LOCALIZATION IN THE GROUND-STATE OF THE ISING-MODEL WITH A RANDOM TRANSVERSE FIELD

We study the zero-temperature behavior of the Ising model in the presence of a random transverse field. The Hamiltonian is given by GRAPHICS where J > 0, x, y-epsilon-Z(d), sigma-1, sigma-3 are the usual Pauli spin 1/2 matrices, and h = {h(x), x-epsilon-Z(d)} are independent identically distributed random variables. We consider the ground state correlation function <sigma-3-(x)-sigma-3-(y)> and prove: 1. Let d be arbitrary. For any m > 0 and J sufficiently small we have, for almost every choice of the random transverse field h and every x-epsilon-Z(d), that <sigma-3-(x)-sigma-3-(y)> lesser-than-or-equal-to C(x,h)e-m\x-y\ for all y-epsilon-Z(d) with C(x,h) < infinity. 2. Let d greater-than-or-equal-to 2. If J is sufficiently large, then, for almost every choice of the random transverse field h, the model exhibits long range order, i.e., GRAPHICS for any x-epsilon-Z(d).