Loop unitary and phase band topological invariant in generic multi-band Chern insulators

TL;DR

This paper generalizes a dynamical topological invariant from two-band to multi-band Chern insulators, linking it to Chern number differences and revealing new multifold fermions in phase bands.

Contribution

It extends the dynamical 3-winding-number to multi-band systems and provides a generic expression based on phase bands and projectors.

Findings

The dynamical 3-winding-number equals the difference of Chern numbers pre- and post-quench.

Derived a phase band expression depending only on phase and projectors.

Discovered a multifold fermion in (k, t) space for a three-band quench.

Abstract

Quench dynamics of topological phases have been studied in the past few years and dynamical topological invariants are formulated in different ways. Yet most of these invariants are limited to minimal systems in which Hamiltonians are expanded by Gamma matrices. Here we generalize the dynamical 3-winding-number in two-band systems into the one in generic multi-band Chern insulators and prove that its value is equal to the difference of Chern numbers between post-quench and pre-quench Hamiltonians. Moreover we obtain an expression of this dynamical 3-winding-number represented by gapless fermions in phase bands depending only on the phase and its projectors, so it is generic for the quench of all multi-band Chern insulators. Besides, we obtain a multifold fermion in the phase band in (k, t) space by quenching a three-band model, which cannot happen for two band models.