Berry Phase and Fidelity in the Dicke model with $A^2$ term

The instability, so-called the quantum-phase-like transition, in the Dicke model with a rotating-wave approximation for finite $N$ atoms is investigated in terms of the Berry phase and the fidelity. It can be marked by the discontinuous behavior of these quantities as a function of the atom-field coupling parameter. Involving an additional field $A^2$ term, it is observed that the instability is not eliminated beyond the characteristic atom-field coupling parameter even for strong interaction of the bosonic fields, contrarily to the previous studies.