A Farewell to Liouvillians

We examine the Liouvillian approach to the quantum Hall plateau transition, as introduced recently by Sinova, Meden, and Girvin [Phys. Rev. B {\bf 62}, 2008 (2000)] and developed by Moore, Sinova and Zee [Phys. Rev. Lett. {\bf 87}, 046801 (2001)]. We show that, despite appearances to the contrary, the Liouvillian approach is not specific to the quantum mechanics of particles moving in a single Landau level: we formulate it for a general disordered single-particle Hamiltonian. We next examine the relationship between Liouvillian perturbation theory and conventional calculations of disorder-averaged products of Green functions and show that each term in Liouvillian perturbation theory corresponds to a specific contribution to the two-particle Green function. As a consequence, any Liouvillian approximation scheme may be re-expressed in the language of Green functions. We illustrate these ideas by applying Liouvillian methods, including their extension to $N_L > 1$ Liouvillian flavors, to random matrix ensembles, using numerical calculations for small integer $N_L$ and an analytic analysis for large $N_L$. We find that behavior at $N_L > 1$ is different in qualitative ways from that at $N_L=1$. In particular, the $N_L = \infty$ limit expressed using Green functions generates a pathological approximation, in which two-particle correlation functions fail to factorize correctly at large separations of their energy, and exhibit spurious singularities inside the band of random matrix energy levels. We also consider the large $N_L$ treatment of the quantum Hall plateau transition, showing that the same undesirable features are present there, too

N-orbital model for a two-dimensional disordered system in a strong magnetic field

An N-orbital model is considered for a two-dimensional disordered system in a strong perpendicular magnetic field. The Hamiltonian is projected on the lowest Landau level. In a field theoretical formulation the large N limit result corresponds to the translationally invariant saddle point for this model. A diagrammatic expansion in powers of 1/N is performed around the large N limit. The conductivity is calculated up to order 1/N2. No mobility edge is found in this expansion. The result is consistent with the non-linear unitary σ-model and independent of magnetic field and disorder strength if the energy is measured in units of the bandwidth.Dans un champ magnétique intense, orthogonal au plan d'un système désordonné bi-dimensionnel, un modèle à N orbitales par site est considéré. L'Hamiltonien est projeté dans le niveau de Landau fondamental. Dans la limite de N grand, le résultat correspond à un col, invariant par translation, d'une théorie des champs. Le développement en puissances de 1/N est effectué diagrammatiquement et la conductivité est calculée jusqu'à l'ordre 1/N2. Dans ce développement on n'observe aucun seuil de mobilité. Le résultat est conforme au modèle σ non linéaire unitaire; il est indépendant du champ magnétique et de l'intensité du désordre si l'on exprime l'énergie en unités de la largeur de bande

N-orbital model for a two-dimensional disordered system in a strong magnetic field

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