Weyl Semimetal and Nonassociative Nambu Geometry

Topological materials are characterized by an electronic band structure with nontrivial topological properties. In this paper, we introduce a basis of operators for the linear space of operators spanned by charge-neutral fermion bilinears. These band projected density operators are constructed using directly the eigenfunctions of the electronic energy band structure and there is no need to assume a flat Berry curvature. As a result, our set of operators has a wider range of validity and is sensitive to physical phenomena which are not detectable in the flat curvature limit. In particular, we show that the Berry monopole configuration of Weyl semimetal give rises to a nonvanishing Jacobiator for these band projected density operators, implying the emergence of nonassociativity at the location of the Weyl nodes. The resulting nonassociativity observes the fundamental identity, the defining property of the Nambu bracket and so one may call this a nonassociative Nambu geometry. We also derive the corresponding uncertainty principle.