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arXiv:2410.00165 · Physical review. B./Physical review. B

Static structure factor and the dispersion of the Girvin-MacDonald-Platzman density mode for fractional quantum Hall fluids on the Haldane sphere

Rakesh K. Dora, Ajit Coimbatore Balram

Published Mar 12, 2025• 6 citations
arXivHTMLPDF

Abstract

We study the neutral excitations in the bulk of the fractional quantum Hall (FQH) fluids generated by acting with the Girvin-MacDonald-Platzman (GMP) density operator on the uniform ground state. Creating these density modulations atop the ground state costs energy, since any density fluctuation in the FQH system has a gap stemming from underlying interparticle interactions. We calculate the GMP density-mode dispersion for various bosonic and fermionic FQH states on the Haldane sphere using the ground-state static structure factor computed on the same geometry. Previously, this computation was carried out on the plane. Analogous to the GMP algebra of the lowest Landau level (LLL) projected density operators in the plane, we derive the algebra for the LLL-projected density operators on the sphere, which facilitates the computation of the density-mode dispersion. Contrary to previous results on the plane, we find that, in the long-wavelength limit, the GMP mode accurately describes the dynamics of the primary Jain states.

DOI
10.1103/physrevb.111.115132

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We study the neutral excitations in the bulk of the fractional quantum Hall (FQH) fluids generated by acting with the Girvin-MacDonald-Platzman (GMP) density operator on the uniform ground state. Creating these density modulations atop the ground state costs energy, since any density fluctuation in the FQH system has a gap stemming from underlying interparticle interactions. We calculate the GMP density-mode dispersion for various bosonic and fermionic FQH states on the Haldane sphere using the ground-state static structure factor computed on the same geometry. Previously, this computation was carried out on the plane. Analogous to the GMP algebra of the lowest Landau level (LLL) projected density operators in the plane, we derive the algebra for the LLL-projected density operators on the sphere, which facilitates the computation of the density-mode dispersion. Contrary to previous results on the plane, we find that, in the long-wavelength limit, the GMP mode accurately describes the dynamics of the primary Jain states.
Collective modes capture the dynamical aspects of fractional quantum Hall (FQH) fluids. Depending on the active degrees of freedom, different types of collective modes can arise in a FQH state. In this work, we consider spinful FQH states in the lowest Landau level (LLL) along the Jain sequence of fillings ν=n/(2n±1)ν{=}n/(2n{\pm}1)ν=n/(2n±1) and compute the Coulomb dispersion of their spin-flip and spin-conserving collective modes in the spherical geometry. We use the LLL-projected density-wave and composite fermion (CF) exciton states as trial wave functions for these modes. To evaluate the dispersion of density-wave states, we derive the commutation algebra of spinful LLL-projected density operators on the sphere, which enables us to extract the gap of the density-wave excitations from the numerically computed density-density correlation function, i.e., the static structure factor, of the FQH ground state. We find that the CF excitons provide an accurate description of the collective modes at all wavelengths, while the density-wave states fail to do so. Specifically, the spin-flip density wave reliably captures the spin-flip collective mode only for the Laughlin and Halperin states, and that too only in the long-wavelength limit. Interestingly, for spin-singlet primary Jain states, the spin-conserving density mode is inaccurate even in the long-wavelength regime. We show that this discrepancy stems from the presence of an additional high-energy spin-conserving parton mode, similar to that found in fully polarized secondary Jain states at ν=n/(4n±1)ν{=}n/(4n{\pm}1)ν=n/(4n±1). We propose an ansatz for this parton mode and compute its Coulomb dispersion in the singlet state at ν=2/5ν{=}2/5ν=2/5. The predicted parton mode can be observed in circularly polarized inelastic light scattering experiments.
