Monday, February 19, 2018

quantum mechanics - Hubbard-Stratonovich transformation and decoupling channels


I'm studying an example of the Hubbard-Stratonovich transformation in Altland and Simons' Condensed Matter Field Theory (2nd ed.), pp. 246-247.



In it they say that...



one is frequently confronted with situations where more than one Hubbard-Stratonovich field is needed to capture the full physics of the problem. To appreciate this point, consider the Coulomb interaction in momentum space. Sint[ˉψ,ψ]=12p1,...,p4ˉψσ,p1ˉψσ,p3V(p1p2)ψσ,p4ψσ,p2δp1p2+p3p4.

In principle, we can decouple this interaction in any of the three channels...



discussed in the previous page. If one chooses to decouple in all three channels then the action becomes ...




Sint[ˉψ,ψ]12p,p,q(ˉψσ,pψσ,p+qV(q)ˉψσ,pψσ,pqˉψσ,pψσ,p+qV(p'p)ˉψσ,pψσ,pˉψσ,pˉψσ,p+qV(p'p)ψσ,pψσ,p+q)





where the first term is decoupled via the




direct channel ρd,qpˉψσ,pψσ,p+q, second in the exchange channel ρx,σσ,qpˉψσ,pψσ,p+q, and third in the Cooper channel ρc,σσ,qpˉψσ,pˉψσ,p+q.




It's generally a good strategy to decouple in all available channels when one is in doubt, then let the mean-field analysis sort out the relevant fields.


My question is, if we choose to decouple the quartic term via 3 different channels (for example) is it necessary to multiply the resulting terms by a factor of 13? This isn't discussed in the textbook and I'm confused by the liberal use of and in the examples.



Answer



No. You should not add a factor of 1/3. As you can see in page 244 of Altland and Simons, the HS transformation is done by multiplying by a unity expressed as a functional integral over an auxiliary field. In this case, they just choose to introduce 3 different fields - 1 for each term.



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