Currently, there is much interest in discovering analytically tractable $\left(3+1\right)$-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 $\left(3+1\right)$-dimensional Abelian topological quantum field theory called the BF model. By employing a mechanism equivalent to Haldane's Chern insulator, we can turn the noninteracting 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 generalized $\left(3+1\right)$-dimensional Thirring model. The low-energy limit of this model is faithfully described by the BF theory. Our approach directly establishes the presence of $\left(2+1\right)$-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.

• Received 8 October 2013
• Revised 22 July 2014

DOI:https://doi.org/10.1103/PhysRevB.90.085114