Toward a new theory of the fractional quantum Hall effect

TL;DR

This paper develops an exact diagonalization approach to study small systems of electrons in strong magnetic fields, challenging the traditional Laughlin state description of the fractional quantum Hall effect and revealing a more complex ground state structure.

Contribution

It introduces an exact diagonalization method for small electron systems and provides new insights into the nature of their ground and excited states beyond the Laughlin model.

Findings

Ground and excited states resemble a sliding Wigner crystal.

Energy gaps vary with magnetic field, appearing and disappearing.

Laughlin state and its excitations do not fully describe the system.

Abstract

The fractional quantum Hall effect was experimentally discovered in 1982. It was observed that the Hall conductivity $\sigma_{yx}$ of a two-dimensional electron system is quantized, $\sigma_{yx}=e^2/3h$, in the vicinity of the Landau level filling factor $\nu=1/3$. In 1983, Laughlin proposed a trial many-body wave function, which he claimed described a ``new state of matter'' -- a homogeneous incompressible liquid with fractionally charged quasiparticles. Here I develop an exact diagonalization theory that allows calculation of the energy and other physical properties of the ground and excited states of a system of $N$ two-dimensional Coulomb interacting electrons in a strong magnetic field. I analyze the energies, electron densities, and other physical properties of the systems with $N\le 7$ electrons, continuously as a function of magnetic field in the range $1/4\lesssim\nu<1$. The results show that both the ground and excited states of the system resemble a sliding Wigner crystal, whose parameters are influenced by the magnetic field. Energy gaps in the many-particle spectra appear and disappear as the magnetic field changes. I also calculate the physical properties of the $\nu=1/3$ Laughlin state for $N\le 8$ and show that neither this state nor its fractionally charged excitations describe the physical reality. The results obtained shed new light on the nature of the ground and excited states in the fractional quantum Hall effect.