Nambu Geometry in Quantum Hall Effect and Topological Insulator

Abstract

Abstract. This short review is a contribution to the conference proceeding of IF-YITP Symposium VI, 2016. We discuss how Nambu geometry emerges in the context of higher dimensional quantum Hall eect or A-class topological insulators [1].

Ground state wave functions for the quantum Hall effect on a sphere and the Atiyah–Singer index theorem

“…. , N. Ignoring interactions 5 At the upper limit of these bounds the magnitude of χ ± is nite but the phase is unde ned, this is a gauge artifact. A well de ned phase is obtained at the south pole by performing the gauge transformation χ ± → e i(m±1)φ χ ± (the ±1 arises from the spin connection).…”

“…In this paper we investigate the quantum Hall effect (QHE) on a sphere from the point of view of the Atiyah-singer index theorem and show how the zero modes relate to Haldane's version of the Laughlin ground state wave function. While the role of topology has long been appreciated in the quantum Hall effect to our knowledge the Atiyah-Singer index theorem has not been exploited to any great extent, except for the case of relativistic 4-component fermions in graphene, [3,4], and in non-commutative geometry in the higher dimensional QHE [5]. The index theorem in our present context of ordinary 2-component non-relativistic electrons was used in [6], though there it was in the context of non-Abelian gauge elds on a torus and the zero modes were not constructed explicitly.…”