Nonperturbative renormalization of Haldane pseudopotentials from the exact two-electron spectrum

Haldane pseudopotentials $V_{|m|}$ provide the effective interaction parameters governing correlated states in the fractional quantum Hall regime. In conventional formulations, these quantities are obtained by projecting the Coulomb interaction onto relative-angular-momentum states within the lowest Landau level, thereby neglecting virtual transitions to higher Landau levels. Here, we formulate a nonperturbative description of effective interactions directly from the exact two-electron spectrum in a magnetic field. By solving the relative-motion problem beyond the lowest-Landau-level approximation, we define renormalized pseudopotentials $V^*_{|m|}$ from the exact eigenenergies and introduce dynamical corrections $\Delta_{|m|}=V^*_{|m|}-V_{|m|}$. The corrections remain systematically negative and depend strongly on both interaction strength and relative angular momentum, reflecting dynamical correlation effects associated with higher-state virtual admixture. The exact results reproduce the perturbative Landau-level-mixing limit at weak coupling while exhibiting substantial deviations in the strong-mixing regime, signaling the breakdown of low-order perturbative expansions. In particular, the short-range interaction channels relevant to Laughlin-type correlations undergo strong renormalization, leading to substantial modification of the effective interaction hierarchy in strongly interacting systems such as ZnO/MgZnO heterostructures. The present formulation establishes a microscopic framework for incorporating nonperturbative Landau-level-mixing effects into effective interaction theories of quantum Hall systems.