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 $k^2$ 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)-$e^2$/κ${\mathrm{kl}}_0^2$, where $l_0$ is the magnetic length and κ the dielectric constant of the medium. At ${\mathrm{kl}}_0$ 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.

• Received 6 January 1989

DOI:https://doi.org/10.1103/PhysRevB.40.1087