One-Dimensional Fermions as Two-Dimensional Droplets Via Chern-Simons Theory

TL;DR

This paper develops a novel theoretical framework connecting one-dimensional fermions to two-dimensional droplet dynamics using Chern-Simons theory, revealing new insights into boundary behavior and symmetries.

Contribution

It introduces a Chern-Simons field theory approach to model 1D fermions as 2D droplets, providing a hydrodynamical and boundary perspective.

Findings

Derived the 1D collective field Hamiltonian from 2D droplet dynamics.

Established a boundary-based hydrodynamical formulation.

Discussed symmetries as properties of curves in two dimensions.

Abstract

Based on the observation that a particle motion in one dimension maps to a two-dimensional motion of a charged particle in a uniform magnetic field, constrained in the lowest Landau level, we formulate a system of one-dimen- sional nonrelativistic fermions by using a Chern-Simons field theory in 2+1 dimensions. Using a hydrodynamical formulation we obtain a two-dimensional droplet picture of one-dimensional fermions. The dynamics involved is that of the boundary between a uniform density of particles and vortices. We use the sharp boundary approximation. In the case of well separated boundaries we derive the one-dimensional collective field Hamiltonian. Symmetries of the theory are also discussed as properties of curves in two dimensions.