Numerical experiments on lattice gas models

Abstract

The two dimensional Ising Model bas never been solved in a finite field.- The critical point exponents,1 however, have all been inferred from the exact solution of Onsager2 in zero field or determined by series expansions.8 It remains to determine the magnetization m(H,T ) for finite H. Recently Mattis and Plischk3 derived rigorous analytic lower bounds to m(H,T ) in terms of the zero field internal energy u(O,T) and the spontaneous magnetization of Yang4 mo(T). As the zero field susceptibility could not be rigorously incorporated into this expression the response to · small fields was much too weak and these analytic bounds do not lie very close to the correct answer. ¶ In this chapter we present the results of numerical computations giving a lower bound to m(H,T) which, except for a small region of the H-T plane, lies within .1% of the correct answer. This lower bound is obtained by dividing the infinite lattice into strips of infinite length and ·width M spins. This is achieved by removing .ferromagnetic bonds and can only lower the magnetization as has been shown by Griffiths .-5. The Kramers-Wannier transfer matrix for . such a strip is a 2N x 2N matrix whose largest eigenvalue, as well as the corresponding eigenvector, we obtain by a simple iterative process described in Section II. ¶ In Section III we introduce a new approximation to the transfer matrix, solvable in zero field, which reproduces co?:completely the critical point behavior of the full Ising Model in zero field. This pseudo transfer matrix has a feature which makes it easier to study numerically.