Phonons in Electron Crystals with Berry Curvature

TL;DR

This work addresses how Berry curvature affects phonons in electron crystals, including anomalous Hall crystals, by deriving a general low-energy phonon action with $m_{ab}$, $\lambda_{abcd}$, $\beta$, and a kineo-elastic coupling $\ell_{abc}$. It introduces a translational gauge twist to connect microscopic ground-state energies to the phonon coefficients, and validates the approach against time-dependent Hartree-Fock (TDHF) in two models: $\lambda$-jellium and rhombohedral multilayer graphene (RMG). The study shows that while the AHC shares some features with magnetic-field Wigner crystals, the low-energy phonon dispersion remains that of the zero-field case due to commuting translations, yet Berry curvature can soften crystals and enhance effective mass, with the kineo-elastic term producing strong nonreciprocity in transverse speeds (notably in RMG). The framework provides a quantitative, parameter-driven route to predict phonon spectra and lattice stability in 2D materials with Berry curvature, guiding interpretation of experiments on moiré systems and valley-polarized crystals.

Abstract

Recent advances in 2D materials featuring nonzero Berry curvature have inspired extensions of the Wigner crystallization paradigm. This paper derives a low-energy effective theory for such quantum crystals, including the anomalous Hall crystal (AHC) with nonzero Chern number. First we show that the low frequency dispersion of phonons in AHC, despite the presence of Berry curvature, resembles that of the zero field (rather than finite magnetic field) Wigner crystal due to the commutation of translation generators. We explain how key parameters of the phonon theory such as elastic constants and effective mass can be extracted from microscopic models, and apply them to two families of models: the recently introduced $λ$-jellium model and a model of rhombohedral multilayer graphene (RMG). In the $λ$-jellium model, we explore the energy landscape as crystal geometry shifts, revealing that AHC can become `soft' under certain conditions. This causes transitions in lattice geometry, although the quantized Hall response remains unchanged. Surprisingly, the Berry curvature seems to enhance the effective mass, leading to a reduction in phonon speed. For the AHC in RMG, we obtain estimates of phonon speed and shear stiffness. We also identify a previously overlooked `kineo-elastic' term in the phonon effective action that is present in the symmetry setting of RMG, and leads to dramatic differences in phonon speeds in opposite directions. We numerically confirm these predictions of the effective actions by time-dependent Hartree-Fock calculations.

Figures (6)

• Figure 1: The numerical procedure for parameter extraction. (a) Different unit cells of the same area are specified by the orientation $\phi$ and modular parameter $\tau$. All inequivalent choices of $\tau$ can be chosen by sampling a fundamental domain of the modular group (one choice is shaded in gray). Solid lines represent the boundaries of the fundamental domain, which correspond to rhombic lattices. The dashed line corresponds to rectangular lattices. The square lattice (orange) corresponds to $\tau=i$ whereas the triangular lattice (purple) corresponds to $\tau=e^{2\pi i/3}$. (b) Boosting the system in the $x$ direction gives the crystal a center-of-mass velocity $v_x$ for its sliding motion.
• Figure 2: Hartree-Fock phase diagram of the $\lambda$-jellium model, which shows the competition between crystalline phases. Fermi liquid phases in the phase diagram are out of the parameter regimes shown here. Colors correspond to Wigner Crystals and Anomalous Hall Crystal. The parameter combinations $(\lambda,r_s) = (0.8,5), (0.8,10), (1.2,5)$ will be studied in more careful detail in a later part of the manuscript; the anomalous Hall crystals take different shapes at those parameter points. All data are computed from SCHF ($18\times18$) with $13$ bands taken, where a triangular unit cell was assumed.
• Figure 3: Plasmon spectrum of (a) the Wigner crystal and (b) the anomalous Hall crystal in the $\lambda$-jellium model. Dots are time-dependent Hartree-Fock spectra computed on a $18\times 18$ system with $7$ bands taken. Red lines come from the effective phonon action with coefficients from linear response. The two methods display excellent quantitative agreement.
• Figure 4: Numerical elastic parameters for the low-energy phonon theory of $\lambda$-jellium. Panels (a-c) show shear stiffness, panels (d-f) show effective mass, and panels (g-i) show velocity. Quantities are scaled by appropriate powers of $r_s$ to make the large $r_s$ limit order unity (see text). (a) Scaled shear stiffness $\mu \cdot r_s$ as a function of $\lambda$ and $r_s$. The gray region suffers from convergence issues, precluding accurate determination of the stiffness. Solid lines mark the phase boundary of different crystals, and the dotted lines surround a region of negative stiffness. (b) Line cuts of scaled shear stiffness against $r_s$. (c) Line cuts of scaled shear stiffness against $\lambda$. Shear stiffness decreases precipitously at the first-order transition between WC and AHC, becoming negative in a small region (see text). (d) The scaled effective mass $m/r_s^2$ plotted against $r_s$ and $\lambda$. (e) Line cut of the scaled effective mass versus $r_s$, showing interaction effects enhances the mass at nonzero $\lambda$. (f) Line cut of the normalized effective mass versus $\lambda$. An intermediate value of $\lambda$ enhances effective mass drastically. (g) Scaled speed of sound $v_t \cdot r_s^{3/2}$ plotted against $r_s$ and $\lambda$. (h) Line cut of the scaled speed of sound against $r_s$. (i) Line cut of the scaled speed of sound against $\lambda$. All data are computed from SCHF ($18\times18$) with $13$ bands taken, where a triangular unit cell was assumed.
• Figure 5: The shape of anomalous Hall crystal (AHC) depends on microscopic parameters. (a,b): for $r_s=5$, the triangular lattice is locally unstable when $\lambda\in[0.5,0.7]$. Non-zero imaginary part in the TDHF collective mode spectrum at $\boldsymbol{q}_1 = \boldsymbol{G}_1/12$ and the negative shear modulus both point to instability. (c): Energy landscape for different lattices parameterized by $\tau$ at $\lambda=0.8, r_s=5$. The energetically preferred lattice is the square lattice where $\tau=i$. (e): Energy landscape for $\lambda=0.8, r_s=10$. The energetically preferred lattice is a rhombic lattice where $\tau=0.2+0.6i$. (f): Energy landscape for $\lambda=1.2, r_s=5$. The energetically preferred lattice is the triangular lattice. All data are computed from SCHF ($24\times24$) with $13$ bands taken.