This is document for the sample program which uses K \(\omega\) library in the ISSP Math Library; this program computes the Green’s function with \(K\omega\). For the details of K \(\omega\) library, See “\(K\omega\) manual” in this package.

Calculation in this program

This program compute the Green’s function

\[\begin{align} G_{i}(z) = \langle i | (z-{\hat H})^{-1}| i \rangle \equiv {\boldsymbol \varphi}_i^{*} \cdot (z-{\hat H})^{-1} {\boldsymbol \varphi}_i, \end{align}\]

where \(| i \rangle\) is a wavefunction, \({\cal H}\) is the Hamiltonian, and \(z\) is a complex frequency.

\({\cal H}\) in the above equation is obtained by either the

following two ways:

  • Input \({\cal H}\) as a file with the MatrixMarket format

  • Construct \({\cal H}\) as a Hamiltonian of the Heisenberg model in this program.

In the computation of the Green’s function, we use either the following two method according to the type of \({\hat H}\) (a real- or a complex- number).

  • \({\hat H}\) of real numbers : Shifted Bi-Conjugate Gradient(BiCG) method

  • \({\hat H}\) of complex numbers : Shifted Conjugate Orthogonal Conjugate Gradient(COCG) method