Mean-field behavior of the quantum Ising susceptibility and a new lace expansion for the classical Ising model

Abstract

The transverse-field Ising model is widely studied as one of the simplest quantum spin systems. It is known that this model exhibits a phase transition at the critical inverse temperature \(\beta _\textrm{c}\), which is determined by the spin-spin couplings and the transverse field \(q\ge 0\). Björnberg Commun. Math. Phys. 323, 329–366 (2013) investigated the divergence rate of the susceptibility for the nearest-neighbor model as the critical point is approached by simultaneously changing the spin-spin coupling \(J\ge 0\) and \(q\) in a proper manner, with fixed temperature. In this paper, we fix J and \(q\) and show that the susceptibility diverges as \(({\beta _\textrm{c}}-\beta )^{-1}\) as \(\beta \uparrow {\beta _\textrm{c}}\) for \(d>4\) assuming an infrared bound on the space-time two-point function. One of the key elements is a stochastic-geometric representation in Björnberg & Grimmett J. Stat. Phys. 136, 231–273 (2009) and Crawford & Ioffe Commun. Math. Phys. 296, 447–474 (2010). As a byproduct, we derive a new lace expansion for the classical Ising model (i.e., \(q=0\)).