Boundedness of Susceptibility in Spin Glass Transition of Transverse Field Mixed $p$-spin Glass Models

Spontaneous symmetry breaking phenomena in the transverse field mixed $p$-spin glass model in finite dimensions are studied with Nishimori’s gauge theory. Useful identities in the gauge theory enable us to study ${\mathbb{Z}}_2$-symmetry breaking. It is proven that ferromagnetic long-range order and spontaneous magnetization at an arbitrary temperature are bounded by those on the Nishimori line in the corresponding classical model. These bounds imply that neither ferromagnetic long-range order nor spontaneous magnetization exists in spin glass phase. It is proven also that the ferromagnetic susceptibility has an upper bound in paramagnetic and spin glass phases.