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arXiv:hep-lat/0112037 · Nuclear Physics B - Proceedings Supplements

Exact finite-size scaling with corrections in the two-dimensional Ising model with special boundary conditions

W. Janke, R. Kenna

Published Mar 1, 2002• 5 citations
arXivHTMLPDF
DOI
10.1016/s0920-5632(01)01889-8

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The Ising model in two dimensions with the special boundary conditions of Brascamp and Kunz is analyzed. Leading and subdominant scaling behavior of the Fisher zeros are determined exactly. The exact finite-size scaling, with corrections, of the specific heat is determined both at critical and effective critical (pseudocritical) points. The shift exponents associated with the scaling of these effective critical points are not the same as the inverse correlation length critical exponent. All corrections to scaling are analytic.
INTRODUCTION Finite-size scaling (FSS) is a well established technique for the extraction of critical exponents from finite volume analyses [1]. Such exponents characterise critical phenomena at a second-order phase transition. The simplest model exhibiting such a transition is the Ising model in two dimensions, which, despite a long history and extensive study, still o#ers new results and insights. Here, we study the model under the special boundary conditions of Brascamp and Kunz [2] to extract new information and to help resolve some hitherto puzzling features of FSS. Let CL (#) be the specific heat at inverse temperature # for a system of linear extent L. FSS of the specific heat is characterized by the location of its peak, #L , its height CL (# L ) and its value at the infinite-volume critical point CL (# c ). The peak position, #L , is a pseudocritical point which typically approaches # c as L ## as |# # c L -# , (1) where # is the shift exponent. In two dimensions,
Finite-size scaling, finite-size corrections, and boundary effects for critical systems have attracted much attention in recent years. Here we derive exact finite-size corrections for the free energy F and the specific heat C of the critical ferromagnetic Ising model on the Mu x 2 Nu square lattice with Brascamp-Kunz (BK) boundary conditions [J. Math. Phys. 15, 66 (1974)] and compare such results with those under toroidal boundary conditions. When the ratio xi/2=(Mu+1)/2 Nu is smaller than 1 the behaviors of finite-size corrections for C are quite different for BK and toroidal boundary conditions; when ln(xi/2) is larger than 3, finite-size corrections for C in two boundary conditions approach the same values. In the limit Nu-->infinity we obtain the expansion of the free energy for infinitely long strip with BK boundary conditions. Our results are consistent with the conformal field theory prediction for the mixed boundary conditions by Cardy [Nucl. Phys. B 275, 200 (1986)] although the definitions of boundary conditions in two cases are different in one side of the long strip.