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

Crossover between ordinary and normal transitions in the presence of a bulk field

Anna Maciołek, A. Drzewiński, A. Ciach

Published Jul 25, 2001• 6 citations
Publisher

Abstract

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.

DOI
10.1103/physreve.64.026123

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arXiv:cond-mat/0001291•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Sep 1, 2000•4 citations
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 films $L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ in a nonvanishing bulk magnetic field H are studied at the bulk critical temperature Tc{T}_{c}Tc​ for two choices of surface fields (a) H1=HL=0{H}_{1}{=H}_{L}=0H1​=HL​=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 fsolv{f}_{\mathrm{solv}}fsolv​ are found and discussed. When H1=0{H}_{1}=0H1​=0 the scaling function for fsolv{f}_{\mathrm{solv}}fsolv​ 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,H=0,H=0, the Casimir amplitude. For the normal transition the scaling function for fsolv{f}_{\mathrm{solv}}fsolv​ has a single minimum near the continuation of the pseudocoexistence (capillary condensation) line, with an amplitude about 100 times the Casimir amplitude.
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.
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.
We consider d=2 Ising strip with surface fields acting on boundary spins. Using the properties of the transfer matrix spectrum we identify two pseudotransition temperatures and show that they satisfy similar scaling relations as expected for real transition temperatures in strips with d>2. The solvation force between the boundaries of the strip is analysed as a function of temperature, surface fields and the width of the strip. For large widths the solvation force can be described by scaling functions in three different regimes: in the vicinity of the critical wetting temperature of 2D semi-infinite system, in the vicinity of the bulk critical temperature, and in the regime of weak surface fields where the critical wetting temperature tends towards the bulk critical temperature. The properties of the relevant scaling functions are discussed.
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.