# Quantum mechanics of composite fermions

**Authors:** Junren Shi  
**arXiv:** [2309.00299](https://arxiv.org/abs/2309.00299)  
**Date:** 2023-09-01  
**Summary page:** https://www.summarizepaper.com/en/paper/2309.00299v1/

## Key points

- Theory of composite fermions is fundamental in studying fractional quantum Hall states
- Consists of two parts: ansatz for constructing wave-functions and HLR theory for predicting responses to perturbations
- Lopez-Fradkin's theory justifies both ansatz and HLR theory based on singular Chern-Simons transformation
- Authors aim to provide alternative basis and unify the two parts into a coherent theory
- Introduce dipole picture, based on quantum mechanics of composite fermions
- Composite fermion states described by holomorphic/anti-holomorphic bivariate wave functions in Bergman space
- Wave equations for composite fermions resemble Schrödinger equation but include drift velocity corrections not present in HLR theory
- General wave-function ansatz based on Pasquier-Haldane's interpretation of dipole picture used to construct many-body wave functions of electrons
- General ansatz and standard ansatz used in previous theories are equivalent for ideal fractional quantum Hall states, but use different representations for composite fermions
- Phenomenological model derived from microscopic Hamiltonian and variational principles applied to justify it
- Provides new perspective on theory of composite fermions through dipole picture framework
- Offers unified understanding of many-body wave functions and responses to external perturbations in fractional quantum Hall states

## Summary

The theory of composite fermions is a fundamental concept in the study of fractional quantum Hall states. It consists of two complementary parts: a standard ansatz for constructing many-body wave-functions of these states, and an effective theory known as the HLR theory, which predicts their responses 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, the authors aim to provide an alternative basis for understanding composite fermions and unify the two parts into a coherent theory. They develop a new approach called the dipole picture, which is based on the quantum mechanics of composite fermions. In this picture, states of a composite fermion are described by bivariate wave functions that are holomorphic (anti-holomorphic) in the coordinate of its constituent electron (vortex). These wave functions are defined in a Bergman space with its weight determined by the spatial profiles of the physical and emergent Chern-Simons magnetic fields. Using a semi-classical phenomenological model and the quantization rules of the Bergman space, the authors establish general wave equations for composite fermions. These wave equations resemble the ordinary Schrödinger equation but include drift velocity corrections that are not present in the HLR theory. To construct many-body wave functions of electrons, they develop a general wave-function ansatz based on Pasquier-Haldane's interpretation of the dipole picture. This involves projecting states of composite fermions solved from the wave equation into a half-filled bosonic Laughlin state of vortices. The authors find that for ideal fractional quantum Hall states, their general ansatz and the standard ansatz used in previous theories are equivalent. However, they use different wave-function representations for composite fermions. To justify their phenomenological model, they derive it from microscopic Hamiltonian and apply general variational principles of quantum mechanics. Overall, this work provides a new perspective on the theory of composite fermions by developing a coherent framework based on the dipole picture. The authors' approach offers a unified understanding of many-body wave functions and responses to external perturbations in fractional quantum Hall states.