Abstract:
Chern-Simons type gauge field is generated by the means of the singular area preserving transformations in the lowest Landau level of electrons forming fractional quantum Hall state. Dynamics is governed by the system of constraints which correspond to the Gauss law in the non-commutative Chern-Simons gauge theory and to the lowest Landau level condition in the picture of composite fermions. Physically reasonable solution to this constraints corresponds to the Laughlin state. It is argued that the model leads to the non-commutative Chern-Simons theory of the QHE and composite fermions.

Abstract:
Using the recently developed approach to quantum Hall physics based on Newton-Cartan geometry, we consider the hydrodynamics of an interacting system on the lowest Landau level. We rephrase the non-relativistic fluid equations of motion in a manner that manifests the spacetime diffeomorphism invariance of the underlying theory. In the massless (or lowest Landau level) limit, the fluid obeys a force-free constraint which fixes the charge current. An entropy current analysis further constrains the energy response, determining four transverse response functions in terms of only two: an energy magnetization and a thermal Hall conductivity. Kubo formulas are presented for all transport coefficients and constraints from Weyl invariance derived. We also present a number of Streda-type formulas for the equilibrium response to external electric, magnetic and gravitational fields.

Abstract:
This paper deals with the comparison between the strong Thomas-Fermi theory and the quantum mechanical ground state energy of a large atom confined to lowest Landau band wave functions. Using the tools of microlocal semiclassical spectral asymptotics we derive precise error estimates. The approach presented in this paper suggests the definition of a modified strong Thomas-Fermi functional, where the main modification consists in replacing the integration over the variables perpendicular to the magnetic field by an expansion in angular momentum eigenfunctions. The resulting DSTF theory is studied in detail in the second part of the paper.

Abstract:
The stripe state in the lowest Landau level is studied by the density matrix renormalization group (DMRG) method. The ground state energy and pair correlation functions are systematically calculated for various pseudopotentials in the lowest Landau level. We show that the stripe state in the lowest Landau level is realized only in a system whose width perpendicular to the two-dimensional electron layer is smaller than the order of magnetic length.

Abstract:
The spin polarization versus temperature at or near a fully filled lowest Landau level is explored for finite-size systems in a periodic rectangular geometry. Our results at $\nu=1$ which also include the finite-thickness correction are in good agreement with the experimental results. We also find that the interacting electron system results are in complete agreement with the results of the sigma model, i.e., skyrmions on a torus have a topological charge of $Q \ge 2$ and the Q=1 solution is like a single spin-flip excitation. Our results therefore provide direct evidence for the skyrmionic nature of the excitations at this filling factor.

Abstract:
We describe the lowest Landau level of a quantum electron star in AdS4. In the presence of a suitably strong magnetic field, the dynamics of fermions in the bulk is effectively reduced from four to two dimensions. These two-dimensional fermions can subsequently be treated using the techniques of bosonization and the difficult many-body problem of building a gravitating, charged quantum star is reduced to solving the sine-Gordon model coupled to a gauge field and a metric. The kinks of the sine-Gordon model provide the holographic dual of the lowest Landau levels of the strongly-coupled d=2+1 dimensional boundary field theory. The system exhibits order one oscillations in the magnetic susceptibility, now arising as a classical effect in the bulk. Moreover, as the chemical potential is varied, we find jumps in the charge density, oscillations in the fractionalised charge density and plateaux in the cohesive charge density

Abstract:
We study the vortex distribution of the wave functions minimizing the Gross Pitaevskii energy for a fast rotating condensate in the Lowest Landau Level (LLL): we prove that the minimizer cannot have a finite number of zeroes thus the lattice is infinite, but not uniform. This uses the explicit expression of the projector onto the LLL. We also show that any slow varying envelope function can be approximated in the LLL by distorting the lattice. This is used in particular to approximate the inverted parabola and understand the role of ``invisible'' vortices: the distortion of the lattice is very small in the Thomas Fermi region but quite large outside, where the "invisible" vortices lie.

Abstract:
We study a rapidly rotating gas of unpolarized spin-1/2 ultracold fermions in the two-dimensional regime when all atoms reside in the lowest Landau level. Due to the presence of the spin degree of freedom both s-wave and p-wave interactions are allowed at ultralow temperatures. We investigate the phase diagram of this system as a function of the filling factor in the lowest Landau level and in terms of the ratio between s- and p-wave interaction strengths. We show that the presence of attractive interactions induces a wide regime of phase separation with formation of maximally compact droplets that are either fully polarized or composed of spin-singlets. In the regime with no phase separation, we give evidence for fractional quantum Hall states. Most notably, we find two distinct singlet states at the filling nu =2/3 for different interactions. One of these states is accounted for by the composite fermion theory while the other one is a paired state for which we identify two competing descriptions with different topological structure. This paired state may be an Abelian liquid of composite spin-singlet Bose molecules with Laughlin correlations. Alternatively, it may be a known non-Abelian paired state, indicated by good overlaps with the corresponding trial wavefunction. By fine tuning of the scattering lengths it is possible to create the non-Abelian critical Haldane-Rezayi state for nu =1/2 and the permanent state of Moore and Read for nu =1. For purely repulsive interactions, we also find evidence for a gapped Halperin state at nu=2/5.

Abstract:
We study the spectral statistics in the center of the lowest Landau band of a 2D disordered system with smooth potential and strong transverse magnetic field. Due to the finite size of the system, the energy range in which there are extended states is finite as well. The behavior in this range can be viewed as the analogue of the Anderson metal-insulator transition for the case of the Hall system. Accordingly, we verify recent predictions regarding the exponent of the asymptotic power law of $\Sigma^2 (\bar N)$, $\gamma$, and that of the stretched exponential dominating the large $s$ behavior of the spacings distribution, $\alpha$. Both the relations, $\alpha = 1- \gamma$, and $\gamma = 1 - {1\over{\nu d}}$ where $\nu$ is the critical exponent of the localization length and $d$ is the dimension, are found to hold within the accuracy of our computations. However, we find that none of several possible models of the entire spacings distribution correctly describes our situation. Finally, for very large $\bar N$, $\bar N > 60$, we find a new regime in which $\Sigma^2 (\bar N)$ behaves as a power law with an unexpectedly large power, $\gamma_1 = 1.38 \pm 0.02$.