  Within a phenomenological holographic model in $(4+1)$-bulk dimensions,
defined by Einstein-gravity with a negative cosmological constant, coupled to a
Dirac-Born-Infeld and a Chern-Simons term, we explore the fate of BF-bound
violation for a probe scalar field and a fluctuation mode of the corresponding
geometry. We assume this simple model to capture the dynamics of a strongly
coupled SU$(N_c)$ gauge theory with $N_f$ fundamental matter, which in the
limit ${\cal O} \left( N_c \right) \sim {\cal O}\left(N_f\right)$ and with a
non-vanishing matter density, is holographically described by an
AdS$_2$-geometry in the IR. We demonstrate that, superconductor/superfluid
instabilities are facilitated and spontaneous breaking of translational
invariance is inhibited with increasing values of $\left(N_f / N_c \right) $.
This is similar, in spirit, with known results in large $N_c$ Quantum
Chromodynamics with $N_f$ quarks and a non-vanishing density, in which the
chiral density wave phase becomes suppressed and superconducting instabilities
become favoured as the number of quarks is increased.

High Energy Physics - Theory