Striped States in Quantum Hall Effect: Deriving a Low Energy Theory from Hartree-Fock

There is growing experimental and theoretical evidence that very clean two dimensional electron systems form unidirectional charge density waves (UCDW) or "striped" states at low temperatures and at Landau level filling fractions of the form $nu = M + x$ with $4 M 10$ an integer and $0.4 x 0.6$. Following previous work, we model the striped state using a Hartree Fock approach. We construct the low energy excitations of the system by making smooth deformations of the stripe edges analogous to the construction of edge state excitations of quantum Hall droplets. These low energy excitations are described as a coupled Luttinger liquid theory, as discussed previously by MacDonald and Fisher (Phys. Rev. B 61, 5724 (2000)). Here, we extend that work and explicitly derive all of the parameters of this low energy theory using a Hartree Fock approach. We also make contact with the equivalent low energy hydrodynamic approach of Fogler and Vinokur (Phys. Rev. Lett. 84, 5828 (2000)) and similarly derive the parameters of this theory. As examples of the use of these results, we explicitly calculate the low-energy excitation spectrum and study tunneling into the striped state.