Singular connection approach to topological phases and resonant optical responses

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.

16 pages (including Appendix), 3 figures; closer to published version

References in corpus (5)

• Orbital magnetization in periodic insulators
• Riemannian geometry of resonant optical responses
• Experimental Measurement of the Quantum Metric Tensor and Related Topological Phase Transition with a Superconducting Qubit
• The Role of Shift Vector in High-Harmonic Generation from Non-Centrosymmetric Topological Insulators under Strong Laser Fields
• Non-Abelian tensor Berry connections in multi-band topological systems