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Journal of Physics A Mathematical and General

Universal amplitude-exponent relations for interfaces and conformal mappings

Andrew O. Parry

Published Aug 21, 1992• 3 citations

Abstract

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.

DOI: 10.1088/0305-4470/25/16/005

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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.

We carefully consider the critical amplitude ratios which may be defined for the local susceptibility at critical wetting transitions. In the strong-fluctuation regime scaling ideas suggest that there are two universal amplitude ratios (denoted R and R z ) which reflect the criticality a ) exactly at the wall and b ) related to the short-distance expansion. R and R z are calculated explicitly in dimension d = 2 and for different mean-field theories where we find that the amplitude relation R = R z is obeyed. In d = 3, corresponding to the marginal dimensionality, R is calculated using a linear renormalization group approach for a rather general model yielding non-universal behaviour. Here r is shown to satisfy a simple amplitude-exponent relation. We argue that future simulation studies of the amplitude ratio for a novel modified Ising model should provide a stringent test of the stiffness-instability theory of Fisher and Jin.

We consider two-dimensional (d = 2) systems with short-ranged microscopic interactions, where interface unbinding (wetting) transitions occur in the limit of vanishing temperature T .For T = 0 the transition is characterized by non-universal critical properties analogous to those established for thermal wetting transitions in d = 3, albeit with a redefined capillary parameter ω.Within a functional renormalization-group treatment of an effective interfacial model, we compute the finite-temperature phase diagram, exhibiting a line of interface unbinding transitions, terminating at T = 0 with an interfacial quantum-critical point.We identify distinct scaling regimes, reflecting the interplay between quantum and thermal interfacial fluctuations.A crossover line marking the onset of the quantum-critical regime is described by the d = 3 interfacial correlationlength exponent ν || .This potentially opens another way to investigate the non-universal character of ν || .On the other hand, the emergent interfacial quantum-critical regime shows no signatures of non-universality.

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.

The authors consider a semi-infinite two-dimensional Ising model with nearest-neighbour couplings that deviate from the bulk critical couplings by Al -2 , where l is the distance from the surface. The surface critical exponents of this system are non-universal. Under a conformal mapping onto a strip of width L, the Al -1 inhomogeneity transforms into A((L/ pi )sin( pi L/L)) -1 . For the square lattice the spectrum of the transfer matrix in the strip geometry is calculated exactly in the extreme anisotropic limit. The analytical results and numerical results for the triangular lattice are compared with predictions of conformal invariance.