(3+1)-dimensional topological quantum field theory from a tight-binding model of interacting spinless fermions

TL;DR

This paper constructs a (3+1)-dimensional lattice model of interacting spinless fermions that reproduces a topological quantum field theory in the low energy limit, linking lattice models to topological field theories.

Contribution

It introduces a 3D tight-binding model that realizes a BF topological quantum field theory from interacting fermions, connecting lattice models with topological field theories.

Findings

Reproduces (3+1)D BF theory from a lattice model.

Establishes a 3D chiral topological insulator via a Haldane-like mechanism.

Detects topological order through fermionic density measurements.

Abstract

Currently, there is much interest in discovering analytically tractable (3+1)-dimensional models that describe interacting fermions with emerging topological properties. Towards that end we present a three-dimensional tight-binding model of spinless interacting fermions that reproduces, in the low energy limit, a (3+1)-dimensional Abelian topological quantum field theory called BF model. By employing a mechanism equivalent to the Haldane's Chern insulator, we can turn the non-interacting model into a three-dimensional chiral topological insulator. We then isolate energetically one of the two Fermi points of the lattice model. In the presence of suitable fermionic interactions, the system, in the continuum limit, is equivalent to a generalised (3+1)-dimensional Thirring model. The low energy limit of this model is faithfully described by the BF theory. Our approach directly establishes the presence of (2+1)-dimensional BF theory at the boundary of the lattice and it provides a way to detect the topological order of the model through fermionic density measurements.