Semiclassical analysis of edge state energies in the integer quantum Hall effect

Abstract

Analysis of edge-state energies in the integer quantum Hall effect is carried out within the semiclassical approximation. When the system is wide so that each edge can be considered separately, this problem is equivalent to that of a one dimensional harmonic oscillator centered at x = x_c and an infinite wall at x = 0, and appears in numerous physical contexts. The eigenvalues E_n(x_c) for a given quantum number n are solutions of the equation S(E,x_c)=π[n+ γ(E,x_c)] where S is the WKB action and 0 < γ < 1 encodes all the information on the connection procedure at the turning points. A careful implication of the WKB connection formulae results in an excellent approximation to the exact energy eigenvalues. The dependence of γ[E_n(x_c),x_c] ≡γ_n(x_c) on x_c is analyzed between its two extreme values $\frac{1}{2}$ as x_c ↦-∞ far inside the sample and $\frac{3}{4}$ as x_c ↦∞ far outside the sample. The edge-state energiesE_n(x_c) obey an almost exact scaling law of the form $E_{n}(x_{c})=4[n+\gamma_{n}(x_{c})]f (\frac{x_{c}}{\sqrt{4n+3}})$ and the scaling function f(y) is explicitly elucidated.