Energy spectrum of a layered system in a strong magnetic field

We investigate the excitation spectrum of two- and three-layer electron systems in a strong perpendicular magnetic field with ν=(1/2 and (1/3, respectively, in each layer. For layer separation z=0 the dispersion relations ω(k) vanish as k² for k-->0, as one expects for Goldstone modes. For z>0, ω(k) behaves as an acoustic mode, vanishing linearly for small k. For large values of k one finds that the dispersion relations have the form Δ(z)-e²/κkl²₀, where l₀ is the magnetic length and κ the dielectric constant of the medium. At kl₀ of order unity, the dispersion relations develop a dip as z is increased. These become soft modes at certain critical values of z, indicating that the system undergoes a phase transition as the layer spacing is increased.