Coadjoint-orbit bosonization of a Fermi surface in a weak magnetic field

We present a bosonized effective field theory for a 2d Fermi surface in a weak magnetic field using the coadjoint orbit approach, which was recently developed as a nonlinear bosonization method in phase space for Fermi liquids and non-Fermi liquids. We show that by parametrizing the phase space with the guiding center and the mechanical momentum, and by using techniques in noncommutative field theory, the physics of Landau levels and Landau level degeneracy ($N_{\Phi}$) naturally arises. For a parabolic dispersion, the resulting theory describes $N_{\Phi}$ flavors of free chiral bosons propagating in \emph{momentum space}. In addition, the action contains a linear term in the bosonic field, which upon mode expansion becomes a topological $\theta$-term. By properly quantizing this theory, we reproduce the well-known thermal and magnetic responses of a Fermi surface, including linear-in-$T$ specific heat, Landau diamagnetism, and the de Haas-van Alphen effect. {In particular, the de Haas-van Alphen effect is shown to be a direct consequence of the topological $\theta$-term.} Our theory paves the way toward understanding correlated gapless fermionic systems in a magnetic field using the powerful approach of bosonization.