Fractional quantum Hall (FQH) states are prototypical examples of strongly interacting topologically ordered systems. In this work, we obtain thermodynamic fits on the plane for the pair correlation function, and its Fourier transform, the static structure factor, of two-component spin-singlet Halperin and Jain FQH fluids by expanding them in the recently introduced basis of the orthogonal associated Laguerre polynomials [Fulsebakke et al., SciPost Phys. 14, 149 (2023), https://doi.org/10.21468/SciPostPhys.14.6.149 ] and ascertaining the expansion coefficients by fitting them to large-system Monte Carlo data evaluated using their trial wavefunctions. In this fitting procedure, aside from constraining the exact short-distance behavior of the wavefunction, we also derive and enforce the sum rules that the long-wavelength expansion of the static structure factor must adhere to. We show that incorporating these constraints is crucial for obtaining numerically stable and accurate values of the long-wavelength Girvin-MacDonald-Platzman (GMP)/symmetric density-wave excitation gap. We further extend this approach to spin-resolved density-correlation functions, enabling the evaluation of the gap of the antisymmetric density-wave mode for these spin-singlet FQH states. Finally, we use the density-correlators to compute variational energies of the states and construct phase diagrams for bilayer FQH systems. These could be relevant for understanding recent bilayer FQH experiments that map out the phase diagram by tuning the interlayer separation and density-imbalance/layer-polarization.
The low-energy physics of fractional quantum Hall (FQH) states-a paradigm of strongly correlated topological phases of matter-to a large extent is captured by weakly interacting quasiparticles known as composite fermions.In this paper, based on numerical simulations and effective field theory, we argue that some high-energy states in the FQH spectra necessitate a different description based on parton quasiparticles.We show that Jain states at filling factor ν ¼ n=ð2pn Æ 1Þ with integers n, p ≥ 2 support two kinds of collective modes: In addition to the well-known Girvin-MacDonald-Platzman (GMP) mode, they host a high-energy collective mode, which we interpret as the GMP mode of partons.We elucidate observable signatures of the parton mode in the dynamics following a geometric quench.We construct a microscopic wave function for the parton mode and demonstrate agreement between its variational energy and exact diagonalization.Using the parton construction, we derive a field theory of the Jain states and show that the previously proposed effective theories follow from our approach.Our results point to partons being "real" quasiparticles which, in a way reminiscent of quarks, become observable only at sufficiently high energies.
We show model wavefunctions for neutral collective modes in fractional quantum Hall (FQH) states have simple analytic forms obtained from judicially reducing the powers of selected pairs in the ground state Jastrow factor. This scheme of "pair excitations" works for the magneto-roton modes of single-component Abelian and non-Abeliean FQH states, as well as neutral fermion mode for the Moore-Read (MR) state. The analytic wavefunctions enable computations in the thermodynamic limit previous inaccessible to numerics, and the long wavelength limit of the neutral energy gap of the magneto-roton modes can be interpreted as fusion of charges in two-dimensional plasma picture, extending the plasma analogy to neutral excitations. A lattice diagrammatic method of representing these many-body wavefunctions and FQH elementary excitations is also presented.
We study fractional quantum Hall states at filling fractions in the Jain sequences using the framework of composite Dirac fermions. Synthesizing previous work, we write an effective field theory consistent with all symmetry requirements, including Galilean invariance and particle-hole symmetry. Employing a Fermi-liquid description, we demonstrate the appearance of the Girvin-Macdonald-Platzman algebra and compute the dispersion relation of neutral excitations and various response functions. Our results satisfy requirements of particle-hole symmetry. We show that while the dispersion relation obtained from the modified random-phase approximation (MRPA) of the Halperin-Lee-Read (HLR) theory is particle-hole symmetric, correlation functions obtained from this scheme are not. The results of the Dirac theory are shown to be consistent with the Haldane bound on the projected structure factor, while those of the MPRA of the HLR theory violate it.
We introduce a quench of the geometry of Landau level orbitals as a probe of nonequilibrium dynamics of fractional quantum Hall (FQH) states. We show that such geometric quenches induce coherent many-body dynamics of neutral degrees of freedom of FQH fluids. The simplest case of mass anisotropy quench can be experimentally implemented as a sudden tilt of the magnetic field, and the resulting dynamics reduces to the harmonic motion of the spin-2 ``graviton'' mode, i.e., the long-wavelength limit of the Girvin-MacDonald-Platzman magnetoroton. We derive an analytical description of the graviton dynamics using the bimetric theory of FQH states and find agreement with exact numerical simulations at short times. We show that certain types of geometric quenches excite higher-spin collective modes, thus establishing their existence in a microscopic model and motivating an extension of geometric theories of FQH states.