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arxiv: 2509.13100 · v3 · pith:PPAHNY7Lnew · submitted 2025-09-16 · ❄️ cond-mat.str-el

Dispersion of collective modes in spinful fractional quantum Hall states on the sphere

Rakesh K. Dora , Ajit C. Balram This is my paper

Pith reviewed 2026-05-21 23:02 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords fractional quantum Hall effectcollective modescomposite fermionsspinful statesdensity wavesparton modesspherical geometryJain sequence
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0 comments X

The pith

Composite fermion excitons accurately describe collective modes in spinful fractional quantum Hall states on the sphere at all wavelengths.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper computes the Coulomb dispersions of spin-flip and spin-conserving collective modes for spinful lowest Landau level states along the Jain sequence of fillings on the sphere. It employs two families of trial wave functions: LLL-projected density-wave states, for which the authors first derive the commutation algebra of spinful projected density operators to extract gaps from the static structure factor, and composite fermion exciton states. The calculation shows that CF excitons match the expected mode dispersions across the full range of wavevectors, whereas density-wave states succeed only in restricted cases such as the long-wavelength limit for Laughlin and Halperin states. In spin-singlet primary Jain states the spin-conserving density mode deviates even at long wavelength, which the authors trace to an extra high-energy spin-conserving parton mode; they supply an explicit ansatz for this mode and evaluate its dispersion at filling factor 2/5. The results clarify which variational states capture the dynamical response of these fluids and indicate how the additional mode could be detected in circularly polarized inelastic light scattering.

Core claim

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. For spin-singlet primary Jain states, the spin-conserving density mode is inaccurate even in the long-wavelength regime because of an additional high-energy spin-conserving parton mode, for which an ansatz is proposed and its Coulomb dispersion is computed at nu = 2/5.

What carries the argument

Composite fermion exciton trial wave functions for the collective modes, together with the commutation algebra of spinful LLL-projected density operators on the sphere that allows extraction of excitation energies from the ground-state static structure factor.

If this is right

  • CF exciton states reproduce both spin-conserving and spin-flip mode dispersions at all wavelengths for the studied Jain-sequence fillings.
  • LLL-projected density-wave states succeed only for the spin-flip mode of Laughlin and Halperin states in the long-wavelength limit.
  • Spin-singlet primary Jain states require an extra high-energy spin-conserving parton mode to account for the spin-conserving collective excitation even at long wavelength.
  • An explicit ansatz for the additional parton mode permits computation of its Coulomb dispersion in the spin-singlet state at filling factor 2/5.
  • The predicted parton mode is observable in circularly polarized inelastic light scattering experiments.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the CF exciton description holds, analogous exciton constructions may be needed to describe collective modes in other families of spinful fractional quantum Hall states.
  • Detection of the extra parton mode would indicate that multi-component parton constructions are required for a complete account of dynamical excitations beyond the primary composite-fermion picture.
  • The spherical-geometry dispersions supply concrete predictions that can be checked by future exact diagonalization studies on larger systems.

Load-bearing premise

The LLL-projected density-wave and CF exciton states are assumed to be sufficiently accurate variational descriptions of the true collective-mode eigenstates at all wavelengths.

What would settle it

Exact diagonalization of the Coulomb Hamiltonian on the sphere yielding collective-mode energies that deviate from the CF-exciton dispersions at intermediate and short wavelengths would falsify the claim of accuracy at all wavelengths.

Figures

Figures reproduced from arXiv: 2509.13100 by Ajit C. Balram, Rakesh K. Dora.

