Topological BF theory of the quantum hydrodynamics of incompressible polar fluids

TL;DR

This paper develops a topological BF theory to describe the quantum hydrodynamics of incompressible polar fluids in 3+1 dimensions, revealing fractional excitations and connections to topological insulators.

Contribution

It introduces a novel topological BF theory framework for 3D polar fluids and extends the Girvin-MacDonald-Platzman algebra to higher dimensions.

Findings

Fractionalized excitations in the hydrodynamic model.

Extension of the algebra to include vortex-density operators.

Connection between the algebra and 3D topological insulators.

Abstract

We analyze a hydrodynamical model of a polar fluid in (3+1)-dimensional spacetime. We explore a spacetime symmetry -- volume preserving diffeomorphisms -- to construct an effective description of this fluid in terms of a topological BF theory. The two degrees of freedom of the BF theory are associated to the mass (charge) flows of the fluid and its polarization vorticities. We discuss the quantization of this hydrodynamic theory, which generically allows for fractionalized excitations. We propose an extension of the Girvin-MacDonald-Platzman algebra to (3+1)-dimensional spacetime by the inclusion of the vortex-density operator in addition to the usual charge density operator and show that the same algebra is obeyed by massive Dirac fermions that represent the bulk of $\mathbb{Z}^{\,}_{2}$ topological insulators in three-dimensional space.