Singular connection approach to topological phases and resonant optical responses

TL;DR

This paper introduces singular connections as a new mathematical tool to analyze topological phases and optical responses in quantum systems, providing a topological invariant related to transition dipoles and optical excitation constraints.

Contribution

It proposes singular connections as an alternative to Berry connections, enabling algebraic computation of topological invariants from transition dipole zeros in quantum band structures.

Findings

Topological invariant computed via singular connections counts zero points of dipole matrix elements.

Invariant provides a lower bound on momenta where electron excitation is forbidden.

Singular connections clarify gauge invariance of optical topological quantities.

Abstract

We introduce a class of singular connections as an alternative to the Berry connection for any family of quantum states defined over a parameter space. We find a natural application of the singular connection in the context of transition dipoles between two bands. We find that the shift vector is nothing but the difference between the singular connection and the connection induced from the Berry connections of involved bands; the gauge invariance of the shift vector is transparent from this expression. We show, using singular connections, that the topological invariant in two dimensions associated with optical transitions between the two bands can be computed, by means of this connection, by algebraically counting the points in the zero locus of a transition dipole matrix element of the two bands involved. It follows that this invariant provides a natural topological lower bound on the number of momenta in the Brillouin zone for which an electron cannot be excited from one Bloch band to the other by absorbing a photon.