to the critical fluctuations, a phenomenon known as the critical Casimir effect. For p = 2, 3 and 50, we study the scaling behaviour of the solvation force along the pseudo-phase coexistence and along the critical and sub-critical isotherms.
Within the framework of the effective-field theory, we examine the phase transitions of a transverse spin-12\frac{1}{2}21​ Ising film. The critical temperatures of the film as a function of the interactions, transverse fields, and film thickness are studied. It is found that for the ratio of the surface interactions to the bulk ones R=Js/J{R=J}_{s}/JR=Js​/J less than a critical value Rc,{R}_{c},Rc​, the critical temperature Tc/J{T}_{c}/JTc​/J of the film is smaller than the bulk critical temperature TcB/J{T}_{c}^{B}/JTcB​/J and as the film thickness L is increased further, Tc/J{T}_{c}/JTc​/J increases and approaches asymptotically TcB/J{T}_{c}^{B}/JTcB​/J for large values of L. However, for R>Rc,Tc/JR>{R}_{c}, {T}_{c}/JR>Rc​,Tc​/J is larger both than the bulk TcB/J{T}_{c}^{B}/JTcB​/J and the surface TcS/J{T}_{c}^{S}/JTcS​/J critical temperatures of the corresponding semi-infinite system and as the film thickness L is increased further, Tc/J{T}_{c}/JTc​/J decreases and approaches asymptotically, for large values of L, the surface magnetic transition TcS/J{T}_{c}^{S}/JTcS​/J observed in the corresponding semi-infinite system. We calculate also some magnetic properties of the film such as the layer magnetizations, their averages and their profiles and the longitudinal susceptibility of the film. The film longitudinal susceptibility still diverges at the film critical temperature as does the bulk longitudinal susceptibility, but its magnitude is reduced. Also there is a rounded peak at the bulk critical temperature when R>Rc.R>{R}_{c}.R>Rc​. The bulk related character of the susceptibility is more pronounced and the surface related character is less pronounced when the film thickness is large.
We study the surface critical behavior of semi-infinite systems belonging to the bulk universality class of the Ising model. Special attention is paid to the local behavior of experimentally relevant quantities such as the order parameter and the correlation function in the crossover regimes between different surface universality classes, where the surface field h1 and the temperature enhancement c can induce additional macroscopic length scales. Starting from the field-theoretical φ4 model and employing renormalization-group improved perturbation theory (ϵ-expansion), explicit results for the local behavior of the correlation and structure function (zero-loop) and the order parameter (one-loop) are derived. Supplementing earlier studies that focused on the special transition, we here pay particular attention to the situation where a large c suppresses the tendency to order in the surface (ordinary transition) but a surface field h1 generates a small surface magnetization m1. Our results are in good agreement with recent phenomenological considerations and Monte Carlo studies devoted to similar questions and, combined with the latter, provide a much more detailed understanding of the local properties of systems with weak surface fields.