Quantum Hall Effect and Quillen Metric

Abstract

Abstract. We study the generating functional, the adiabatic curvature and the adiabatic phase for the integer quantum Hall effect (QHE) on a compact Riemann surface. For the generating functional we derive its asymptotic expansion for the large flux of the magnetic field, i.e., for the large degree k of the positive Hermitian line bundle L k . The expansion consists of the anomalous and exact terms. The anomalous terms are the leading terms of the expansion. This part is responsible for the quantization of the adiabatic transport coefficients in QHE. We then identify the non-local (anomalous) part of the expansion with the Quillen metric on the determinant line bundle, and the subleading exact part with the asymptotics of the regularized spectral determinant of the Laplacian for the line bundle L k , at large k. Finally, we show how the generating functional of the integer QHE is related to the gauge and gravitational (2+1)d Chern-Simons functionals. We observe the relation between the Bismut-Gillet-Soulé curvature formula for the Quillen metric and the adiabatic curvature for the electromagnetic and geometric adiabatic transport of the integer Quantum Hall state. Then we relate the adiabatic phase in QHE to the eta invariant and show that the geometric part of the adiabatic phase is given by the Chern-Simons functional.

2D gravitational Mabuchi action on Riemann surfaces with boundaries

“…The heat kernel for the Laplace operator then simply is the product of the heat kernel for the circle and the heat kernel of the interval as just given in the previous example, with the obvious replacements π → a, b: 18) with the corresponding torus obviously having periods 2a and 2b. Poisson resummation or equivalently the modular transformation formula for θ 3 yields…”

Fractional quantum numbers via complex orbifolds

“…The first is explicit and universal, that is, it is, up to an integral factor, the canonical symplectic form on the space of Aharonov-Bohm fluxes and is quantized, therefore providing a connection to the Integer Quantum Hall effect (IQHE) [35,5,7]. For a deeper analysis of the IQHE, see [19], where the generating functional, the adiabatic curvature and the adiabatic phase for the IQHE is studied on a compact Riemann surface. The second piece in the formula is a complete derivative, hence it does not affect charge transport.…”

Role of the spin connection in quantum Hall effect: A perspective from geometric quantization

“…There has recently been a lot of research elucidating the effective action for the quantum Hall effect on manifolds of different geometries and topologies [1]- [6]. This was partly motivated by the fact that, even though from the experimental point of view, we may only be interested in spaces of trivial topology, nontrivial geometry and topology can shed light on various physical quantities such as transport coefficients.…”

“…The term involving just the electromagnetic field and the mixed term involving both electromagnetic and gravitational fields have been known for a long time [2,3]. The addition of the purely gravitational part and the generalization to include higher Landau levels revealed an interesting curiosity [4,5,6]. Apart from the gravitational framing anomaly, the electromagnetic field and the spin connection of the manifold combine in a particular way [4].…”