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I'm trying to simulate the degenerative Anderson model. So depending on an energy difference first orbital and afterwards spin magnetism occurs. First i try to solve an easier ansatz with a limitation of only two orbitals to $l_z = \pm 1$. In this case i end up with a set of 4 equations:

$\Delta \tan^{-1} (1/\pi * n_{m \sigma}) = E_0-E_f+(U-J)n_{\overline{m}\sigma}+U(n_{1\overline{\sigma}}+n_{2\overline{\sigma}})$

Where here $\sigma$ is the direction of the spin of certain electron in the orbital $m$, here only $1,2$ and the overline always is the opposite. U is a Hubbard potential and $J$ an intraatomic exchange. So we gathered a self consistent set of non-linear equation and i want to find solution in terms of $n_{m \sigma}$ satisfying all 4 equations. I already did some simulation and were able to plot it on dependancy of the energy difference with respect to the fermi energy, which is my parameter. So i change it in a certain range and one finds magnetic solution for $E_0 < E_f$.

Is there a good method to solve such a problem with not too much calculation power? And in the end i want to play around with both potentials and see how the number of electron is changing for each situation.

Thanks for any advice,


PS: i already asked a question concerning to my code i wrote in SO, but i was rethinking the method of mine for solving this problem. I did an iterative attempt and reformulated the equation for a root searching algorithm and varied the energy difference $E_0 - E_f$. The programming stuff is on ipython plus a result of one simualtion run.

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I suggest you try asking this on scicomp.stackexchange.com –  David Ketcheson Jan 13 '12 at 20:12
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