Towards a Holographic Quark Matter Crystal

We construct the gravity dual of $d=4$, $\mathcal{N}=4$, SU(N_\rm{c}) super Yang-Mills theory, coupled to N_\rm{f} flavors of dynamical quarks, at non-zero temperature $T$ and non-zero quark density N_\rm{q}. The supergravity solutions possess a regular horizon if $T>0$ and include the backreaction of N_\rm{c} color D3-branes and N_\rm{f} flavor D7-branes with N_\rm{q} units of electric flux on their worldvolume. At zero temperature the solutions interpolate between a Landau pole singularity in the ultraviolet and a Lifshitz geometry in the infrared. At high temperature the thermodynamics is directly sensitive to the Landau pole, whereas at low temperature it is not, as expected from effective field theory. At low temperature and sufficiently high charge density we find thermodynamic and dynamic instabilities towards the spontaneous breaking of translation invariance.