Collective modes ofCP3skyrmion crystals in quantum Hall ferromagnets

Abstract

The two-dimensional electron gas (2DEG) in a bilayer quantum Hall system can sustain an interlayer coherence at filling factor $\ensuremath{\nu}=1$ 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 $\ensuremath{\nu}=1$, 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 $({\ensuremath{\Delta}}_{SAS})$, Zeeman coupling $({\ensuremath{\Delta}}_{Z})$, or electrical bias $({\ensuremath{\Delta}}_{b})$, these textured crystal states can involve spin, pseudospin, or both intertwined. This last case is a ``$C{P}^{3}$ 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 ${\ensuremath{\Delta}}_{SAS}\ensuremath{\rightarrow}0$, the state evolves into a meron crystal and supports an extra gapless [U(1)] mode in addition to the phonon. For a $C{P}^{3}$ skyrmion crystal, we find a U(1) gapless mode in the presence of nonvanishing symmetry-breaking fields ${\ensuremath{\Delta}}_{SAS}$, ${\ensuremath{\Delta}}_{Z}$, and ${\ensuremath{\Delta}}_{b}$. 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.