Chiral Gravitons on the Lattice

Abstract

Chiral graviton modes are elusive excitations arising from the hidden quantum geometry of fractional quantum Hall states. It remains unclear, however, whether this picture extends to lattice models, where continuum translations are broken and additional quasiparticle decay channels arise. We present a framework in which we explicitly derive a field theory incorporating lattice chiral graviton operators within the paradigmatic bosonic Harper-Hofstadter model. Extensive numerical evidence suggests that chiral graviton modes persist away from the continuum, and are well captured by the proposed lattice operators. We identify geometric quenches as a viable experimental probe, paving the way for the exploration of chiral gravitons in near-term quantum simulation experiments.

Figures (8)

• Figure 1: Schematic representations of (a) the Harper-Hofstadter model and (b) the graviton operators we propose. The lattice model has tunneling rates $J_x$ and $J_y$ along perpendicular directions, and magnetic flux $\phi$ per plaquette. Chiral graviton operators are expressed as the sum of different correlated hopping terms centered at a given site.
• Figure 2: Graviton weights $I_\pm(\omega)$ for: (a) $12\times8$ torus with $n_\phi=1/8$ flux; (b) $12\times6$ torus with $n_\phi=1/6$ flux; and (c) $8\times8$ torus with $n_\phi=1/4$ flux. (d) Low-energy spectrum for a $12\times6$ torus. The red circle indicates the energy scale of the graviton-mode, $\omega_\mathrm{G}\approx0.54$. All results extracted from ED methods. Spectral densities obtained by taking $n_\mathrm{Krylov}=1000$ Krylov states, and a Lorentzian broadening factor of $\gamma=0.001$.
• Figure 3: Graviton mode on a FQHE droplet for $n_\phi=1/6$ with $N=5$ particles. (a) Expectation value of the density in the ground state, $\ev{n_\mathbf{r}}$. (b) Graviton response obtained through time evolution with a Lorentzian broadening of $\gamma=0.05$. Total duration of $T=100/J$, performed in time steps of $\delta t=0.1/J$. MPS results obtained with a maximum bond dimension $\chi_\mathrm{max}=300$, attaining truncation errors of the order $\varepsilon_\mathrm{trunc}\sim10^{-6}$.
• Figure 4: Anisotropy study for $n_\phi=1/8$ flux. (a) Ground state phase diagram characterization. Plot shows the behavior of energy gap, fidelity, and variance of the graviton number as a function of the anisotropy $\eta$ in the $k_x=0$ momentum sector. ED results for a $10\times8$ torus. (b) Quench dynamics on a $8\times8$ torus. Real time behavior of the monitored observables. The inset shows the Fourier transform obtained after a time evolution $T=600/J$, with a damping $\gamma=5/T$. (c) Snapshots of the two-particle function $C_{2p}(\mathbf{r})$ showing the clockwise precession of the correlation hole during the quench.
• Figure S1: More on graviton weights. Low-energy spectral densities $I_\pm(\omega)$ for fluxes (a) $n_\phi=1/7$ and (b) $n_\phi=1/5$. (c) High-energy behavior for flux $n_\phi=1/8$. (d) Integrated density over frequencies. ED results on tori. Spectral densities obtained by taking $n_\mathrm{Krylov}=1000$ states. The Lorentzian broadening is set to $\gamma=0.001$ for panels (a) and (b), but is lowered to $\gamma=0.01$ for plots (c) and (d).