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Berry phase and quantum metric for Bloch electrons

Jean-Noël Fuchs


In this talk, I will try to answer the following question : what is the effective Hamiltonian describing a Bloch electron restricted to a single band, but in in the presence of other bands and of external electromagnetic fields ? I will first review Peierls’ answer involving the Peierls’ substitution and leading to well-known semiclassical equations of motion (cf. Ashcroft-Mermin). Then I will argue that, although this answer leads to many successes, it is too naive as it only involves the band energy spectrum but completely forgets the corresponding eigenvectors. Eventually, I will turn to the modern understanding – which goes back to Thouless, Berry and Haldane – of the existence of a geometric structure hidden within band theory (namely a Berry phase and a quantum metric). This will allow us to derive a modified effective single-band Hamiltonian. The corresponding semiclassical equations of motion (cf. Qian Niu and collaborators) include extra terms such as the Karplus-Luttinger anomalous velocity and the orbital magnetic moment. If time permits, I will end by showing how the quantum Hall effect appears in hindsight as electric transport in a peculiar band insulator in zero external magnetic field.

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