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 ν=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 ν=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 (Δ_SAS) , Zeeman coupling (Δ_Z) , or electrical bias (Δ_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 Δ_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 Δ_SAS , Δ_Z , and Δ_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.