We obtain the exact solution for a two-dimensional rectangular Ising ferromagnet forming a corner with a surface field applied to the spins on edges. We establish the existence of the filling transition and give the condition for the filling temperature. We discuss the basic properties of the transition.
The critical 2-dimensional Ising model is studied with four types boundary conditions: free, fixed ferromagnetic, fixed antiferromagnetic, fixed double antiferromagnetic. Using Bond Propagation algorithms with surface fields, we obtained the free energy, internal energy and specific heat numerically on square lattices with square shape and various combinations of the four types boundary conditions. The numerical data are analyzed with finite size scaling. The bulk, edge and corner terms are extracted very accurately. The exact results are conjectured for the corner logarithmic term in the free energy, the edge and corner logarithmic terms in the internal energy and specific heat. The corner logarithmic terms in the free energy agree with the conformal field theory very well.
The interface method is used to derive the order-disorder critical temperature of the two-dimensional Ising model on a rectangular lattice with ferromagnetic interactions in the x direction and antiferromagnetic interactions in the y direction. Employing this approximation in two different ways gives different critical temperatures Tc for nonzero uniform magnetic fields H. However, both of the phase boundaries Tc(H) reduce to the exact results available at H=0 and T=0. .AE