Suppose we have the bulk of a topological insulator, in $2+1$ dimensions, described by a quadratic Hamiltonian in the fermion field operators, namely $$H=\sum_{i,j}\psi_i^{*} h_{ij}\psi_j$$ (the Hamiltonian is regularized on a lattice, here $i,j$ denote lattice and other possible degrees of freedom, $$\{\psi_i,\psi_j^{*}\}=\delta_{ij}$$ 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, $$\int [D\psi][D\psi^{*}]\exp(iS(\psi,\psi^*,A))=\exp(iS_{\text{eff}}(A)),$$ 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.
Best Answer
I think I have figured it out.
We are looking for the topological response, hence it is enough to look at the response of any system described by a Hamiltonian adiabatically connected to the one we started with. In particular, we can use one with a flat spectrum: $H=\int\frac{d^2p}{(2\pi)^2}\psi^{*}(p) H(p)\psi(p)$ in which the relation $H(p)=U(p)\text{diag}(-I_{k},I_{n-k})U^{*}(p)$ holds locally. Here $k$ is the number of occupied bands and $U(k)=[v_1(p),...,v_{n}(p)]$ is an orthogonal matrix of eigenvectors of $H(p)$. In particular, the $k\times n$ matrix $S(p)=[v_1(p),...,v_{k}(p)]$ describes, locally, the occupied bundle. We can also write $H(p)=P^{\perp}(p)-P(p)$, where $P(p)=S(p)S^{*}(p)$ is the projector onto the fiber of the occupied bundle over $p$. The curvature of the bundle can be described as an endomorphism of the trivial bundle $\mathbb{T}^2\times \mathbb{C}^n$ by the expression $\Omega=PdP\wedge dPP=dP\wedge P^{\perp}dP$. The later expression will be useful in computing the effective action. The idea is that we start with $Z_0=\text{Det}(G_0^{-1})$ and then perform the minimal coupling to an external gauge field. In a phase space representation ($(p,x)$-representation), and assuming that the external gauge field varies in scales larger than the system's typical scale, we can write the inverse of the new Green's function as $G^{-1}(p,x)\approx G_0^{-1}(p) -e A_{\mu}(x)\partial G_0^{-1}/\partial p_ \mu(p)$. Writing $\Sigma(p,x)=-eA_{\mu}(x)\partial G_0^{-1}/\partial p_ \mu(p)$ we have $\text{Det}(G^{-1})\approx Z_0\text{Det}(I+G_0\Sigma)$. The effective action is then obtained by the formal expansion $\log(\text{Det}(I+G_0\Sigma))=\text{Tr}\log(I+G_0\Sigma)\approx \text{Tr}(G_0\Sigma)-\frac{1}{2}\text{Tr}(G_0\Sigma G_0\Sigma)$. The partition function should be Gauge invariant, hence, we will only look for such terms. In $2+1$ dimensions, besides the Maxwell term, we are allowed to have the Chern-Simons term $S_{CS}(A)=(1/4\pi)\int A\wedge dA$. We are looking for the quadratic terms (hence, in the second term of the expansion written before) which have $A_\mu(x)$ and $\partial_\mu A_\nu(x)$. The products under the functional trace are in fact convolutions and, when we perform the transformation to the mixed position-momentum representation, we get a twisted product expansion, namely the Moyal product: $\int d^3x_2 A(x_1,x_2)B(x_2,x_3)\rightarrow A(p,x)B(p,x)+(i/2)\{A,B\}_{\text{PB}}(p,x)+...$, where $\{.,.\}_{\text{PB}}$ is the Poisson Bracket. The first two terms of the expansion are enough for us. If we look for the mentioned terms we end up with the contribution $(ie^{2}/2)(1/3!)(\int d^3p/(2\pi)^3 \text{tr} (G_0\partial G_0^{-1}/\partial p_\mu G_0\partial G_0^{-1}/\partial p_\nu G_0\partial G_0^{-1}/\partial p_\lambda) \varepsilon_{\mu\nu\lambda}) \int A\wedge dA$. By performing the integral over the frequency $p_0$, the later can be shown to be equal to $2\pi i\sigma_{H} S_{CS}(A)$, with $\sigma_{H}=(e^{2}/2\pi)\times \int_{\mathbb{T}^2}\text{tr}(i\Omega/2\pi)\equiv (e^{2}/2\pi)\times c_1$ ($c_1$ denotes the first Chern number of the occupied bundle), as expected.