Theory of four-dimensional fractional quantum Hall states

We propose a pseudo-potential Hamiltonian for the Zhang-Hu’s generalized fractional quantum Hall states to be the exact and unique ground states. Analogously to Laughlin’s quasi-hole (quasi-particle), the excitations in the generalized fractional quantum Hall states are extended objects. They are vortex-like excitations with fractional charges +(−)1/m³ in the total configuration space CP³. The density correlation function of the Zhang-Hu states indicates that they are incompressible liquid.     

Anomalous hydrodynamics of fractional quantum Hall states

We propose a comprehensive framework for quantum hydrodynamics of the fractional quantum Hall (FQH) states. We suggest that the electronic fluid in the FQH regime can be phenomenologically described by the quantized hydrodynamics of vortices in an incompressible rotating liquid. We demonstrate that such hydrodynamics captures all major features of FQH states, including the subtle effect of the Lorentz shear stress. We present a consistent quantization of the hydrodynamics of an incompressible fluid, providing a powerful framework to study the FQH effect and superfluids. We obtain the quantum hydrodynamics of the vortex flow by quantizing the Kirchhoff equations for vortex dynamics.     

Model fractional quantum Hall states and Jack polynomials

We describe an occupation-number-like picture of fractional quantum Hall states in terms of polynomial wave functions characterized by a dominant occupation-number configuration. The bosonic variants of single-component Abelian and non-Abelian fractional quantum Hall states are modeled by Jack symmetric polynomials (Jacks), characterized by dominant occupation-number configurations satisfying a generalized Pauli principle. In a series of well-known quantum Hall states, including the Laughlin, Read-Moore, and Read-Rezayi, the Jack polynomials naturally implement a "squeezing rule" that constrains allowed configurations to be restricted to those obtained by squeezing the dominant configuration. The Jacks presented in this Letter describe new trial uniform states, but it is yet to be determined to which actual experimental fractional quantum Hall effect states they apply.     

Nonlinear quantum shock waves in fractional quantum Hall edge states

Using the Calogero model as an example, we show that the transport in interacting nondissipative electronic systems is essentially nonlinear and unstable. Nonlinear effects are due to the curvature of the electronic spectrum near the Fermi energy. As is typical for nonlinear systems, a propagating semiclassical wave packet develops a shock wave at a finite time. A wave packet collapses into oscillatory features which further evolve into regularly structured localized pulses carrying a fractionally quantized charge. The Calogero model can be used to describe fractional quantum Hall edge states. We discuss perspectives of observation of quantum shock waves and a direct measurement of the fractional charge in fractional quantum Hall edge states.     

Disorder and the fractional quantum Hall effect

The fractional quantum Hall effect is studied in the 2D electron gas of four GaAs-Al_xGa_1?xAs heterostructures. Localization due to disorder, known to give rise to the wide integral quantum Hall plateaus, is demonstrated to inhibit the fractional effect, which is observed only in the higher mobility samples.     

Edge states tunneling in the fractional quantum Hall effect: integrability and transport

This is a short review of nonperturbative techniques that have been used in the past 5 years to study transport out of equilibrium in low dimensional, strongly interacting systems of condensed matter physics. These techniques include massless factorized scattering, the generalization of the Landauer Büttiker approach to integrable quaisparticles, and duality. The case of tunneling between edges in the fractional quantum Hall effect is discussed in details. To cite this article: H. Saleur, C. R. Physique 3 (2002) 685–695.     

Topology and fractional quantum Hall effect

Starting from Laughlin-type wave functions with generalized periodic boundary conditions describing the degenerate ground state of a quantum Hall system we explicitly construct r-dimensional vector bundles. It turns out that the filling factor ν is given by the topological quantity c₁/r where c₁ is the first Chem number of these vector bundles. In addition, we managed to proof that under physical natural assumptions the stable vector bundles correspond to the experimentally dominating series of measured fractional filling factors ν = n/(2pn±1). Most remarkably, due to the very special form of the Laughlin wave functions the fluctuations of the curvature of these vector bundles converge to zero in the limit of infinitely many particles which shows a new mathematical property. Physically, this means that in this limit the Hall conductivity is independent of the boundary conditions which is very important for the observability of the effect. Finally, we discuss the relation of this result to a theorem of Donaldson.