  Light-matter interacting systems involving multi-level atoms are appealing
platforms for testing equilibrium and dynamical phenomena. Here, we explore a
tricritical Dicke model, where an ensemble of three-level systems interacts
with a single light mode, through two different approaches: a generalized
Holstein-Primakoff map, and a treatment using the Gell-Mann matrices. Both
methods are found to be equivalent in the thermodynamic limit of an infinite
number of atoms. In equilibrium, the system exhibits a rich phase diagram where
both continuous and discrete symmetries can be spontaneously broken. We
characterize all the different types of symmetries according to their scaling
behaviors. Far from the thermodynamic limit, considering just a few tens of
atoms, the system already exhibits features that could help characterize both
second and first-order transitions in a potential experiment. Importantly, we
show that the tricritical behavior is preserved when dissipation is taken into
account, moreover, the system develops a steady-state phase diagram with
various regions of bistability, all of them converging at the tricritical
point. Having multiple stable normal and superradiant phases opens prospective
avenues for engineering interesting steady states by a clever choice of initial
states and/or parameter quenching.
