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

Appendix F
KPM
The recursion relations of the polynomials of first and second kind, Eq.(5.34)
and Eq.(5.35), allow for a simple iteration procedure: The core of KPM is the iterative
construction of the states |α n 〉 = T n ( ˜H)|α〉, described by the following steps[WWAF06],
|α 0 〉 = |α〉, (F.1)
|α 1 〉 = ˜H|α 0 〉, (F.2)
|α n+1 〉 = 2 ˜H |α n 〉−|α n−1 〉, (F.3)
where |α〉 is the starting vector. For instance, to calculate the local DOS at site i, our
starting vector would be the site-occupation vector |i〉. Then the coefficients for the T n (E)
polynomial are given by µ n = 〈i|i n 〉 and ρ i (E) is reconstructed by using Eq.(5.32). Consequently,
the most time consuming part is the matrix-vector multiplication H|α n 〉. Because
H is sparse in our case, a very efficient multiplication is done by decomposing the operator
• in a matrix A which contains the information about connected sites, which comprises
their site number and the type of hopping between them
• and a matrix SOH which contains all hopping matrices which are two-dimensional
due to SOC.
The type of hopping is coded in a number h ij , e.g. h ij = 0 can be defined as “hopping from
i to j ⇐⇒ hopping in the positive x-direction”. To give an example: The site i is connected
with site j = A[i][2 ∗k] 1 and the type of hopping is h ij = A[i][2∗k +1], with k being an
1 we use C-type of writing matrices
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