  We develop a method to detect instabilities leading to nematic phases in
strongly coupled metallic systems. We do so by adapting the well-known
Pomeranchuk technique to a weakly coupled system of fermions in a curved
asymptotically AdS bulk. The resulting unstable modes are interpreted as
corresponding to instabilities on the dual strongly coupled holographic metal.
We apply our technique to a relativistic $3+1$-dimensional bulk with generic
quartic fermionic couplings, and explore the phase diagram at zero temperature
for finite values of the fermion mass and chemical potential, varying the
couplings. We find a wide region of parameters where the system is stable,
which is simply connected and localized around the origin of coupling space.
