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TRIQS / triqs_0.x

DEPRECATED -- This is the repository of the older versions of TRIQS
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Question: Reference for tail algebra #74

Closed johfer closed 12 years ago

johfer commented 12 years ago

Dear TRIQS-Team,

thanks for sharing this great code! In particular I find the built-in tail algebra for Green's functions very useful so that all operations on the Greens functions are also reflected in the tail coefficients. I wonder whether any of you can give me the respective reference where this is written down? For instance the inversion in TailGF::invert seems kind of tricky. I've seen something related in some work by Oudovenko, but not quite yet. Could you give me the exact reference?

Thanks a lot,

Johannes Ferber

mferrero commented 12 years ago

Hi Johannes,

Sorry for this late answer. I don't really have a reference for the tail algebra. In the end, it is not so tricky. For example, for the inverse, you would proceed like this. Imagine that G has a tail

a_1 / omega + a_2 / omega^2 + a_3 / omega^3

Now suppose you know the inverse Ginv and it has a tail

\sum_n b_n / omega^n

The idea is to write that G * Ginv = 1 and to get coupled equations for the tail coefficients b{n}. In doing so, you will realize that the lowest order in Ginv has to be omega^3. Any higher order is allowed and in principle non-zero. So you have to set a cutoff, e.g. omega^10. You then have to solve the equation for the b{n}. For example, b{-3} = 1 / a{3}. Then you must have b{-4} a{3} + b{-3} a{2} = 0 which you solve for b{-4}. And so on for the other b{n} up to the chosen cutoff.

I hope this helps!

johfer commented 12 years ago

Hi Michel,

ah, I see.

Thanks a lot, best regards,

Johannes