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Journal of Physics A Mathematical and General

Finite-size effects at 2D Ising critical points via conformal mapping

Peter H. Kleban, Gulden Akinci, Reinhard Hentschke, Kenneth R. Brownstein

Published Feb 21, 1986• 27 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/19/3/024

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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.
Using a bond-propagation algorithm, we study the finite-size behavior of the critical two-dimensional Ising model on a finite triangular lattice with free boundaries in five shapes: triangular, rhomboid, trapezoid, hexagonal, and rectangular. The critical free energy, internal energy, and specific heat are calculated. The accuracy of the free energy reaches 10(-26). Based on accurate data on several finite systems with linear size up to N=2000, we extract the bulk, surface, and corner parts of the free energy, internal energy, and specific heat accurately. We confirm the conformal field theory prediction that the corner free energy is universal and find logarithmic corrections in higher-order terms in the critical free energy for the rhomboid, trapezoid, and hexagonal systems, which are absent for the triangular and rectangular systems. The logarithmic edge corrections due to edges parallel or perpendicular to the bond directions in the internal energy are found to be identical, while the logarithmic edge corrections due to corresponding edges in the specific heat are different. The corner internal energy and corner specific heat for angles π/3, π/2, and 2π/3 are obtained, as well as higher-order corrections. Comparing with the corner internal energy and corner specific heat we previously found on a rectangle of the square lattice [Phys. Rev. E 86, 041149 (2012)], we conclude that the corner internal energy and corner specific heat for the rectangular shape are not universal.
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 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.