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

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 $\beta_{\mathrm{c}}(q)$, where $q$ is the strength of the transverse field. Bj\"ornberg [Commun. Math. Phys., 232 (2013)] investigated the divergence rate of the susceptibility for the nearest-neighbor model as the critical point is approached by simultaneously changing $q$ and the spin-spin coupling $J$ in a proper manner, with fixed temperature. In this paper, we prove that the susceptibility diverges as $(\beta_{\mathrm{c}}(q)-\beta)^{-1}$ as $\beta\uparrow\beta_{\mathrm{c}}(q)$ 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 Bj\"ornberg & Grimmett [J. Stat. Phys., 136 (2009)] and Crawford & Ioffe [Commun. Math. Phys., 296 (2010)]. As a byproduct, we derive a new lace expansion for the classical Ising model (i.e., $q=0$).