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 ${\beta }_{\mathrm{c}}$, which is determined by the spin-spin couplings and the transverse field $q\ge 0$. Björnberg [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 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 $({\beta }_{\mathrm{c}}-\beta {)}^{-1}$ as $\beta \uparrow {\beta }_{\mathrm{c}}$ 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 (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., $q=0$).

34 pages, 22 figures