Topological response of gapped fermions to a $\text{U}(1)$ Gauge field

Gauge field. The well-known Hall response is obtained by identifying the Chern-Simons term in the effective action with the correct coefficient. We extend the result to

2d+1

dimensions in which the response is associated to a Chern-Simons term with a coefficient related to a characteristic class coming from topological band theory. We comment on the bulk-to-boundary principle which arises naturally when one considers the theory on a manifold with boundary.

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The condition of having n large enough is necessary since otherwise there can be maps which are not homotopic but lead to isomorphic vector bundles. One can show that for n large enough the set of isomorphism classes of complex vector bundles over a compact manifold M, Vectr(M), is isomorphic to the set of homotopy classes of maps from the manifold to the Grassmannian of k planes in Cn, i.e. Vectr(M) ∼= [M, Grr(Cn )], see for e.g.16. This leads to the good notion of topological equivalence of Hamiltonians - gapped Hamiltonians are considered to be topologically equivalent if, by adding enough flat bands (hence the large n), they can be continuously deformed into each other17. This is related to the notion of stable equivalence in K-theory

[10]