Random Current Representation for Transverse Field Ising Model

Random current representation (RCR) for transverse field Ising models (TFIM) has been introduced in [14]. This representation is a space-time version of the classical RCR exploited by Aizenman et. al. [1,3,4]. In this paper we formulate and prove corresponding space-time versions of the classical switching lemma and show how they generate various correlation inequalities. In particular we prove exponential decay of truncated two-point functions at positive magnetic fields in the z-direction and address the issue of the sharpness of the phase transition.