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Physical review. B, Condensed matter

Capillary length of a planar interface from low temperatures to the critical point: An Isingd=2 strip

J. Stecki

Published Mar 15, 1993• 18 citations
Publisher

Abstract

The correlation length controlling the long-range decay of the density-density correlation function in the direction parallel to the interface is computed exactly, for T\ensuremath{\le}${\mathit{T}}_{\mathit{c}}$, in the two-dimensional Ising M\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty} strip periodic in the long direction with a (+/-) boundary condition in the other direction which creates a planar strongly fluctuating interface. A crossover from long-range capillary-wave-like behavior to short-range bulk correlations above ${\mathit{T}}_{\mathit{c}}$, a new scaling function, and corrections to scaling at ${\mathit{T}}_{\mathit{c}}$ are found.

DOI
10.1103/physrevb.47.7519

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The correlation length controlling the long-range decay of the density-density correlation function in the direction parallel to the interface is computed exactly, for T\ensuremath{\le}Tc{\mathit{T}}_{\mathit{c}}Tc​, in the two-dimensional Ising M\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty} strip periodic in the long direction with a (+/-) boundary condition in the other direction which creates a planar strongly fluctuating interface. A crossover from long-range capillary-wave-like behavior to short-range bulk correlations above Tc{\mathit{T}}_{\mathit{c}}Tc​, a new scaling function, and corrections to scaling at Tc{\mathit{T}}_{\mathit{c}}Tc​ are found.
By solving partial differential equations for the correlation functions implied by conformal invariance, we obtain exact results at the critical point for the bulk six-point function 〈\ensuremath{\sigma}(1)...\ensuremath{\sigma}(6)〉 of the two-dimensional Ising model and the three-point functions 〈\ensuremath{\sigma}(1)\ensuremath{\sigma}(2)\ensuremath{\sigma}(3)〉, 〈\ensuremath{\sigma}(1)\ensuremath{\mu}(2)\ensuremath{\mu}(3)〉 of the semi-infinite two-dimensional Ising model with fixed boundary spins. Here \ensuremath{\sigma} and \ensuremath{\mu} denote order (spin) and disorder variables, respectively. We also derive the explicit form of the magnetization profile at the bulk critical temperature in an Ising strip of infinite length and finite width, with spin-up boundary conditions on one edge and spin-down boundary conditions on the other.
We examine the derivation and use of a short-ranged (d-1)-dimensional interface Hamiltonian to describe properties of a d-dimensional liquid-vapor or Ising system near the critical point. We argue that such a simplified description, which ignores bulk excitations (``bubbles'' of the opposite phase) and multiple-valued interface configurations (``overhangs'') is valid only on length scales larger than the bulk correlation length ${\ensuremath{\xi}}_{B}$. Such excitations with wavelengths up to order ${\ensuremath{\xi}}_{B}$ are essential for a correct description of the critical fluctuations, and preclude the use of an interface Hamiltonian to study bulk critical properties. This is explicitly demonstrated in d=2 by showing that bubbles and overhangs are relevant operators and we argue that this is true in any dimension. (However, these contributions do not necessarily affect the formal perturbation expansion about the degenerate case d=1, as carried out by Wallace and Zia.) This viewpoint is implicit in the physical picture Widom used to derive scaling laws relating interface and bulk critical properties. The long-wavelength fluctuations accurately described by an interface Hamiltonian produce a ``wandering'' of the interface, but this plays no important role in the critical behavior and can be reconciled with Widom's picture. We examine several modifications of the usual Ising model for which in certain limits a single-valued description becomes exact. Such models either exhibit no bulk critical behavior at all, even if the surface tension \ensuremath{\sigma} vanishes, or have critical properties in a different universality class from the usual Ising-model (liquid-vapor) critical point.
According to finite-size scaling theory, the correlation length in an Ising strip with infinite length and a width of L lattice constants has the form ${\ensuremath{\xi}}_{L}$(T)=Lf(c(T-Tc{T}_{c}Tc​)L) in the scaling regime L\ensuremath{\gg}1,\ensuremath{\Vert}T-Tc{T}_{c}Tc​\ensuremath{\Vert}/Tc{T}_{c}Tc​\ensuremath{\ll}1. Here f(x) is a universal scaling function, which does, however, depend on the boundary conditions at the edges of the strip. Utilizing the correspondence between the two-dimensional Ising model and the one-dimensional quantum Ising model in a transverse field, we calculate the explicit form of the function f(x) for spin-spin and energy-energy correlations in strips with periodic, free, fixed, and mixed free-fixed boundary conditions.
The author considers the finite-size (FS) scaling behaviour of the transverse correlation lengths xi /sub /// (n) for an interface confined in a strip geometry exactly at a second-order fluctuation-dominated wetting transition. He shows that the dimensionless amplitude ratios A/sub /// (n) = Sigma L 2 /k B T xi /sub /// (n) (with Sigma the surface stiffness coefficient) are related to the short-distance expansion critical exponent theta by A/sub /// (n) = pi 2 n(n+ theta -1)/2 for all fluctuation-dominated wetting transitions, provided the binding potential V(y) in the capillary-wave model is conformally mapped from the semi-infinite plane. He argues that the existence of this amplitude-exponent relation reflects the conformal invariance of the capillary-wave model Lagrange density and lists a number of one-point functions which can be shown to exhibit this invariance explicitly.
The scaling function at the wetting transition for the two-point energy-density correlation function is computed exactly for two-dimensional Ising models with interfaces in the [10] and [11] directions. The two-point separations perpendicular and parallel to the wall are scaled by ${\ensuremath{\xi}}_{\ensuremath{\perp}}\ensuremath{\sim}{({T}_{w}\ensuremath{-}T)}^{\ensuremath{-}1}$ and ${\ensuremath{\xi}}_{\ensuremath{\parallel}}\ensuremath{\sim}{({T}_{w}\ensuremath{-}T)}^{\ensuremath{-}2}$, respectively; and the energy-density operator scales as ${T}_{w}\ensuremath{-}T$. The resulting scaling function is independent of the microscopic couplings, boundary interactions, wall orientation, and specifics of the operator. Furthermore, the result is simple and consists of Gaussians and error functions, suggestive of random walk.
Off-diagonal profiles of local densities (e.g. order parameter or energy density) are calculated at the bulk critical point, by conformal methods, on a strip with transverse coordinate v, for different types of boundary conditions (free, fixed and mixed). Such profiles, which are defined by the non-vanishing matrix element of the appropriate operator between the ground state and the corresponding lowest excited state of the strip Hamiltonian, enter into the expression of two-point correlation functions on a strip. They are of interest in the finite-size scaling study of bulk and surface critical behaviour since they allow the elimination of regular contributions. The conformal profiles, which are obtained through a conformal transformation of the correlation functions from the half-plane to the strip, are in agreement with the results of a direct calculation, for the energy density of the two-dimensional Ising model.