Inter-Landau-Level Pair Tunneling Rigidifies the Long-Wavelength Structure Factor of the {\nu}=1/3 Laughlin State

Abstract

A textbook chain of reasoning, originating with Girvin, MacDonald, and Platzman, predicts that Landau-level mixing (LLM) at nu=1/3 should soften the long-wavelength structure factor S(L): virtual excitations suppress V_1, the magnetoroton gap closes, and the single-mode approximation then forces S(L) to increase. We test this chain on the Haldane sphere at nu=1/3, N=6, 2Q=15. (i) Perturbatively, via first-order Schrieffer-Wolff dressing of the LLL Laughlin state, fed Lowdin-renormalised pseudopotentials: the dressed state gives Delta S(1) at kappa=10 of approximately +21%, consistent with the GMP prediction (~19% softening of V_1^eff). (ii) Non-perturbatively, via rigorous two-Landau-level (LLL+SLL) exact diagonalisation using a repaired DiagHam 2LL module that retains, for the first time on the sphere, the full inter-LL scattering channels. The response reverses sign: S(1) rigidifies monotonically from 0.118 at kappa->0 to 0.080 at kappa=10, a 32% suppression. Perturbative and non-perturbative results thus disagree in sign, with a gap of ~53 percentage points between +21% (SW) and -32% (full ED). Channel-mask ablation localises the reversal to the DD<->UU pair-tunneling channel (98% of the shift). A four-way decomposition shows the three-body coherent content overpowers the perturbative admixture by ~2.5x. Repaired code: https://github.com/Cannan-1/DiagHamOnSphereOpenSource (v2.0).