where ϵ ϵ = 2 E / E 0 and α α = ϕ ϕ ( B ) / ϕ ϕ 0 , ϕ ϕ ( B ) = B a 2 is proportional to the magnetic flux through a lattice cell and ϕ ϕ 0 = 2 π π ℏ ℏ / q is the magnetic flux quantum. The flux ratio α α can also be expressed in terms of the magnetic length l m = ℏ ℏ / e B , such that α α = ( 2 π π ) − − 1 ( a / l m ) 2 .