Collective modes of $C{P}^3$ 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 $\left({\Delta }_{SAS}\right)$, Zeeman coupling $\left({\Delta }_Z\right)$, or electrical bias $\left({\Delta }_b\right)$, 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 ${\Delta }_{SAS}\to 0$, 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 ${\Delta }_{SAS}$, ${\Delta }_Z$, 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.

Figure 1

The 16 possible transitions between the single-particle energy levels in the UCS are divided in the GRPA into 3 uncoupled subsystems involving 8, 4, and 4 transitions.

Figure 2

Dispersion relations of the collective excitations in the UCS at $\nu =1$ as computed in the GRPA. Parameters are $d∕\ell =1$, ${\Delta }_Z∕(e^2∕\kappa \ell )=0.2$, and (a) ${\Delta }_b∕(e^2∕\kappa \ell )=0$, ${\Delta }_{SAS}=0$, (b) ${\Delta }_b∕(e^2∕\kappa \ell )=0.1$, ${\Delta }_{SAS}=0$, (c) ${\Delta }_b∕(e^2∕\kappa \ell )=0.4023$, ${\Delta }_{SAS}=0$, and (d) ${\Delta }_b∕(e^2∕\kappa \ell )=0.4$, ${\Delta }_{SAS}∕(e^2∕\kappa \ell )=0.02$. The straight lines give the frequency of the nondispersive modes. The modes 5, 11, and 13 correspond to the dispersions ${\omega }_5\left(q\right)$, ${\omega }_{11}\left(q\right)$, and ${\omega }_{13}\left(q\right)$ derived in the text. The abbreviations SW and SWPF refer to the spin-wave and spin-wave with pseudospin-flip modes (see text).

Figure 3

(Color online) Spin texture in the $x\text{−}y$ plane (vectors) and density in a unit cell of a skyrme lattice in (a) the SLA and (b) the meron configurations at $\nu =1.30$ with ${\Delta }_Z∕(e^2∕\kappa \ell )=0.01$ and ${\Delta }_Z∕(e^2∕\kappa \ell )=0$, respectively. In (a), $S_z=-\frac{1}{2}$ at the position of the skyrmions and each skyrmion carries charge $q=-e$. In (b), $S_z=-\frac{1}{2}$ at the positions of the four merons at the corner of the unit cell and at the position of the meron at the center, while $S_z=+\frac{1}{2}$ at the positions of the other merons. Each meron carries a charge $q=-\frac{e}{2}$. Contours with dark (red) regions indicate a high density.

Figure 4

(Color online) Dispersion relation of the collective modes of the skyrme crystal for (a) a SLA configuration with $\nu =1.30$ and ${\Delta }_Z∕(e^2∕\kappa \ell )=0.008$ and (b) a meron configuration similar to that of Fig. 3b but with $\nu =1.38$ and ${\Delta }_Z∕(e^2∕\kappa \ell )=0$. The dispersion is computed along the path $\Gamma \text{−}M\text{−}X\text{−}\Gamma$ in the Brillouin zone, as shown by the inset in (b). The symbols indicate from what response function each frequency comes. The softening of the modes seen in (b) is due to the meron crystal becoming unstable at $\nu =1.36$ with respect to a biskyrmion crystal.

Figure 5

(Color online) Pseudospin texture in the $x\text{−}y$ plane (vectors) and value of $S_z$ (contours) in a unit cell of a bimeron crystal in a SLA configuration with $\nu =1.20$ and ${\Delta }_{SAS}∕(e^2∕\kappa \ell )=0.02$. The full black circles indicate the position of the maximum in the bimeron density. Dark (red) contours correspond to $S_z=+\frac{1}{2}$, while gray (blue) contours are for $S_z=-\frac{1}{2}$. Note that the pseudospin texture for the bimeron at the center has a global phase rotation with respect to the pseudospin texture of the bimerons at the four corners of the unit cell.

Figure 6

(Color online) Collective mode spectrum of a bimeron crystal in the SLA configuration of Fig. 5 with $\nu =1.2$, ${\Delta }_{SAS}∕(e^2∕\kappa \ell )=0.02$, and $d∕\ell =0.1$. The path in the Brillouin zone is as indicated in the inset of Fig. 4b and the symbols are as indicated in the legend of that same figure.

Figure 7

(Color online) Collective mode spectrum of a pseudospin-meron crystal in a SLA configuration similar to that of Fig. 3b with $\nu =1.2$, ${\Delta }_{SAS}∕(e^2∕\kappa \ell )=0$, and $d∕\ell =0.8$. The path in the Brillouin zone is as indicated in the inset of Fig. 4b and the symbols are as indicated in the legend of that figure.

Figure 8

(Color online) Collective mode spectrum of a spin-polarized meron crystal with the following parameters: $\nu =0.8$, ${\Delta }_Z∕(e^2∕\kappa \ell )=0.004$, ${\Delta }_{SAS}∕(e^2∕\kappa \ell )=0.0002$, ${\Delta }_b∕(e^2∕\kappa \ell )=0$, and $d∕\ell =0.3$.

Figure 9

(Color online) Collective mode spectrum of a crystal of $C{P}^3$ skyrmions along path -$M\text{−}X$- in the Brillouin zone. The parameters are $\nu =0.8$, ${\Delta }_Z∕(e^2∕\kappa \ell )=0.004$, ${\Delta }_{SAS}∕(e^2∕\kappa \ell )=0.0002$, ${\Delta }_b∕(e^2∕\kappa \ell )=0$, and $d∕\ell =1.0$.

Figure 10

(Color online) Collective mode spectrum of a crystal of $C{P}^3$ skyrmions with strong tunneling. The parameters are $\nu =0.8$, ${\Delta }_Z∕(e^2∕\kappa \ell )=0.008$, ${\Delta }_{SAS}∕(e^2∕\kappa \ell )=0.0044$, ${\Delta }_b∕(e^2∕\kappa \ell )=0$, and $d∕\ell =1.0$.

Figure 11

(Color online) Collective mode spectrum of a crystal of symmetric skyrmions. The parameters are $\nu =0.8$, ${\Delta }_Z∕(e^2∕\kappa \ell )=0.008$, ${\Delta }_Z∕(e^2∕\kappa \ell )=0.006$, ${\Delta }_b∕(e^2∕\kappa \ell )=0$, and $d∕\ell =1.0$.

Figure 12

(Color online) Collective mode spectrum of a crystal of $C{P}^3$ skyrmions with a finite bias. The parameters are $\nu =0.8,{\Delta }_Z∕(e^2∕\kappa \ell )=0.01,{\Delta }_{SAS}∕(e^2∕\kappa \ell )=0.0002,{\Delta }_b∕(e^2∕\kappa \ell )=0.20$, and $d∕\ell =1.0$.