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We study the quantum phase transition of the Dicke model in the classicaloscillator limit, where it occurs already for finite spin length. In contrastto the classical spin limit, for which spin-oscillator entanglement diverges atthe transition, entanglement in the classical oscillator limit remains small.We derive the quantum phase transition with identical critical behavior in thetwo classical limits and explain the differences with respect to quantumfluctuations around the mean-field ground state through an effective model forthe oscillator degrees of freedom. With numerical data for the full quantummodel we study convergence to the classical limits. We contrast the classicaloscillator limit with the dual limit of a high frequency oscillator, where thespin degrees of freedom are described by the Lipkin-Meshkov-Glick model. Analternative limit can be defined for the Rabi case of spin length one-half, inwhich spin frequency renormalization replaces the quantum phase transition.

Quantum phase transition in the Dicke model with critical and non-critical entanglement