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arXiv:2111.10593 · Physical Review Letters

Multiple Magnetorotons and Spectral Sum Rules in Fractional Quantum Hall Systems

Dung Xuan Nguyen, F. D. M. Haldane, Edward H. Rezayi, Dam Thanh Son, Kun Yang

Published Jun 13, 2022• 31 citations
arXivHTMLPDF

Abstract

We study, numerically, the charge neutral excitations (magnetorotons) in fractional quantum Hall systems, concentrating on the two Jain states near quarter filling, $\ensuremath{\nu}=2/7$ and $\ensuremath{\nu}=2/9$, and the $\ensuremath{\nu}=1/4$ Fermi-liquid state itself. In contrast to the $\ensuremath{\nu}=1/3$ states and the Jain states near half filling, on each of the two Jain states $\ensuremath{\nu}=2/7$ and $\ensuremath{\nu}=2/9$ the graviton spectral densities show two, instead of one, magnetoroton peaks. The magnetorotons have spin 2 and have opposite chiralities in the $\ensuremath{\nu}=2/7$ state and the same chirality in the $\ensuremath{\nu}=2/9$ state. We also provide a numerical verification of a sum rule relating the guiding center spin $\overline{s}$ with the spectral densities of the stress tensor.

DOI
10.1103/physrevlett.128.246402

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We study, numerically, the charge neutral excitations (magnetorotons) in fractional quantum Hall systems, concentrating on the two Jain states near quarter filling, \ensuremathν=2/7\ensuremath{\nu}=2/7\ensuremathν=2/7 and \ensuremathν=2/9\ensuremath{\nu}=2/9\ensuremathν=2/9, and the \ensuremathν=1/4\ensuremath{\nu}=1/4\ensuremathν=1/4 Fermi-liquid state itself. In contrast to the \ensuremathν=1/3\ensuremath{\nu}=1/3\ensuremathν=1/3 states and the Jain states near half filling, on each of the two Jain states \ensuremathν=2/7\ensuremath{\nu}=2/7\ensuremathν=2/7 and \ensuremathν=2/9\ensuremath{\nu}=2/9\ensuremathν=2/9 the graviton spectral densities show two, instead of one, magnetoroton peaks. The magnetorotons have spin 2 and have opposite chiralities in the \ensuremathν=2/7\ensuremath{\nu}=2/7\ensuremathν=2/7 state and the same chirality in the \ensuremathν=2/9\ensuremath{\nu}=2/9\ensuremathν=2/9 state. We also provide a numerical verification of a sum rule relating the guiding center spin s‾\overline{s}s with the spectral densities of the stress tensor.
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.
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.
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.
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.
We numerically investigate the collective excitation of spin-conserving and spin-reversed configuration of rotating diluted ultra-cold dipolar Bose gas. Rotating trapped Bose gas produces a fictitious magnetic field perpendicular to the trapping harmonic potential, which exhibits strongly correlated fractional quantum Hall states. We consider the long-range dipole-dipole interaction and compute the low lying excitations spectrum for the three fractions of the first Jain series ν=1/2,1/4,1/6ν= 1/2, 1/4, 1/6ν=1/2,1/4,1/6. We find that for both the spin-conserving and spin-reversed excitation the gap between the fundamental mode and the higher excitation mode increases upon increase in the filling fraction. The fundamental modes and the next higher-energy mode of excitation spectra for each of the three fractions show the presence of double roton for spin-conserving configuration only. Finally we complement our observation by calculating the spectral weight for the fundamental mode of excitation spectra which show the momenta at which the spectral weight exhibits the maxima shifts towards the lower momenta for both the excitations. Our observation for the spectral weight could be related with the inelastic Raman scattering which may be useful for the future experimental study to detect the excitation in ultracold system.
We consider spin-polarized electrons in a single Landau level on a torus. The quantum Hall problem is mapped onto a one-dimensional lattice model with lattice constant $2\ensuremath{\pi}/{L}_{1}$, where L1{L}_{1}L1​ is a circumference of the torus (in units of the magnetic length). In the Tao-Thouless limit ${L}_{1}\ensuremath{\rightarrow}0$, the interacting many-electron problem is exactly diagonalized at any rational filling factor $\ensuremath{\nu}=p/q\ensuremath{\le}1$. For odd qqq, the ground state has the same qualitative properties as a bulk $({L}_{1}\ensuremath{\rightarrow}\ensuremath{\infty})$ quantum Hall hierarchy state and the lowest-energy quasiparticle excitations have the same fractional charges as in the bulk. These states are the ${L}_{1}\ensuremath{\rightarrow}0$ limits of the Laughlin and Jain wave functions for filling fractions where these exist. We argue that the exact solutions generically, for odd qqq, are continuously connected to the two-dimensional bulk quantum Hall hierarchy states---i.e., that there is no phase transition as ${L}_{1}\ensuremath{\rightarrow}\ensuremath{\infty}$ for filling factors where such states can be observed. For even-denominator fractions, a phase transition occurs as L1{L}_{1}L1​ increases. For $\ensuremath{\nu}=1/2$ this leads to the system being mapped onto a Luttinger liquid of neutral particles at small but finite L1{L}_{1}L1​; this then develops continuously into the composite fermion wave function that is believed to describe the bulk $\ensuremath{\nu}=1/2$ system. The analysis generalizes to non-Abelian quantum Hall states.