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Physical review. B, Condensed matter

Effects of finite geometry and boundary conditions on scattering functions at two-dimensional critical points

Peter H. Kleban, Reinhard Hentschke

Published Aug 1, 1986• 11 citations
Publisher

Abstract

Using the consequences of conformal invariance we examine the effects of finite size and boundary conditions on the scattering line shape for two-dimensional systems at bulk critical points. The "apparent exponent" behavior observed for free boundary conditions is modeled analytically, and the influence of the finite geometry on the overall line shape is discussed. For fixed boundary conditions critical effects are strongly suppressed.

DOI
10.1103/physrevb.34.1980

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Using the consequences of conformal invariance we examine the effects of finite size and boundary conditions on the scattering line shape for two-dimensional systems at bulk critical points. The "apparent exponent" behavior observed for free boundary conditions is modeled analytically, and the influence of the finite geometry on the overall line shape is discussed. For fixed boundary conditions critical effects are strongly suppressed.
Using recent results by Cardy (1984) based on conformal invariance of critical correlation functions the authors calculate universal results for scattering functions S(k), susceptibilities, correlation lengths and specific heat correction terms for finite Ising systems in two dimensions with circular, elliptical and rectangular shapes and free boundary conditions. Results show the effects of critical order on these quantities. For a circle, S(k) decays as 1/k 2- eta for a large range of intermediate k values with an 'apparent' exponent eta =0.09. The probable influence of end, edge and domain wall effects in the rectangular geometry is discussed. Application of the results to experimental systems and other theoretical models is discussed.
We show how the apparent power-law scaling of classical two-dimenional structure factors in bounded domains at criticality is related to fixed and free boundary conditions on the order parameter. This enables us to predict the behavior of the corresponding apparent scaling dimensions based on a simple symmetry analysis. With the use of correlation functions from conformal field theory, our predictions are confirmed with exact calculations for multicritical Ising models. Effects of random boundary fields are also discussed.
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 critical behavior of site percolation confined to the interior of a parabola is investigated. At the percolation threshold, conformal mapping is used to derive analytical expressions for the pair correlation function and the local order parameter in the case of free and fixed boundary conditions, respectively. Numerical results obtained by intensive Monte Carlo simulations are in excellent agreement with the analytical expressions. The numerical data for the off-critical local order parameter with free boundary conditions show that correction to scaling terms are important and the leading correction exponent is estimated to be \ensuremath{\epsilon}\ensuremath{\simeq}0.18.