Origin of the anomalous L=2 static-structure-factor behavior

Determine why the projected static structure factor of the composite-fermion Fermi liquid behaves anomalously at angular momentum L=2, including whether the anomaly is related to curvature effects in the spherical geometry.

Background

The paper studies the long-wavelength projected static structure factor Sˉ(q)\bar{S}(\mathbf{q}) of the composite-fermion Fermi liquid using composite-fermion diagonalization in spherical geometry. In the thermodynamic extrapolation, the authors omit the L=2L=2 data point because it does not lie on the smooth sequence formed by the remaining even-angular-momentum data with L≥4L\geq4.

The supplementary analysis reports that the L=2L=2 result remains separated from the other data across system sizes and filling factors and differs appreciably from the zeroth-order composite-fermion result. The authors explicitly state that the reason for this anomaly is unknown, while suggesting that curvature effects may be responsible.

References

Unfortunately, the precise behavior of the static structure factor in the long-wavelength limit for the microscopic CFFL remains unclear and, in fact, appears to be in tension with field-theoretic predictions, which is why we have not attempted to fit its pair-correlation function and determine its planar energies in the thermodynamic limit, as we do for other incompressible FQH states.

— Displacement field stabilizes even-denominator and partonic fractional quantum Hall states in the $\mathcal{N}{=}2$ Landau levels of Bernal-stacked bilayer graphene  (2609.19379 - Dora et al., 16 Sep 2026) in Section 3.1, paragraph following Eq. (19)

We conjecture that the hole-conjugates of all the parton states of our interest are chiral. We will test this conjecture and show numerical evidence in support of it.

— Displacement field stabilizes even-denominator and partonic fractional quantum Hall states in the $\mathcal{N}{=}2$ Landau levels of Bernal-stacked bilayer graphene  (2609.19379 - Dora et al., 16 Sep 2026) in Appendix B, Section titled “The leading coefficients in the static structure factor of the parton states”

For the hole-conjugate of these two non-Abelian states, we conjecture that ${\rm var}(s){=}0$, and for the other topological quantum numbers, we will use the values that are consistent with Eq.~eq: topological_numbers_PH_transformation.

— Displacement field stabilizes even-denominator and partonic fractional quantum Hall states in the $\mathcal{N}{=}2$ Landau levels of Bernal-stacked bilayer graphene  (2609.19379 - Dora et al., 16 Sep 2026) in Appendix B, Section titled “The leading coefficients in the static structure factor of the parton states”

We do not know, however, why $S(\mathbf{q})$ behaves anomalously for $L=2$, although it is likely related to curvature effects.

— Universality of long-wavelength behavior of composite-fermion Fermi liquid  (2608.19604 - Makki et al., 20 Aug 2026) in Supplementary Material, discussion following Figs. S1 and S2