Rigorous status of the conjectured upper bound on the sixth-order coefficient $\bar S_6$ of the guiding-center structure factor of chiral fractional quantum Hall states

The guiding-center static structure factor of an incompressible fractional quantum Hall (FQH) state expands at small wavenumber as $\bar S(\bm k)=\bar S_4(k\ell)^4+\bar S_6(k\ell)^6+\cdots$, where $\ell$ is the magnetic length. The quartic coefficient $\bar S_4$ obeys the Haldane lower bound, saturated by ideal conformal-block wavefunctions. Field-theoretic treatments of chiral FQH states additionally predict $\bar S_6$ exactly in terms of topological data: the guiding-center shift, the chiral central charge, and the orbital-spin variance var(s). Kumar and Haldane (SciPost Phys. 16, 117 (2024)) computed $\bar S_4$ and $\bar S_6$ with infinite-cylinder density-matrix renormalization group and found that the predicted value is attained by model wavefunctions but approached only from below by generic ground states, suggesting the field-theory expression acts as an upper bound. The open problem is the rigorous status of this conjecture: prove or refute that the field-theory expression bounds $\bar S_6$ from above for generic chiral FQH ground states in the two-dimensional limit, delimiting the class of states for which it holds, and establish the microscopic meaning of the var(s) ingredient, which currently has no interpretation in the composite-boson description of the lowest Landau level. No work that proves, refutes, or delimits the conjecture is known in the literature to date.

Background

In a fractional quantum Hall (FQH) state, electrons in a two-dimensional plane partially fill a Landau level, the discrete energy levels of a charged particle in a strong perpendicular magnetic field. Within the lowest Landau level (LLL), the electron dynamics is governed by the guiding center coordinates, the slowly varying part of the electron position whose two components do not commute, so that all intra-level correlations are encoded in the guiding-center density operator. The guiding-center static structure factor $\bar S(\bm k)$ is the Fourier transform of the guiding-center density-density correlation function. Translational and inversion symmetry force its long-wavelength expansion to begin at fourth order,

$$\bar S(\bm k)=\bar S_4(k\ell)^4+\bar S_6(k\ell)^6+\cdots,$$

with $\ell=\sqrt{\hbar/(eB)}$ the magnetic length. Haldane showed that the tensor determining the quartic term obeys a lower bound (the Haldane bound) that for rotation-invariant states reads $\bar S_4\ge\pi|\bar\eta_H/(eB)|$, where $\bar\eta_H$ is the guiding-center contribution to the Hall viscosity, the dissipationless viscous response of a quantum Hall fluid to shear strain. Physically, $\bar S_4$ measures the total spectral weight of spin-2 (quadrupolar) guiding-center excitations, the long-wavelength limit of the magneto-roton (the gapped neutral collective density excitation of an FQH fluid) that behaves as an emergent graviton, a gapped chiral spin-2 collective mode probed by circularly-polarized Raman scattering. Formally, $\bar S_4$ is the metric (fluctuation) part and $\bar\eta_H$ the Berry-curvature (adiabatic-response) part of the fluid's response to area-preserving shape deformations, and the positive semidefiniteness of that response yields the bound. Model wavefunctions such as the Laughlin and Moore-Read states, which are dense zero-energy ground states of special model Hamiltonians built from conformal-field-theory correlators, saturate the bound.

A second, sharper property has been conjectured from field theory. Curved-space computations of the gravitational anomaly — the response of the FQH fluid to curved geometry, encoded in a gravitational Chern-Simons term and also called the framing anomaly — and the bimetric effective theory, a field theory of FQH states in which the fluid's internal metric is a dynamical variable (Can, Laskin and Wiegmann; Gromov and Son), predict that for chiral FQH states the sixth-order coefficient is fixed by topological data alone:

$$\bar S_6=\pm\frac{\nu(3\bar{\mathcal S}^2+1)-\bar c+12\nu\,\mathrm{var}(s)}{96},$$

where $\nu$ is the filling fraction, $\bar{\mathcal S}$ is the guiding-center shift (the excess flux needed to place the state on a sphere, with the Landau-orbit contribution removed; it differs from the conventional Wen-Zee shift $\mathcal S=\nu^{-1}N_e-N_\phi$, with $N_e$ the electron number and $N_\phi$ the number of flux quanta, by the Landau-orbit contribution), $\bar c\equiv c_-\!-\!\nu$ with $c_-$ the chiral central charge of the edge theory (the difference between the central charges of the right- and left-moving one-dimensional edge modes, fixed by the gravitational anomaly of the edge), and $\mathrm{var}(s)=\frac{s^TK^{-1}s}{t^TK^{-1}t}-\left(\frac{s^TK^{-1}t}{t^TK^{-1}t}\right)^2$ is the orbital-spin variance, a combination of the $K$-matrix, charge vector $t$, and spin vector $s$ that label an Abelian topological order, in which quasiparticle types are integer vectors $l$ with electric charge $t^TK^{-1}l$ and mutual exchange statistics $2\pi\,l^TK^{-1}l'$; the $+$ or $-$ sign applies to a chiral state or its particle-hole (PH) conjugate, the state obtained by filling the complement of the occupied orbitals within the same Landau level. The bimetric theory itself warns that this universality is subtle: the relevant gravitational Chern-Simons coefficient is fixed only indirectly by consistency with the Girvin-MacDonald-Platzman (GMP) density algebra (the algebra obeyed by LLL-projected density operators), the contributions of higher-spin guiding-center modes were never computed explicitly, and equality is not expected generically.

