How to generate the NCA input file

In general case, the matrix $F^\alpha_{nm}=\langle n\vert d_\alpha\vert m\rangle$ is a full matrix between the Hilbert subspaces of $n$ and $n+1$ electrons. The best choice of base for solving NCA equations is the local eigenbase, where the impurity energies $E_{mn}=\delta_{mn} E_m$ are diagonal. In this case, one needs to write the $F$ matrix in local eigenbase as well. First, one can write $F$ in direct base, where it is very simple and mostly zero. However, one needs to pay some attention to the fermionic sign that appears due to the fermionic nature of the electron operator $d_\alpha$. Then, one can transform $F$ to the local eigenbase.

The matrix $F$ is greatly simplified when the hybridization $\Delta_{\alpha \b }$ is diagonal and Coulomb repulsion is of the form $U_{\alpha \b }n_\alpha n_\b$ since in this case, the direct base is also the local eigenbase. For this case, the program called generate generates the necessary base and calculates the $F$ matrix, which is necessary to run the NCA code.

Let me give a simple example of what the program generate does. In the two band case, with the crystal field splitting, the local states are

$$\begin{array}{l\vert c\vert l\vert l\vert l} \char93 & Stat... ...+E_2+3 U\\ 8 & 2 2 & 4 & 1 & 2 E_1+2 E_2 +6 U\\ \end{array}$$ (25)

where $n$ is the total number of electrons on the impurity, $Deg$ is degeneracy of the state and a state $2 1$ represents an atomic state with 2 electrons in the first band and 1 electron in the second. To generate this output, one needs to give the following data to the program

• number of baths : 2
• degeneracy of the first bath : 2
• degeneracy of the second bath: 2

In the program generate, user can then take only some part of the atomic states (to speed up the calculation) or all of the states. For example, if one is interested in the $\langle n\rangle\sim 1$ and $U$ is large, there is no point of keeping the state with 4 electrons on the local level since the probability for the state number 8 is very small. The program generate than calculates the $F$ matrix and writes it to a file which needs to be given to the NCA program. If one is interested in the SU(N) problem, it is best to choose

• number of baths : 1
• degeneracy of the bath : N .