  We study charged Dirac fermions on an AdS$_2\times R^2$ background with a
non-zero magnetic field. Under certain boundary conditions, we show that the
charged fermion can make the background unstable, resulting in spontaneously
formation of a vortex lattice. We observe that an electric field emerges in the
back-reacted solution due to the vortex lattice constructed from spin polarized
fermions. This electric field may be extended to the UV boundary which leads to
a finite charge density. We also discuss corrections to the thermodynamic
functions due to the lattice formation.
