Deconstructing the Liouvillian approach to the quantum Hall plateau transition - Oganesyan, V; Chalker, J; Sondhi, S (2003)
We examine the Liouvillian approach to the quantum Hall plateau transition, as introduced recently by Sinova, Meden, and Girvin [Phys. Rev. B 62, 2008 (2000)] and developed by Moore, Zee, and Sinova [Phys. Rev. Lett. 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 reexpressed in the language of Green functions. We illustrate these ideas by applying Liouvillian methods (including their extension to NL > 1 Liouvillian flavors) to random matrix ensembles, using numerical calculations for small integer NL and an analytical analysis for large NL. We find that the behavior at N L > 1 is different in qualitative ways from that at NL = 1. In particular, the NL = ∞ 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-NL treatment of the quantum Hall plateau transition, showing that the same undesirable features are present there, too. We suggest that failings of this kind are likely to be generic in Liouvillian approximation schemes.