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Quaternion Generalization of the Laughlin state and the Three dimensional fractional QHE

By A. V. Balatsky


ABSTRACT: The 3D state of strongly correlated electrons is proposed, which in the external magnetic field B exhibits the fractional quantum Hall effect, with the zero temperature conductivity tensor σij = (e 2 /h)(1/m) ∑ k ǫ ijkB k / | B |. The analog of Landau and Laughlin states in 3D are given using quaternion coordinates as generalization of complex coordinates. We discuss the notion of the fractional statistics in 3D introduced recently by Haldane. PACS No. 05.-30.- d; 72.20.- i.The concept of “anyons ” or “fractional statistics ” 1 particles has been intensively investigated recently. Soon after experimental discovery of the Fractional QHE (FQHE) 2 Laughlin proposed variational wave function which describes the incompressible electron liquid in 2D in external magnetic field at fractional filling factors, which naturally leads to the “fractional statistics ” of the quasiparticles 3. The holomorphic structure of the Laughlin state rely essentially on the fact that the coordinate space of electron liquid is 2D. Mathematically 2D space allows the existence of the abelian representations of the braid group instead of permutation group in higher dimensions 4. Physically Wilczek’s construction of anyons as the bound state of particles with fluxes is very natural namely in 2D space. However we were led to a conclusion that some aspects of the 2D FQHE state can be generalized on to strongly correlated states of electrons in higher dimensions, namely the 3D. I will present the variational wave function of the isotropic 3D electron liquid subjected in to strong magnetic field B. This state can be considered as a generalization of the Laughlin state on 3D electron liquid. The conductivity tensor of this state generalizes the FQHE conductivity: σij = e2

Year: 2013
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