Two-Dimensional Ising Model in a Finite Magnetic Field

We present the results of numerical calculations giving accurate estimates of the magnetization of the two-dimensional Ising model on a square lattice. Moreover, we argue that these results are strict lower bounds to the correct magnetization $M(H,T)$. The estimates are obtained by dividing the infinite lattice into finite strips of width between two and nine spins and infinite length. The largest eigenvalue and the corresponding eigenvector of the transfer matrix are then obtained by an iterative process. The estimates of $M(H,T)$ converge to the correct answer for the infinite lattice everywhere except for a small region in the $T\ensuremath{-}H$ plane. We also compute isotherms and critical isobar for the corresponding lattice gas. Finally, we propose a new approximation to the transfer matrix, exactly solvable in two dimensions for $H=O$, which reproduces exactly the critical-point behavior of the full Ising model.