Kumar and Haldane tested this numerically. Using infinite-cylinder density-matrix renormalization group (iDMRG), a tensor-network method that reaches the two-dimensional limit through cylinders of growing circumference, they computed $\bar S_4$ and $\bar S_6$ for Laughlin, Moore-Read, and Jain composite-fermion (electrons bound to an even number of flux quanta) $\nu=2/5$ ground states of model Hamiltonians and of realistic Hamiltonians (Coulomb interactions, and Haldane pseudopotentials — the two-body interaction energies in each relative-angular-momentum channel — perturbing the model interactions). They found: the equality holds for conformal-block wavefunctions (also verified analytically for the Moore-Read state on curved backgrounds by Dwivedi and Klevtsov); for generic chiral ground states $\bar S_6$ lies slightly but systematically below the predicted value (about 4% below at $\nu=1/3$ Coulomb, 1–2% below at $\nu=2/5$, the only case analyzed with $\mathrm{var}(s)\neq 0$); and perturbing away from the model Hamiltonian only lowers $\bar S_6$ from its zero-perturbation maximum, while $\bar S_4$ rises above the Haldane bound. By the spectral sum rules (exact constraints on the frequency-integrated weight of dynamical response functions) of Golkar, Nguyen and Son and of Nguyen, Haldane, Rezayi, Son and Yang, non-saturation of the Haldane bound accompanies a non-maximally-chiral graviton (both spin-2 chiralities present). The authors therefore conjectured that the field-theory expression is an upper bound on $\bar S_6$ for generic chiral FQH ground states and stated that a theoretical justification of this conjecture is an interesting problem for future work. One ingredient of the conjectured expression additionally lacks any microscopic meaning: the orbital-spin variance $\mathrm{var}(s)$, or equivalently the particle-hole-odd combination $\nu[(S_{\rm GC}/p)^2+\mathrm{var}(s)]=(s-t/2)^TK^{-1}(s-t/2)$ (with $p$ the numerator of $\nu=p/q$ in lowest terms and $S_{\rm GC}$ the guiding-center spin of the composite boson), for which the authors state they are unaware of any interpretation in the composite-boson framework (each electron bound to an even number of flux quanta, forming a condensate described by a Ginzburg-Landau-Chern-Simons theory) used in the same paper.

Problem Statement

Establish the rigorous status of the conjectured upper bound on the sixth-order guiding-center correlation coefficient of chiral FQH states, together with — as a related, separately answerable open question raised by the same expression — the microscopic content of its least-understood ingredient.

1. Prove, refute, or precisely delimit the conjecture of Kumar and Haldane that, for gapped, rotation-invariant, chiral (in the source's terms, at least weakly maximally chiral, meaning that the low-lying part of the entanglement spectrum — the eigenvalues of the reduced density matrix across a spatial cut, which mirror the chiral edge modes — contains contributions of only one propagation direction) ground states of generic two-body Hamiltonians within the lowest Landau level, taken in the two-dimensional thermodynamic limit, the coefficient $\bar S_6$ in $\bar S(\bm k)=\bar S_4(k\ell)^4+\bar S_6(k\ell)^6+\cdots$ is bounded above by the field-theory value

$$\bar S_6^{\rm FT}=\pm\frac{\nu(3\bar{\mathcal S}^2+1)-\bar c+12\nu\,\mathrm{var}(s)}{96},$$

with the sign chosen for a chiral state or its particle-hole conjugate, and with equality attained by the ideal conformal-block (model) wavefunctions, the saturating cases known at present. The delimitation is to cover exactly the class of states for which the inequality holds or fails — including multi-component Abelian states (states with several electron species, described by a multi-component $K$-matrix) with non-zero var(s), non-Abelian states, and particle-hole conjugates — and to remain consistent with the known saturating cases (Laughlin, Moore-Read) and the published non-saturating magnitudes.

2. As a related open question noted by the same source, establish the microscopic and physical interpretation of the orbital-spin variance $\mathrm{var}(s)=\frac{s^TK^{-1}s}{t^TK^{-1}t}-\left(\frac{s^TK^{-1}t}{t^TK^{-1}t}\right)^2$, equivalently of the particle-hole-odd combination $\nu[(S_{\rm GC}/p)^2+\mathrm{var}(s)]=(s-t/2)^TK^{-1}(s-t/2)$, with $p$ the numerator of $\nu=p/q$ in lowest terms and $S_{\rm GC}$ the composite-boson guiding-center spin, equivalently fixed by the right-hand side of this identity, within the composite-boson description of the lowest Landau level, in which framework no interpretation is currently known. The var(s) term is an ingredient of the conjectured bound; the interpretation question remains open regardless of whether the bound is proved, refuted, or delimited.

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