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arXiv:cond-mat/0004361 · Physical Review Letters

Local Functional Models of Critical Correlations in Thin Films

Andrew O. Parry, E. D. Macdonald, C. Rascón

Published Nov 6, 2000• 1 citation
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Abstract

Recent work on local functional theories of critical inhomogeneous fluids and Ising-like magnets has shown them to be a potentially exact, or near exact, description of universal finite-size effects associated with the excess free energy and scaling of one-point functions in critical thin films. This approach is extended to predict the two-point correlation function G in critical thin films with symmetric surface fields in arbitrary dimension d. In d = 2 we show there is exact agreement with the predictions of conformal invariance for the complete spectrum of correlation lengths xi((n)) as well as the detailed position dependence of the asymptotic decay of G. In d = 3 and d>/=4 we present new numerical predictions for the universal finite-size correlation length and scaling functions determining the structure of G across the thin film. Highly accurate analytical closed form expressions for these universal properties are derived in arbitrary dimension.

DOI
10.1103/physrevlett.85.4108

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Recent work on local functional theories of critical inhomogeneous fluids and Ising-like magnets has shown them to be a potentially exact, or near exact, description of universal finite-size effects associated with the excess free energy and scaling of one-point functions in critical thin films. This approach is extended to predict the two-point correlation function G in critical thin films with symmetric surface fields in arbitrary dimension d. In d = 2 we show there is exact agreement with the predictions of conformal invariance for the complete spectrum of correlation lengths xi((n)) as well as the detailed position dependence of the asymptotic decay of G. In d = 3 and d>/=4 we present new numerical predictions for the universal finite-size correlation length and scaling functions determining the structure of G across the thin film. Highly accurate analytical closed form expressions for these universal properties are derived in arbitrary dimension.
This article outlines a tutorial review of some recent progress in the theory of phase transitions and criticality. Topics mentioned are: |φ|6 field theories in (d=3)-dimensional space which correspond to the limits of tricriticality in spherical models; the desirability of finding real systems displaying critical endpoints, etc., in a magnetic field; scaling theories for first-order magnetic transitions in block and cylinder geometries, including spin-wave contributions, ϵ=4 - d expansions for finite-size effects at and near criticality; the universal correlation length amplitude for critical-point correlations in a finite cylinder and its evaluation in d = 2 dimensions on the basis of conformal covariance; the predictions of conformal covariance for other finite-size and surface critical-point properties; the implications, stemming from field theory, of conformal covariance and unitarity for general criticality in two dimensions, including infinitely many sets of exact critical exponents for Ising-like, three-state Potts-like, … systems; the discovery of new, exactly soluble two-dimensional, “restricted solid-on-solid” models which display all the predicted conformally covariant multicritical points and a second infinite set as well!
Thermal and magnetic effects in a system consisting of thin layers of coupled Ising spins with S=1/2 and S=1 are considered. The specific heat and the correlation length display maxima at two different temperatures. It is discussed in what sense these maxima can be interpreted as a finite-size rounding of a thermodynamic singularity associated with a phase transition. The connection with ordinary, extraordinary, and special surface phase transitions is made. In two dimensions (2D), the surface critical exponents are calculated from conformal invariance. The bulk and surface finite-size scaling of the order-parameter profiles at the transition points is discussed. In 2D, an exact scaling function for the profiles is suggested through conformal invariance arguments for the (extra)ordinary transition.
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.
Anisotropy effects on the finite-size critical behavior of a two-dimensional Ising model on a general triangular lattice in an infinite-strip geometry with periodic, antiperiodic, and free boundary conditions (bc) in the finite direction are investigated. Exact results are obtained for the scaling functions of the finite-size contributions to the free energy density. With ξ(>) the largest and ξ(<) the smallest bulk correlation length at a given temperature near criticality, we find that the dependence of these functions on the ratio ξ(<)/ξ(>) and on the angle parametrizing the orientation of the correlation volume is of geometric nature. Since the scaling functions are independent of the particular microscopic realization of the anisotropy within the two-dimensional Ising model, our results provide a limited verification of universality. We explain our observations by considering finite-size scaling of free energy densities of general weakly anisotropic models on a d-dimensional film (i.e., in an L×∞(d-1) geometry) with bc in the finite direction that are invariant under a shear transformation relating the anisotropic and isotropic cases. This allows us to relate free energy scaling functions in the presence of an anisotropy to those of the corresponding isotropic system. We interpret our results as a simple and transparent case of anisotropic universality, where, compared to the isotropic case, scaling functions depend additionally on the shape and orientation of the correlation volume. We conjecture that this universality extends to cases where the geometry and/or the bc are not invariant under the shear transformation and argue in favor of validity of two-scale factor universality for weakly anisotropic systems.
The critical behaviour of thin films containing quenched random impurities and inhomogeneities is investigated by the renormalization-group method to the one-loop order within the framework of the n-component φ4-model. The finite-size crossover in impure films has been considered on the basis of the fundamental relationship between the effective dimensionality Deff and the characteristic lengths of the system. The fixed points, their stability properties and the critical exponents are obtained and analyzed, using an ε̃=(4−Deff)-expansion near the effective spatial dimensionality Deff of the fluctuation modes in D-dimensional hyperslabs with two types of quenched impurities: point-like impurities with short-range random correlations and extended (linear) impurities with infinite-range random correlations along the small-size spatial direction. The difference between the critical properties of infinite systems and films is demonstrated and investigated. A new critical exponent, describing the scaling properties of the thickness of films with extended impurities has been derived and calculated. A special attention is paid to the critical behaviour of real impure films (D=3).
We consider variational (density functional) models of fluids confined in parallel-plate geometries (with walls situated in the planes z = 0 and z = L respectively) and focus on the structure of the pair correlation function . We show that for local variational models there exist two non-trivial identities relating both the transverse Fourier transform and the zeroth moment at different positions and . These relations form an algebra which severely restricts the possible form of the function . For the common situations in which the equilibrium one-body (magnetization/number density) profile exhibits an odd or even reflection symmetry in the plane, the algebra simplifies considerably, and is used to relate the correlation function to the finite-size excess free energy . We rederive non-trivial scaling expressions for the finite-size contribution to the free energy at bulk criticality and for systems where large-scale interfacial fluctuations are present. Extensions to non-planar geometries are also considered.