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arXiv:2607.06267

Chiral Graviton Modes in Non-Abelian lattice Fractional Quantum Hall states

Zeno Bacciconi, Min Long, Hernan Xavier, Hongyu Lu, Marcello Dalmonte, Zi Yang Meng

cond-mat.quant-gascond-mat.mes-hallcond-mat.str-elquant-phSubmitted Jul 7, 2026• Announced Jul 8, 2026• 0 citations
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Abstract

Synthetic quantum matter provides a highly tunable route to fractional quantum Hall physics beyond the constraints of conventional electronic materials. However, previous theoretical studies have mostly focused on their ground state properties. It remains unclear to what extent such platforms could reveal key excitation properties of fractional quantum Hall states. Here, we study charge-neutral collective excitations in a non-abelian lattice fractional quantum Hall state realized in the bosonic Harper-Hofstadter model at unity filling factior, realizing a Moore-Read ground state. Combining full exact diagonalization, band-projected exact diagonalization, and matrix-product-state simulations, we demonstrate the existence of a long-lived chiral graviton mode, probed by chiral 3-body correlators, for the first time on lattice non-Abelian states. The graviton signal is topological sector-independent and could be observed via geometric quenches in small open droplets directly relevant to current cold-atom experiments, while other neutral modes, such as the magnetoroton and neutral fermion, are less resolved at presently achievable volumes.

Comments
8+7 pages, 3+11 figures in main+supplementary material. Comments are welcome

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Synthetic quantum matter provides a highly tunable route to fractional quantum Hall physics beyond the constraints of conventional electronic materials. However, previous theoretical studies have mostly focused on their ground state properties. It remains unclear to what extent such platforms could reveal key excitation properties of fractional quantum Hall states. Here, we study charge-neutral collective excitations in a non-abelian lattice fractional quantum Hall state realized in the bosonic Harper-Hofstadter model at unity filling factior, realizing a Moore-Read ground state. Combining full exact diagonalization, band-projected exact diagonalization, and matrix-product-state simulations, we demonstrate the existence of a long-lived chiral graviton mode, probed by chiral 3-body correlators, for the first time on lattice non-Abelian states. The graviton signal is topological sector-independent and could be observed via geometric quenches in small open droplets directly relevant to current cold-atom experiments, while other neutral modes, such as the magnetoroton and neutral fermion, are less resolved at presently achievable volumes.
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.
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 graviton modes in near-term quantum simulation experiments.
The fractional quantum Hall fluid is a two-dimensional quantum fluid of electrons subject to a strong magnetic field at low temperatures.Neutral excitations in a fractional quantum Hall droplet define the incompressibility gap of the topological phase.Among these states, there are some specific modes in the long-wavelength limit that can be understood as "spin-2 gravitons".In this thesis, we will introduce a set of analytical results for the energy gap of the graviton modes for model Hamiltonians in the thermodynamic limit, which is governed by a well-defined and universal characteristic tensor.These results can help to construct model Hamiltonians for the graviton modes of different FQH phases and elucidate a hierarchical structure of conformal Hilbert spaces (null spaces of model Hamiltonians) containing the graviton modes and their corresponding ground states.An isomorphism can be defined for these conformal Hilbert spaces, and the mapping between them can be regarded as a more rigorous and general reinterpretation of the composite fermionization of FQH states, with naturally emergent composite fermions (each consisting of one electron and an even number of fluxes).The results from exact diagonalization will be shown, which confirm that for gapped phases, low-lying neutral excitations can undergo a phase transition even when the ground state remains in the same phase.Furthermore, the gaplessness of the Gaffnian state could be testified based on this formalism with further numerical experiments.Recently there has been numerical evidence implying the signature of multiple graviton modes.We will introduce the microscopic theory for the emergence of multiple gravitons in fractional quantum Hall droplets based on composite fermionization and the well-defined particle-hole conjugate within a specific conformal Hilbert space.This reveals the dynamical nature of this phenomenon and provides theoretical insights into the chirality and the "merging and splitting" behaviors of the graviton modes.The experimental relevance of multiple graviton modes will be discussed.The microscopic theory of gravitons can also provide valuable insights into the field-theoretical approaches.
Employing the state-of-the-art time-dependent variational principle (TDVP) algorithm, we compute the spectra of charge-neutral excitations in the ν=1/2ν=1/2ν=1/2 (bosonic) \updated{ and 1/31/31/3 (fermionic) fractional Chern insulator (FCI)} on the Haldane honeycomb lattice model. The magnetoroton visualized from the dynamic density structure factor acquires a minimum gap at finite momentum that can go soft with increasing interaction and give rise to a charge density wave (CDW) at the same wavevector. As the system approaches the FCI-to-CDW transition point, we observe a pronounced sharpening of the roton mode, suggesting that the magnetoroton behaves more like a quasiparticle as it softens. Notably, this occurs while the single-particle gap remains finite. Besides the magnetoroton at finite momentum, we also construct quadrupolar chiral operators in a discrete lattice and resolve the chiral graviton mode around the ΓΓΓ point of the Brillouin zone. Furthermore, we show the different chiralities of the gravitons of FCIs with opposite-sign Hall conductance for the first time.
Abstract Fractional quantum Hall states have been observed at filling factor ν = 3/4 in GaAs hole system and bilayer graphene. In theoretical bootstrap analysis, it was revealed that non-Abelian topological orders with Ising anyons can be realized at ν = 3/4, which exhibit 12 fold ground state degeneracy on the torus. The properties of ν = 3/4 states can be analyzed using two complementary approaches. In the first one, they are treated as particle–hole conjugate of ν = 1/4 Moore–Read types states. In the second one, they are mapped to composite fermions with reverse flux attachment at effective filling factor 3/2, whose integral part realizes an integer quantum Hall state and the fractional part realizes ν = 1/2 Moore–Read type states. For bilayer graphene with appropriate Landau level mixing, numerical calculations have found 12 quasi-degenerate ground states on the torus at ν = 3/4. Chiral graviton spectral functions of these states have one low energy peak with negative chirality and one high energy peak with positive chirality. This points to a specific member of the Moore–Read type states and agrees with the deduction based on daughter states.
Recent advances in engineered quantum platforms have enabled the realization of bosonic Laughlin states at ν=1/2ν=1/2ν=1/2 and brought non-Abelian topological phases within experimental reach. A central theoretical challenge is to develop a unified framework connecting lattice fractional quantum Hall (FQH) model states, exact parent Hamiltonians, and experimentally accessible interactions. We systematically construct Hermitian parent Hamiltonians for which continuum lowest Landau level (LLL) model states sampled on lattice sites are exact zero modes. Using these model manifolds as quantitative references, we find that, within the projected lattice LLL at the flux densities studied, short-range density interactions stabilize Laughlin ground-state manifolds at ν=1/3ν=1/3ν=1/3 and 1/41/41/4, while two-body onsite repulsion supports a Moore--Read triplet at ν=1ν=1ν=1. Full multiband calculations reveal a sharp contrast: the Laughlin manifolds remain robust against interband mixing on the studied tori, whereas the Moore--Read triplet becomes less spectrally isolated and eventually undergoes a level crossing with competing states near the interband scale. Our framework provides a theoretical foundation for using lattice model states to guide the search for experimentally accessible Abelian and non-Abelian FQH states.