Scrambling in the Dicke model*

The scrambling rate $\lambda_L$ associated with the exponential growth of out-of-time-ordered correlators can be used to characterize quantum chaos. Here we use the Majorana Fermion representation of spin $1/2$ systems to study quantum chaos in the Dicke model. We take the system to be in thermal equilibrium and compute $\lambda_L$ throughout the phase diagram to leading order in $1/N$. We find that the chaotic behavior is strongest close to the critical point. At high temperatures $\lambda_L$ is nonzero over an extended region that includes both the normal and super-radiant phases. At low temperatures $\lambda_L$ is nonzero in (a) close vicinity of the critical point and (b) a region within the super-radiant phase. In the process we also derive a new effective theory for the super-radiant phase at finite temperatures. Our formalism does not rely on the assumption of total spin conservation.