Phase Transitions in the Ising Model with Traverse Field

Abstract

The Ising model perturbed by a small transverse field is shown to have a phase transition by two methods. With the first method, using a Peierls' contour argument, we are only able to show that spontaneous magnetization occurs with the transverse field goint to 0 as -1/3. With the second method, which used reflection positivity, long range order is shown to occur for a small transverse field independent of temperature.