Commensurability Transition and Stripe Phases in the Ginzburg-Landau Theory

We phenomenologically describe the thermodynamics of charge and spin density waves in doped high-$T_{c}$ oxides. We have explicitly calculated stable non-homogeneous solutions in the incompressible spin driven stripe phase, where stripes are static soliton-like charge density waves (CDW), and in the vicinity of their critical point, where CDW's become harmonic. Our phase diagram points to a commensurability transition separating the low (LI) and high (HI) incommensurable phases. Besides, we demonstrate by rigorous group symmetry arguments that the stripe criticality is compatible with a second order phase transition.