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arXiv:2608.11133

Sum rules and density-wave modes in spin-singlet fractional quantum Hall fluids

Ritajit Kundu, Rakesh K. Dora, Dung Xuan Nguyen, Ajit Coimbatore Balram

Published Aug 11, 2026• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2608.11133

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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.
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.
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.
Model wavefunctions are essential for studying fractional quantum Hall (FQH) phases. They reveal the intricate connections between model FQH states, conformal field theory, Jack polynomials, and pseudopotential Hamiltonians. However, most model wavefunctions studied so far are in the continuum. In this work, we construct model wavefunctions and the corresponding exact parent Hamiltonians for the entire ZkZ_kZk​ Read--Rezayi series on a lattice. Our model wavefunctions obey a generalized clustering rule implemented on the lattice, which we find is encoded in the Macdonald polynomials providing a unified algebraic description of our lattice-specific model states. This ensures that the model states are exact zero-energy ground states of local density interactions and exhibit an infinitely large gap in the particle entanglement spectrum. Numerical studies of the Laughlin and Moore--Read series find the expected zero-mode and one-flux quasihole counting and positive energy gaps at the system sizes studied, and the available finite-size scaling suggests nonzero thermodynamic gaps. Tuning the model interaction range can drive a phase transition out of the FQH phase. Our work not only advances the theoretical understanding of the underlying mathematical structure of lattice model FQH states but also is potentially useful for designing exotic many-body phases in engineered quantum platforms.
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.
We provide a detailed description of a product rule structure of the monomial (Slater) expansion coefficients of bosonic (fermionic) fractional quantum Hall (FQH) states derived recently, which we now extend to spin-singlet states. We show that the Haldane-Rezayi spin-singlet state can be obtained without exact diagonalization through a differential equation method that we conjecture to be generic to other FQH model states. The product rule symmetries allow us to build approximations of FQH states that exhibit increasing overlap with the exact state (as a function of system size) even though our approximation omits more than half of the Hilbert space. We show that the product rule is valid for any FQH state that can be written as an expectation value of parafermionic operators.