Microscopic derivation of Dirac composite fermion theory: Aspects of noncommutativity and pairing instabilities

TL;DR

This paper derives the Dirac composite fermion theory for half-filled Landau levels from first principles, explores noncommutative field theory aspects, and analyzes pairing instabilities influenced by particle-hole symmetry breaking.

Contribution

It provides a microscopic derivation of the Dirac composite fermion theory and introduces a noncommutative field-theoretical framework that extends previous models.

Findings

Weak p-wave pairing in the lowest Landau level due to particle-hole symmetry breaking.

Strong p-wave pairing in the second Landau level with nearly flat dispersion.

Reproduction of Son's theory in the commutative limit with additional physically motivated terms.

Abstract

Building on previous work [N. Read, Phys. Rev. B 58, Z. Dong and T. Senthil, 16262 (1998); Phys. Rev. B 102, 205126 (2020)] on the system of bosons at filling factor $\nu = 1$, we derive the Dirac composite fermion theory for a half-filled Landau level from first principles and applying the Hartree-Fock approach in a preferred representation. On the basis of the microscopic formulation, in the long-wavelength limit, we propose a noncommutative field-theoretical description, which in a commutative limit reproduces the Son's theory, with additional terms that may be expected on physical grounds. The microscopic representation of the problem is also used to discuss pairing instabilities of composite fermions. We find that a presence of a particle-hole symmetry breaking leads to a weak (BCS) coupling $p$-wave pairing in the lowest Landau level, and strong coupling $p$-wave pairing in the second Landau level that occurs in a band with nearly flat dispersion, a third power function of momentum.