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Quantum mechanics of composite fermions | | Junren Shi
; | | Date: |
1 Sep 2023 | | Abstract: | 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. | | Source: | arXiv, 2309.00299 | | Services: | Forum | Review | PDF | Favorites | |
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