Quantum mechanics of composite fermions

Résumé : The theory of composite fermions consists of two complementary parts: a standard ansatz for constructing many-body wave-functions of various fractional quantum Hall states, and an effective theory (the HLR theory) for predicting responses of these states to external perturbations. Conventionally, both the ansatz and the HLR theory are justified by Lopez-Fradkin's theory based on the singular Chern-Simons transformation. In this work, we aim to provide an alternative basis and unify the two parts into a coherent theory by developing quantum mechanics of composite fermions based on the dipole picture. We argue that states of a composite fermion in the dipole picture are naturally described by bivariate wave functions which are holomorphic (anti-holomorphic) in the coordinate of its constituent electron (vortex), defined in a Bergman space with its weight determined by the spatial profiles of the physical and the emergent Chern-Simons magnetic fields. Based on a semi-classical phenomenological model and the quantization rules of the Bergman space, we establish general wave equations for composite fermions. The wave equations resemble the ordinary Schr\"odinger equation but have drift velocity corrections not present in the HLR theory. Using Pasquier-Haldane's interpretation of the dipole picture, we develop a general wave-function ansatz for constructing many-body wave functions of electrons by projecting states of composite fermions solved from the wave equation into a half-filled bosonic Laughlin state of vortices. It turns out that for ideal fractional quantum Hall states the general ansatz and the standard ansatz are equivalent, albeit using different wave-function representations for composite fermions. To justify the phenomenological model, we derive it from the microscopic Hamiltonian and the general variational principle of quantum mechanics.