Fractional quantum numbers via complex orbifolds

Abstract

This paper studies both the conductance and charge transport on 2D orbifolds in a strong magnetic field. We consider a family of Landau Hamiltonians on a complex, compact 2D orbifold Y that are parametrised by the Jacobian torus J(Y ) of Y . We calculate the degree of the associated stable holomorphic spectral orbibundles when the magnetic field B is large, and obtain fractional quantum numbers as the conductance and a refined analysis also gives the charge transport.A key tool studied here is a nontrivial generalisation of the Nahm transform to 2D orbifolds.

Fractional quantum numbers, complex orbifolds and noncommutative geometry

“…In [29] the transport coefficients associated to the eigensections of low lying eigenvalues are considered for any compact Riemann surface where the magnetic field B is a large integer. In earlier work [25], the transport coefficients associated to the eigensections of low lying eigenvalues are considered for any compact complex 2D orbifold where the magnetic field B is a large fraction.…”

“…The goal of this paper is to remove the topological constraints on the magnetic field (either integrality or rationality) in earlier works [3,29,25] in the literature. To achieve this, we instead study the conductance and charge transport on the universal homology orbi-covering space Z of 2D orbifolds in a strong magnetic field.…”

“…These results were extended by Prieto [28], who computed the transport coefficients associated to the eigensections of low lying eigenvalues of the Schrödinger operator. In our previous work [25], we calculated the transport coefficients associated to the eigensections of low lying eigenvalues for any compact complex 2D orbifold where the magnetic field B is a large fraction.…”

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This paper studies the conductance on the universal homology covering space Z of 2D orbifolds in a strong magnetic field, thereby removing the integrality constraint on the magnetic field in earlier works [3,29,25] in the literature. We consider a natural Landau Hamiltonian on Z and study its spectrum which we prove consists of a finite number of lowlying isolated points and calculate the von Neumann degree of the associated holomorphic spectral orbibundles when the magnetic field B is large, and obtain fractional quantum numbers as the conductance.

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