Two-dimensional Ising model in random magnetic fields

The statistical mechanics of finite $L\times L$ Ising square lattices in a field $\pm h$ of random sign is investigated numerically by a modified recursive transfer matrix method for $6<~L<~16$. Our results are consistent with the absence of a spontaneous magnetization for $h\ne 0$ even in the ground state. The singularities occurring at $T=0\mathrm{}$ in the range $0<~h<~4J$, $J$ being the exchange constant, are discussed in terms of a cluster expansion. For nonzero $T$, less than the critical temperature of the pure two-dimensional Ising model, and $h$ sufficiently small, the system exhibits a nonzero spin-glass order parameter of the Mattis type although there is no magnetization. The ferromagnetic correlation function becomes long ranged for $h\to 0$ and is calculated from the domain-wall density.