Itinerant Spin Dynamics in Structures of ... - Jacobs University

96 Chapter 5: Spin Hall Effect
Using the recursion relations of the polynomials,
T 0 (x) = 1, T −1 (x) = T 1 (x) = x,
T m+1 (x) = 2xT m (x)−T m−1 (x), (5.34)
and correspondingly for the polynomials of the second kind
U 0 (x) = 1, U −1 (x) = 0,
U m+1 (x) = 2xU m (x)−U m−1 (x), (5.35)
one can calculate the expansion coefficients µ n iteratively. Replacing the variable x by the
Hamiltonian one can calculate various spectral quantities. The simplest example is the
calculation of the spectral density,
with the coefficients given by
ρ(E) = 1 D
µ n =
∫ 1
−1
D−1
∑
k=0
δ(E −E k ), (5.36)
dxρ(x)T n [x] (5.37)
= 1 D Tr[T n( ˜H)], (5.38)
where ˜H is the rescaled Hamiltonian with all D eigenvalues inside the interval [−1,1]. The
efficiency of the procedure is not yet evident. This changes if one realizes the following
aspects:
• Self averaging properties allow for replacing the trace over the operator by a relatively
small number R ≪ D of random vectors
|r〉 =
D−1
∑
i=0
ζ ri |i〉, (5.39)
where the amplitudes ζ ri = e iφ are random phases on site i. This makes the effort for
the calculation of M coefficients µ n linear in D.
• The most time consuming operation in this procedure is the matrix-vector multiplication
(AppendixF). Due to the fact that the number of neighbors which a site has
in the presented systems, the full multiplication can be replaced by sparse-matrix