We will consider a critical behavior of the susceptibility $\chi(\beta)$ with respect to the inverse temperature $\beta$ for the quantum Ising ferromagnet. As a first step, we followed the established way for the classical Ising model [M.~Aizenman, \emph{Commun. Math. Phys.} \textbf{86}, 1982] and derived an inequalities for $\chi(\beta)$ via the Suzuki-Trotter transformation (see, e.g., [K., RIMS Kôkyûroku \textbf{2116}, 2019] or [S. Handa, Ph.D. thesis, 2019]). At that time, however, we had used the monotonicity for $\chi(\beta)$ with respect to $\beta$ with an extra condition. Our next aim is to drop such condition and show the infrared bound for the quantum Ising model.

In this talk, I will show some attempts to prove the critical behavior via another way: the graphical representation in terms of the space-time Ising model which originates from [J.E.~Björnberg and G.R.~Grimmett, \textit{J. Stat. Phys.}~\textbf{136}, 2009]. This is based on a joint work with Satoshi Handa (Fujitsu Laboratories Ltd.) and Akira Sakai (Hokkaido University).