Collective modes of CP3 skyrmion crystals in quantum Hall ferromagnets

The two-dimensional electron gas (2DEG) in a bilayer quantum Hall system can sustain an interlayer coherence at filling factor nu=1 even in the absence of tunneling between the layers. This system, which can be described as a quantum Hall pseudospin ferromagnet, has low-energy charged excitations which may carry textures in real spin or pseudospin. Away from filling factor nu=1, a finite density of these is present in the ground state of the 2DEG and forms a crystal. Depending on the relative size of the various energy scales, such as tunneling (Delta(SAS)), Zeeman coupling (Delta(Z)), or electrical bias (Delta(b)), these textured crystal states can involve spin, pseudospin, or both intertwined. This last case is a "CP3 skyrmion crystal." In this paper, we present a comprehensive numerical study of the collective excitations of these textured crystals using the generalized random-phase approximation. For the pure spin case, at finite Zeeman coupling the state is a skyrmion crystal with a gapless phonon mode and a separate goldstone mode that arises from a broken U(1) symmetry. At zero Zeeman coupling, we demonstrate that the constituent skyrmions break up, and the resulting state is a meron crystal with four gapless modes. In contrast, a pure pseudospin-skyrme crystal at finite tunneling has only the phonon mode. For Delta(SAS)-> 0, the state evolves into a meron crystal and supports an extra gapless [U(1)] mode in addition to the phonon. For a CP3 skyrmion crystal, we find a U(1) gapless mode in the presence of nonvanishing symmetry-breaking fields Delta(SAS), Delta(Z), and Delta(b). In addition, a second mode with a very small gap is present in the spectrum. We present dispersion relations for the different low-energy modes of these various crystals as well as their physical interpretations.