  Motivated by recent observations of symmtry broken phases in lightly-doped
multilayer graphene, we investigate magnetic phase transitions in a generalized
electron gas model with four-component electron spin. This model simplifies the
problem with a parabolic dispersion band, abstracting away the details of the
graphene band structure to focus solely on the effects of the Coulomb
interaction. We report four findings: 1) In the Hartree-Fock approximation, we
observe that the paramagnetic state undergoes a sequence of density-driven,
first-order phase transitions, progressively depopulating electrons from each
spin component until achieving complete polarization within a very narrow
density window where $1.2<r_s<2$ ($r_s$ being the electron gas parameter). 2)
Further incorporating the correlation energy via the Bohm-Pines random-phase
approximation shows that the cascade of transitions obtained within
Hartree-Fock approximation is replaced by a single ferromagnetic phase
transition at $r_s = 6.12$. 3) The disappearance of cascade is due to the
correlation energy difference between the four-component paramagnetic state and
symmetry-broken phases, which is nearly an order of magnitude more negative
than the corresponding Hartree-Fock energy difference for $1.2 < r_s < 2$. 4)
The transition from the paramagnetic state to the fully polarized state at
$r_s=6.12$ is governed by the balance between exchange and correlation
energies, a competition that cannot be captured by mean-field approximations to
models featuring effective (density-dependent) delta-function interactions,
such as the Stoner model. We use the insights from our model to comment on the
phase diagram of multilayer graphene electron gas.
