  The well-known Landauer-Buttiker (LB) picture used to explain the quantum
Hall effect uses the concept of (chiral) edge states that carry the current. In
their seminal 1992 article, Chklovskii, Shklovskii and Glazman (CSG) showed
that the LB picture does not account for some very basic properties of the gas,
such as its density profile, as it lacks a proper treatment of the
electrostatic energy. They showed that, instead, one should consider alternated
stripes of compressible and incompressible phases. In this letter, we revisit
this issue using a full solution of the quantum-electrostatic problem of a
narrow ballistic conductor, beyond the CSG approach. We recover the LB channels
at low field and the CSG compressible/incompressible stripes at high field. Our
calculations reveal the existence of a third "hybrid" phase at intermediate
field. This hybrid phase has well defined LB type edge states, yet possesses a
Landau level pinned at the Fermi energy as in the CSG picture. We calculate the
magneto-conductance which reveals the interplay between the LB and CSG regimes.
Our results have important implications for the propagation of edge
magneto-plasmons.
