Topological phonons in anomalous Hall crystals

Abstract

Recent experiments on few-layer graphene structures have reported indirect signatures of anomalous Hall crystals (AHCs), but the need for a top gate to stabilize the phase precludes direct imaging of the emergent electronic lattice. This situation necessitates the investigation of alternative signatures of AHCs. The gapless phonons of the emergent electronic lattice provide a clear distinction from conventional quantum Hall states, but it may be difficult to disentangle these phonons from the plethora of other possible low-lying modes. Intriguingly, the quantum geometry of the underlying electronic ground state can imprint on the collective modes, possibly leading the phonons themselves to be topological. Were this the case, the resulting neutral chiral edge modes would provide a further signature of an AHC. Using time-dependent Hartree-Fock, we compute the spectra of collective modes of Wigner crystals (WCs) and AHCs arising in minimal models and study the topology of the phonons and low-lying excitons. Across the WC to AHC transition, we observe a series of band inversions among collective modes, producing topological phonons and excitons, and a sharp sign change in the phonon Chern number upon entering the AHC phase. We conclude by discussing the relevance of collective mode topology to experiments on candidate systems for AHCs.

Figures (2)

• Figure 1: The collective mode spectrum along high-symmetry lines of the infinite Chern band model with $r_s=10$ and (a) $\mathcal{B}A_{\text{1BZ}}=\pi/4$, (b) $\mathcal{B}A_{\text{1BZ}}=3\pi/4$, (c) $\mathcal{B}A_{\text{1BZ}}=5\pi/4$, and (d) $\mathcal{B}A_{\text{1BZ}}=2\pi$. The lines indicate bands of collective modes, where line color and style are used to depict energetically isolated groups of bands, and the hatched region marks the non-interacting particle-hole continuum. The trace of the non-Abelian Berry curvature of the isolated groups of bands in (a-d) are plotted in (e-h), respectively. The panels of (e-h) are numbered in order of increasing energy of the bands, such that (x.1), (x.2), (x.3), and (x.4) correspond to the solid purple, dashed blue, dot-dashed turquoise, and dot-dot-dashed green lines, respectively. The Berry curvatures in (e.3-e.4) and (f.3-f.4) are scaled up by a factor of ten for visibility. The higher energy excitons of the AHC in (d) are too dense to resolve the Berry curvature, so we only plot the lowest three sets of bands in this case, leaving panel (h.4) blank.
• Figure 2: The collective mode spectrum along high-symmetry lines of the $\lambda-$jellium model for $r_s=10$ with (a) $\lambda=0.59$ (WC) and (d) $\lambda=2.65$ (AHC). The solid thick lines indicate the phonon modes, the dashed thick line is the lowest energy exciton, and the thin gray lines mark higher energy excitons. The trace of the non-Abelian Berry curvature of the phonons is plotted in (b, e) for $\lambda=0.59$ and $\lambda=2.65$, respectively, and the Berry curvature of the lowest energy excitons is similarly plotted in (c, f). The insets denote the Chern numbers of the bands. In panels (g-i) we plot the collective mode spectrum, phonon Berry curvature, and lowest exciton Berry curvature for the AHC with $\lambda=2.65$ including an external periodic potential of strength $V_0=2.5\times10^{-3}$.