  How the mixed eigenstates vary with approaching the semiclassical limit in
mixed-type many-body quantum systems is an interesting but still less known
question. Here, we address this question in the Dicke model, a celebrated
many-body model that has a well defined semiclassical limit and undergoes a
transition to chaos in both quantum and classical case. Using the Husimi
function, we show that the eigenstates of the Dicke model with mixed-type
classical phase space can be classified into different types. To quantitatively
characterize the types of eigenstates, we study the phase space overlap index,
which is defined in terms of Husimi function. We look at the probability
distribution of the phase space overlap index and investigate how it changes
with 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 the
behavior of the relative proportion of mixed eigenstates. Our findings shed
more light on the properties of eigenstates in mixed-type many-body systems and
suggest that the principle of uniform semiclassical condensation of Husimi
functions should also be valid for many-body quantum systems.