Figure 1
Figure 1. Figure 1: FIG. 1. Comparison of the dispersion of the spin-wave mode [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Schematic of the spin-flip CF excitons included in the CF-diagonalization-based spin-flip CF exciton gap computation [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Top panels: Comparison of the dispersion of the spin-wave mode in primary Jain states computed following different [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a), the SW dispersion at ν=2/3 has a negative energy roton, which is absent at ν=1/3. This negative energy spin-flip roton at ν=2/3 can be understood using CF theory as arising from the CF-cyclotron energy low￾ering transition of a CF from the occupied 1↑-ΛL to the empty 0↓-ΛL. In contrast, at ν=1/3, the spin-flip inter￾ΛL transitions either increase or leave constant the CF cyclotron energy; hence, no ne… view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Schematic of the ( [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (a)]. For L=0, this state has the same S as the ground state [with S z one lower than that of the ground state] while for L≥1 it has one spin lower than the S of the ground state, and thus forms the SW. Similar to the fully polarized states, owing to Larmor’s theorem, the SW dispersion should be gapless in the long-wavelength limit for PP states too. The spin-flip CFE (n↑−1) →(n↓) has the same CF cyclotron… view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Spin-flip dispersion for the LLL Coulomb interaction in the bosonic Halperin-(2 [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. LLL Coulomb spin-wave dispersion in the partially [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. LLL Coulomb dispersion of the spin-conserving charge-neutral mode in the bosonic (2 [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Spin-conserving charge neutral LLL Coulomb gap [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Comparison of the [PITH_FULL_IMAGE:figures/full_fig_p026_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Comparison of the fitted and computed pair-correlation function [PITH_FULL_IMAGE:figures/full_fig_p029_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. LLL Coulomb GMP gap computed on planar and [PITH_FULL_IMAGE:figures/full_fig_p030_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Spin-flip and spin-conserving density-wave gaps in the [PITH_FULL_IMAGE:figures/full_fig_p032_14.png] view at source ↗
read the original abstract

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 $\nu{=}n/(2n{\pm}1)$ 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 $\nu{=}n/(4n{\pm}1)$. We propose an ansatz for this parton mode and compute its Coulomb dispersion in the singlet state at $\nu{=}2/5$. The predicted parton mode can be observed in circularly polarized inelastic light scattering experiments.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper computes Coulomb dispersions of spin-flip and spin-conserving collective modes for spinful FQH states along the Jain sequence ν = n/(2n ± 1) in the LLL on the sphere. LLL-projected density-wave and CF exciton trial states are used; the spinful projected density-operator algebra is derived to extract density-wave gaps directly from the ground-state static structure factor. The central claims are that CF excitons accurately describe the modes at all wavelengths while density-wave states fail (especially the spin-conserving mode in spin-singlet primary Jain states), with the discrepancy at ν = 2/5 attributed to an additional high-energy spin-conserving parton mode whose ansatz and dispersion are presented.

Significance. If the variational trial states faithfully represent the low-lying eigenstates, the work supplies useful numerical dispersions on the sphere, a technically sound algebraic route from structure factor to density-wave gaps, and a concrete parton-mode proposal that could be tested in circularly polarized inelastic light scattering. The parameter-free algebraic extraction and the explicit ansatz for the extra mode are positive features. The significance is reduced by the absence of direct benchmarks against exact eigenstates.

major comments (2)
  1. [Abstract and §5] Abstract and §5 (numerical results): the claim that 'CF excitons provide an accurate description of the collective modes at all wavelengths' is load-bearing for the headline conclusion, yet no wave-function overlaps with exact-diagonalization eigenvectors or direct comparisons of variational energies to exact mode energies are reported for accessible system sizes (N ≲ 12) where ED is feasible; without these, the asserted superiority over density-wave states and the necessity of the extra parton mode rest on unverified variational quality.
  2. [§4.2] §4.2 (spin-singlet Jain states at ν = 2/5): the post-hoc attribution of the long-wavelength discrepancy between the spin-conserving density-wave dispersion and the CF-exciton result to a distinct high-energy parton mode would be strengthened by an independent check, such as the variational energy of the parton ansatz relative to the CF exciton or a small-system overlap with the exact second excited state; the current presentation risks interpreting variational error as evidence for a new mode.
minor comments (2)
  1. [Figure captions and §3] Figure captions and §3: the dispersion plots would be clearer if the expected long-wavelength analytic limits (e.g., from Girvin-MacDonald-Platzman or single-mode approximation) were overlaid as reference curves.
  2. [§2] Notation in §2: the explicit form of the LLL-projected parton-mode ansatz should be written in the main text (rather than only in the supplement) to allow readers to reproduce the variational calculation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below and have incorporated revisions to strengthen the validation of our claims.

