Chiral Graviton Modes in Fermionic Fractional Chern Insulators

TL;DR

Chiral graviton modes, once thought to be unique to continuum FQH liquids, are shown to persist as long-lived excitations in lattice FCIs through a unified lattice-stress tensor and quadrupolar-density formalism. By constructing explicit lattice operators and proving their LL limit compatibility, the authors demonstrate an adiabatic connection between FQH and FCI gravitons along a controlled HL–CB path, using ED and MPS to track energy, chirality, and lifetime across the transition. They quantify intrinsic decay rates, show gravitons remain discernible inside the magnetoroton continuum, and reveal the decay is lattice-induced but finite, not vanishing in the thermodynamic limit. The work provides a concrete microscopic framework for detecting and characterizing geometric collective modes in FCIs and suggests dynamical spectroscopy routes for experimental verification in cold-atom and solid-state moiré systems.

Abstract

Chiral graviton modes are hallmark collective excitations of Fractional Quantum Hall (FQH) liquids. However, their existence on the lattice, where continuum symmetries that protect them from decay are lost, is still an open and urgent question, especially considering the recent advances in the realization of Fractional Chern Insulators (FCI) in transition metal dichalcogenides and rhombohedral pentalayer graphene. Here we present a comprehensive theoretical and numerical study of graviton-modes in fermionic FCI, and thoroughly demonstrate their existence. We first derive a lattice stress tensor operator in the context of the fermionic Harper-Hofstadter(HH) model which captures the graviton in the flat band limit. Importantly, we discover that such lattice stress-tensor operators are deeply connected to lattice quadrupolar density correlators, readily generalizable to generic Chern bands. We then explicitly show the adiabatic connection between FQH and FCI chiral graviton modes by interpolating from a low flux HH model to a Checkerboard lattice model that hosts a topological flat band. In particular, using state-of-the-art matrix product state and exact diagonalization simulations, we provide strong evidence that chiral graviton modes are long-lived excitations in FCIs despite the lack of continuous symmetries and the scattering with a two-magnetoroton continuum. By means of a careful finite-size analysis, we show that the lattice generates a finite but small intrinsic decay rate for the graviton mode. We discuss the relevance of our results for the exploration of graviton modes in FCI phases realized in solid state settings, as well as cold atom experiments.

Figures (22)

• Figure 1: Chiral graviton operator on lattice and adiabatic connection between FQH and FCI. (a) Representation of the chiral lattice operator structure which captures the graviton dynamics in generic FCI phases. The definition of the operator is given in Eq. \ref{['eq:definition_Onn']}. In the figure, we plot $O^\pm_{nn}$ on a single site. The color represents the chiral phase factor $e^{\pm 2 i \theta_\delta}$ attached to the density quadrupolar operator and the size of the circle labels the weigth $f_G(|\delta|)$. (b-c) Top layer depicts the lattice geometries used for FQH-like states on low flux HH models ($n_\phi=1/8$) and FCI states on a CB lattice. The bottom layer shows the MPS numerically evaluated density-density static correlations $\langle n_r n_{r'} \rangle / \langle n_{r'} \rangle \langle n_r \rangle$ featuring a "correlation hole" which is then artistically represented in the middle layer. (d) Computed spectral function of the chiral graviton operator defined in panel (a) across the adiabatic interpolation from FQH to FCI limits (same data are also shown in Fig. \ref{['fig:adiabatic_graviton']} (a)). The two-magnetoroton energy scale is shown as a dotted gray line.
• Figure 2: Full ED spectra of $O^\pm_{nn}$ and $O^\pm_{s}$ at different fluxes. Chiral graviton full ED spectra on the fermionic $\nu=1/3$ Harper Hofstadter model with (a,b) lattice stress tensor operator $O^\pm_s$ and (c,d) density-density operator $O^\pm_{nn}$. Two fluxes are considered, in (a,c) on the left $n_\phi=1/8$ while in (b,d) on the right $n_\phi=1/4$. $N=4$ for all panels and broadening $\eta=0.002$. Vertical dotted line is twice the energy gap given by the magnetoroton energy $E_{mr}$. Interaction parameters are $V=2$ and $r_c=2$.
• Figure 3: Continuum limit of $O^\pm_{nn}$ and $O^\pm_s$. (a) Projected ED graviton spectra of the Harper-Hofstadter model at $N=8$ particles for the lattice stress tensor (dashed lines) and the density-density operator (full lines). Different fluxes $n_\phi$ are shown. Inset: Overlap distance between the two graviton mode operators as the continuum limit $n_\phi\to 0$ is approached (dotted line is $\propto n_\phi^3$). (b) MPS result of graviton spectrum for $n_\phi = 1/4,1/6,1/8$ fluxes obtained from cylinders of the size $L_x\times L_y = 24\times 6,~48\times 9,~36\times 8$ cylinders using density-density operator with full and dashed line labels the chiralities of the graviton. We choose $\eta=0.16n_\phi$ in both panels so that the ratio broadening over graviton energy remains roughly constant.
• Figure 4: Life time analysis on the Harper Hofstadter model. Analysis on the lifetime of the graviton mode from finite size ED spectra on the $\nu=1/3$ Harper Hofstadter model at different fluxes $n_\phi$. (a) Example of how the spectra changes for different regularization parameters $\eta$. FWHM($=2\Gamma_{tot}$) of the smallest and largest value of $\eta$ are shown by double-arrows. (b) Dependence of $\Gamma_{tot}$ on $\eta$ for different fluxes $n_\phi$ for $N=10$ particles. The dashed black line represents the infinite lifetime scenario $\Gamma_{tot}=\eta$, while the $\eta=0$ points are the estimates for $\Gamma_G$. (c) Graviton estimated intrinsic decay rate $\Gamma_G=\Gamma_{tot}(\eta^*)-\eta^*$ (see Eq. \ref{['eq:lifetime_av']} and Eq. \ref{['eq:lifetime_err']}) relative to its ennergy $\omega_G$ as a function of $n_\phi$ for $N=6,8,10$. At $n_\phi=0$ we also show results for continuum LLL with $V_1$ pseudopotential interaction (cross marker and arrow).
• Figure 5: Interpolation from HH$^*_{\frac{1}{8}}$ model to a CB lattice. (a) Square lattice Hofstadter model with $1/8$ flux per plaquette. In the position of the green star, we insert $-2\pi$ flux to make the total unit cell (dashed black line) have zero net flux. The A and B sublattices of the checkerboard sites are encircled in red and blue, respectively. (b) First Brillouin zone of the lattice with high symmetry points labeled. (c) The band gap $\Delta E$ and band width $W$ of the lowest band as we tune from the the HH$^*_{\frac{1}{8}}$ limit to the CB limit. The energy is in units of $t=1$. (d) Berry curvature along the high symmetry point at different interpolation ratios $R$.