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How the mixed eigenstates vary with approaching the semiclassical limit inmixed-type many-body quantum systems is an interesting but still less knownquestion. Here, we address this question in the Dicke model, a celebratedmany-body model that has a well defined semiclassical limit and undergoes atransition to chaos in both quantum and classical case. Using the Husimifunction, we show that the eigenstates of the Dicke model with mixed-typeclassical phase space can be classified into different types. To quantitativelycharacterize the types of eigenstates, we study the phase space overlap index,which is defined in terms of Husimi function. We look at the probabilitydistribution of the phase space overlap index and investigate how it changeswith increasing system size, that is, when approaching the semiclassical limit.We show that increasing the system size gives rise to a power-law decay in thebehavior of the relative proportion of mixed eigenstates. Our findings shedmore light on the properties of eigenstates in mixed-type many-body systems andsuggest that the principle of uniform semiclassical condensation of Husimifunctions should also be valid for many-body quantum systems.

Mixed eigenstates in the Dicke model: Statistics and power-law decay of the relative proportion in the semiclassical limit