Suppose we have the bulk of a topological insulator, in $2+1$ dimensions, described by a quadratic Hamiltonian in the fermion field operators, namely $H=\sum_{i,j}\psi_i^{*} h_{ij}\psi_j$$$H=\sum_{i,j}\psi_i^{*} h_{ij}\psi_j$$ (the Hamiltonian is regularized on a lattice, here $i,j$ denote lattice and other possible degrees of freedom, $\{\psi_i,\psi_j^{*}\}=\delta_{ij}$$$\{\psi_i,\psi_j^{*}\}=\delta_{ij}$$ and we can also assume translation invariance). If we couple this system to an external $\text{U}(1)$ gauge field $A$ and consider the effective theory resulting from integrating over the fermion fields, namely, $\int [D\psi][D\psi^{*}]\exp(iS(\psi,\psi^*,A))=\exp(iS_{\text{eff}}(A))$,$$\int [D\psi][D\psi^{*}]\exp(iS(\psi,\psi^*,A))=\exp(iS_{\text{eff}}(A)),$$ then, in the long wavelength limit, one expects a Chern-Simons term, $\int A\wedge dA$, to appear in the effective action, with a coefficient proportional to the first Chern number of the occupied Bloch bundle (namely the bundle of eigenspaces of the Hamiltonian written in momentum space). This yields the Quantum Hall Effect in two dimensions as the response gives a quantized Hall conductivity.
I see how this Chern-Simons term appears if one couples a massive Dirac fermion in 2+1 dimensions to an external $\text{U}(1)$ gauge field and then considers the quadratic term in the expansion of the functional determinant of the resulting Dirac operator. But in the previous case, I don't know exactly how to approach it and I was wondering if it is possible through path integral methods.
