It follows from Eq. (4.31) that the

It follows from Eq. (4.31) that the incompressible bands of constant ν = νp do not contribute to j. The reservoir injects the current into the compressible bands at one edge of the conductor only (for which the sign of ∂ν/∂x is such that j moves away from the reservoir). The edge channel with index p = 1, 2, . . . , P is defined as that compressible band that is flanked by incompressible bands at filling factors νp and νp−1. The outermost band from the center of the conductor, which is the p = 1 edge channel, is included by defining formally ν0 ≡ 0. The arrangement of alternating edge channels and compressible bands is illustrated in Fig. 95a. Note that different edges may have a different series of edge channels at the same magnetic field value, depending on the smoothness of the potential V at the edge (which, as discussed before, determines the incompressible bands that exist at the edge). This is in contrast to the situation in the integer QHE, where a one-to-one correspondence exists between edge channels and bulk Landau levels (Section IV.A.2). In the fractional QHE an infinite hierarchy of energy gaps exists, in principle, corresponding to an infinite number of possible edge channels, of which only a small number (corresponding to the largest energy gaps) will be realized in practice. The current Ip = (e/h)_μ(νp − νp−1) injected into edge channel p by the reservoir follows directly from Eq. (4.31) on integration over x. The total current I through the wire is I = PP p=1 IpTp, if a fraction Tp of the injected current Ip is transmitted to the reservoir at the other end of the wire (the remainder returning via the opposite edge). For the conductance G ≡ eI/_μ, one thus obtains the generalized Landuer formula for a twoterminal conductor,482