read point-by-point responses

  1. Referee: [Abstract and §5] Abstract and §5 (numerical results): the claim that 'CF excitons provide an accurate description of the collective modes at all wavelengths' is load-bearing for the headline conclusion, yet no wave-function overlaps with exact-diagonalization eigenvectors or direct comparisons of variational energies to exact mode energies are reported for accessible system sizes (N ≲ 12) where ED is feasible; without these, the asserted superiority over density-wave states and the necessity of the extra parton mode rest on unverified variational quality.

    Authors: We agree that direct benchmarks against exact diagonalization for small systems would provide stronger evidence for the accuracy of the CF exciton trial states. In the revised manuscript we will add wave-function overlaps between the CF exciton states and the exact eigenstates, together with comparisons of variational energies to exact mode energies, for all accessible system sizes up to N ≈ 12. These additions will directly support the superiority of CF excitons over density-wave states and the identification of the additional parton mode. revision: yes

  2. Referee: [§4.2] §4.2 (spin-singlet Jain states at ν = 2/5): the post-hoc attribution of the long-wavelength discrepancy between the spin-conserving density-wave dispersion and the CF-exciton result to a distinct high-energy parton mode would be strengthened by an independent check, such as the variational energy of the parton ansatz relative to the CF exciton or a small-system overlap with the exact second excited state; the current presentation risks interpreting variational error as evidence for a new mode.

    Authors: We acknowledge the value of an independent verification to distinguish a genuine additional mode from possible variational error. In the revision we will report the variational energy of the proposed parton ansatz relative to the CF exciton state at ν = 2/5. For the smallest accessible system sizes we will also compute the overlap of the parton ansatz with the exact second excited state. These calculations will be included to substantiate the attribution of the discrepancy to a distinct high-energy spin-conserving parton mode. revision: yes

Circularity Check

0 steps flagged

No significant circularity; algebraic extraction and variational ansatzes remain independent

full rationale

The paper derives the commutation algebra of spinful LLL-projected density operators on the sphere to extract density-wave gaps directly from the ground-state static structure factor; this step is a self-contained algebraic identity independent of any mode trial function. CF exciton and density-wave states are introduced as separate variational ansatzes whose dispersions are computed explicitly, while the additional parton-mode ansatz is proposed to interpret a quantitative mismatch rather than being fitted or defined to reproduce the input data by construction. No equation reduces to its own inputs, no uniqueness theorem is imported from self-citation, and no ansatz is smuggled via prior work. The derivation chain therefore stands on explicit computation and the new algebra rather than tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on standard lowest-Landau-level projection and spherical-geometry modeling assumptions plus one newly postulated entity introduced to resolve a numerical discrepancy.

axioms (2)
  • domain assumption Lowest Landau level projection remains valid for the collective-mode wave functions at the fillings considered.
    All trial states are constructed inside the LLL; this is invoked when the authors state they use LLL-projected density-wave and CF exciton states.
  • domain assumption The spherical geometry with its curvature corrections accurately captures the long-wavelength physics of the planar FQH system for these states.
    All dispersions are computed in spherical geometry; the algebra derivation and structure-factor extraction rely on this choice.
invented entities (1)
  • high-energy spin-conserving parton mode no independent evidence
    purpose: Accounts for the observed inaccuracy of the spin-conserving density-wave description in spin-singlet primary Jain states.
    Introduced after noting that the density mode is inaccurate even at long wavelength for singlet states; analogous to modes previously identified in fully polarized secondary Jain states.

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Works this paper leans on

130 extracted references · 130 canonical work pages · 1 internal anchor

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