Quantum Hall effect and Quillen metric

TL;DR

This paper explores the mathematical structures underlying the integer quantum Hall effect, linking the generating functional, adiabatic curvature, and phase to advanced geometric concepts like the Quillen metric and Chern-Simons functionals.

Contribution

It establishes a novel connection between the generating functional in QHE and the Quillen metric, spectral determinants, and Chern-Simons theory, providing new insights into the geometric nature of QHE.

Findings

Asymptotic expansion of the generating functional for large magnetic flux

Identification of the anomalous part with the Quillen metric

Relation of the adiabatic phase to the eta invariant and Chern-Simons functional

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-Soul\'e 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.