Commensurate lock-in in holographic non-homogeneous lattices

We consider the spontaneous formation of striped structures in a holographic model which possesses explicit translational symmetry breaking, dual to an ionic lattice with spatially modulated chemical potential. We focus on the perturbative study of the marginal modes which drive the transition to a phase exhibiting spontaneous stripes. We study the wave-vectors of the instabilities with largest critical temperature in a wide range of backgrounds characterized by the period and the amplitude of the chemical potential modulation. We report the first holographic observation of the commensurate lock-in between the spontaneous stripes and the underlying ionic lattice, which takes place when the amplitude of the lattice is large enough. We also observe an incommensurate regime in which the amplitude of the lattice is finite, but the preferred stripe wave-vector is different from that of the lattice.