Higher-bracket structure of density operators in Weyl fermion systems and topological insulators

Abstract

We study the algebraic structure of electron density operators in gapless Weyl fermion systems in 𝑑=3,5,7,⋯ spatial dimensions and in topological insulators (without any protecting symmetry) in 𝑑=4,6,8,⋯ spatial dimensions. These systems are closely related by the celebrated bulk-boundary correspondence. Specifically, we study the higher bracket—a generalization of commutator for more than two operators—of electron density operators in these systems. For topological insulators, we show that the higher-bracket algebraic structure of density operators structurally parallels with the Girvin-MacDonald-Platzman algebra (the 𝑊1+∞ algebra), the algebra of electron density operators projected onto the lowest Landau level in the quantum Hall effect. By the bulk-boundary correspondence, the bulk higher-bracket structure mirrors its counterparts at the boundary. Specifically, we show that the density operators of Weyl fermion systems, once normal-ordered with respect to the ground state, their higher bracket acquires a 𝑐-number part. This part is an analog of the Schwinger term in the commutator of the fermion current operators. We further identify this part with a cyclic cocycle, which is a topological invariant and an element of Connes' noncommutative geometry.