Response functions and Spectrum of Collective Excitations of Fractional Quantum Hall Effect Systems

TL;DR

This paper calculates electromagnetic response functions and collective excitation spectra of fractional quantum Hall systems using fermion Chern-Simons theory, confirming key symmetries and sum rules, and exploring quasiparticle modes and superfluid properties.

Contribution

It provides a semiclassical framework for response functions and collective modes in FQHE, including the effects of fluctuations and application to screening and anyon superfluidity.

Findings

Response functions obey Galilean and gauge invariance.

Spectrum includes quasiparticle-quasihole bound states.

Two collective modes coalesce at the cyclotron frequency.

Abstract

We calculate the electromagnetic response functions of a Fractional Quantum Hall system within the framework of the fermion Chern-Simons theory for the Fractional Hall Effect (FQHE) which we developed before. Our results are valid in a semiclassical expansion around the average field approximation (AFA). We reexamine the AFA and the role of fluctuations. We argue that, order-\-by-\-order in the semiclassical expansion, the response functions obey the correct symmetry properties required by Galilean and Gauge Invariance and by the incompressibility of the fluid. In particular, we find that the low-momentum limit of the semiclassical approximation to the response functions is exact and that it saturates the $f$-sum rule.We obtain the spectrum of collective excitations of FQHE systems in the low-momentum limit. We find a rich spectrum of modes which includes a host of quasiparticle- quasihole bound states and, in general,two collective modes coalescing at the cyclotron frequency. The Hall conductance is obtained from the current-density correlation function, and it has the correct value already at the semiclassical level. We applied these results to the problem of the screening of external charges and fluxes by the electron fluid, and obtained asymptotic expressions of the charge and current density profiles, for different types of interactions. Finally, we reconsider the anyon superfluid within our scheme and derive the spectrum of collective modes for interacting hard-core bosons and semions. In addition to the gapless phase mode, we find a set of gapped collective modes.