Critical behavior of the two-dimensional Ising model with a slit.

Abstract

The two-dimensional Ising model with a slit is studied. The slit free energy is defined, in which the bulk term, edge terms, and corner terms other than that of the slit are canceled. The bond propagation algorithm is used to calculate the free energy, internal energy, and heat capacity numerically. At the critical point, the slit free energy has a logarithmic correction L^{-1}lnL, in addition to the correction lnL, where L is the typical length of the system. The fitted coefficients of these terms agree with the conformal field theory very well. The integral constant and extrapolation length, which are pending in the conformal field theory, are obtained. In the slit internal energy, the leading term is proportional to L. There is a correction term, ln^{2}L, that has a geometry-independent coefficient, the exact value of which is conjectured to be -sqrt[2]/(2π^{2}). The leading term in the slit heat capacity is proportional to L^{2}. There is also a ln^{2}L term; however, its coefficient depends on the geometry. In the infinitely long strip limit, its exact value is conjectured to be 5/π^{2}. Near the critical point, for different size lattices, the rescaled slit free energies, internal energies, and heat capacities collapse onto the corresponding universal curves. According to the scaling and conformal field theory, the leading term of the slit free energy should be -c/8ln|t| for L|t|≫1 and |t|≪1, where t is the reduced temperature. The amplitude c/8 is universal, and c is the central charge. Our results validate these predictions qualitatively. For L|t|≫1 and |t|≪1, the slit internal energy and heat capacity scale as t^{-1} and t^{-2}, respectively.