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arXiv:hep-lat/9209006

Capillary Waves in Binder's Approach to the Interface Tension

U.‐J. Wiese

Published Sep 3, 1992• 2 citations
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Abstract

In Binder's approach the reduced interface tension sigma of the Ising model in the broken phase is determined from the finite volume effects of the partition function Z(M) at fixed total magnetization M. For small |M| the partition function of a system of size L^d with periodic boundary conditions is dominated by configurations with two interfaces, such that Z(M) ~ exp(- 2 sigma L^{d-1}). Capillary wave fluctuations of the interfaces correct this result to Z(M) ~ exp(- 2 sigma L^{d-1}) with x = -1. The knowledge of the pre-exponential behavior allows an improved fit of numerical data, and a determination of the interface stiffness.

DOI
10.48550/arxiv.hep-lat/9209006

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In Binder's approach the reduced interface tension sigma of the Ising model in the broken phase is determined from the finite volume effects of the partition function Z(M) at fixed total magnetization M. For small |M| the partition function of a system of size L^d with periodic boundary conditions is dominated by configurations with two interfaces, such that Z(M) ~ exp(- 2 sigma L^{d-1}). Capillary wave fluctuations of the interfaces correct this result to Z(M) ~ exp(- 2 sigma L^{d-1}) with x = -1. The knowledge of the pre-exponential behavior allows an improved fit of numerical data, and a determination of the interface stiffness.
We study interfaces with periodic boundary conditions in the low-temperature phase of the improved Blume-Capel model on the simple cubic lattice. The interface free energy is defined by the difference of the free energy of a system with antiperiodic boundary conditions in one of the directions and that of a system with periodic boundary conditions in all directions. It is obtained by integration of differences of the corresponding internal energies over the inverse temperature. These differences can be computed efficiently by using a variance reduced estimator that is based on the exchange cluster algorithm. The interface tension is obtained from the interface free energy by using predictions based on effective interface models. By using our numerical results for the interface tension σ and the correlation length ξ obtained in previous work, we determine the universal amplitude ratios R_{2nd,+}=σ_{0}f_{2nd,+}^{2}=0.3863(6), R_{2nd,-}=σ_{0}f_{2nd,-}^{2}=0.1028(1), and R_{exp,-}=σ_{0}f_{exp,-}^{2}=0.1077(3). Our results are consistent with those obtained previously for the three-dimensional Ising model, confirming the universality hypothesis.
In analogy with a recently proposed multicanonical ensemble we introduce an ensemble where the partition function is simulated with a term in the action containing a varying magnetic field. Using this ensemble we demonstrate on lattices with periodic boundary conditions that it is possible to enhance the appearance of order-order interfaces by many orders of magnitude. To perform a stringent test of the method we consider the D=2 Ising model at \ensuremath{\beta}=0.5 and simulate square lattices up to size 100\ifmmode\times\else\texttimes\fi{}100. By a finite-size scaling analysis, the order-order interface tension per unit area is obtained. Our best infinite-volume extrapolation is in excellent agreement with Onsager's exact result.
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.
The interface between domains of opposite magnetization in the 3D Ising model near the critical temperature displays universal finite-size effects which can be described in terms of a gaussian model of capillary waves. It turns out that these finitesize corrections depend rather strongly on the shape of the lattice. This prediction, which has no adjustable parameters, is tested and accurately verified for various lattice shapes by means of numerical simulations with a cluster algorithm. This supports also a long-standing conjecture on the finite-size effects in Wilson loops of Lattice Gauge Theories. 1
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.
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.