Dynamics of the Wigner crystal of composite particles

Conventional wisdom has long held that a composite particle behaves just like an ordinary Newtonian particle. In this paper, we derive the effective dynamics of a type-I Wigner crystal of composite particles directly from its microscopic wave function. It indicates that the composite particles are subjected to a Berry curvature in the momentum space as well as an emergent dissipationless viscosity. While the dissipationless viscosity is the Chern-Simons field counterpart for the Wigner crystal, the Berry curvature is a feature not presented in the conventional composite fermion theory. Hence, contrary to general belief, composite particles follow the more general Sundaram-Niu dynamics instead of the ordinary Newtonian one. We show that the presence of the Berry curvature is an inevitable feature for a dynamics conforming to the dipole picture of composite particles and Kohn's theorem. Based on the dynamics, we determine the dispersions of magnetophonon excitations numerically. We find an emergent magnetoroton mode which signifies the composite-particle nature of the Wigner crystal. It occurs at frequencies much lower than the magnetic cyclotron frequency and has a vanishing oscillator strength in the long-wavelength limit.

Figure 1

Variational ground-state energies of CPWCs relative to that of the ordinary WC. Red points indicate the results of Ref. [8]. The phase boundaries between CPWC phases with different values of $m$ are determined by comparing the energies, and indicated by dashed vertical lines. Inset: Configuration of the simulation cell.

Figure 2

Emergent gauge field. (a) Distribution of the emergent gauge field in the Brillouin zone for four representative filling factors [see legends in (b)] with different values of $m$. (b) Decay of the dissipationless viscosity coefficient $\mathrm{\Delta }{\mathcal{B}}_e\left({\mathit{R}}_i^0\right)$ in the real space. $R$ denotes the distance between two particles, and $a$ is the lattice constant. (c) Filling factor $\nu$ dependence of the emergent gauge field at the $K$ point of the Brillouin zone (circle-solid line) and the mean-field value $\mathrm{\Delta }B$ defined in (43) (triangle-dashed line).

Figure 3

Phonon dispersions of type-I CPWCs. (a–d) Phonon dispersions for a few representative filling factors. Both results using the projected dynamic matrix (solid lines) and the unprojected one (dotted lines) are shown. (e) Filling factor $\nu$ dependence of phonon energies of the upper (U) and the lower (L) branch at high-symmetry points of the Brillouin zone, including $K$ point, $M$ point, as well as $\mathrm{\Gamma }$ point (evaluated at $\mathit{q}=0.01\mathit{K}$). $e^2/\epsilon l$ ($\approx 4.3\sqrt{B\left[\mathrm{T}\right]}\mathrm{meV}$ for GaAs) is the Coulomb energy scale. Error bars for the phonon energies near the $\mathrm{\Gamma }$ point are shown.

Figure 4

Oscillator strength of the emergent magnetoroton mode for a few representative filling factors, in units of ${\omega }_{\mathit{q}}/{\omega }_c$, where ${\omega }_{\mathit{q}}$ is the frequency of the mode, and ${\omega }_c$ is the cyclotron frequency.