Random Current Representation for Transverse Field Ising Model

Random current representation (RCR) for transverse field Ising models (TFIM) has been introduced in [15]. 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 z-direction and address the issue of the sharpness of phase transition. 1. The model and the results In what follows, we shall, for brevity, consider translation invariant models on Z . Specifically, let TN be the d-dimensional lattice torus of linear size N and J = {Jij = Ji−j} is a finite range irreducible translation invariant interaction. Let h ≥ 0, ρ > 0, λ ≥ 0 and 0 ≤ β ≤ ∞. The quantum Hamiltonian we are going to consider is of the form, −HN = ρ 2 ∑ i,j Jij σ̂ z i σ̂ z j + h ∑