Introduction To Quantum Field Theory In Condensed Matter Physics by Henrik Bruus Karsten Flensberg

Author:Henrik Bruus, Karsten Flensberg
Language: eng
Format: epub
Tags: physics, condensed matter theory, many-body physics, FreeScience.info, lecture notes
Published: 2001-03-26T16:00:00+00:00

k 2 |»A; F k| % k 2 + k - ki| « A; F

Figure 10.6: (a) The non-crossing wigwam diagrams, one inside the other, where k x and k 2 can take any value on the spherical shell of radius k F and thickness Ak « 1/1. The phase space is O a oc (inkpAk) 2 . (b) The crossing wigwam diagram has the same restrictions for k x and k 2 as in (a) plus the constraint that |k + k 2 — k 1 | sa k F . For fixed k 2 the variation of k x within its Fermi shell is restricted to the intersection between this shell and the Fermi shell of k + k 2 — k l5 i.e. to a ring with cross section l/l 2 and radius sa k p . The phase space is now Oft oc (47rA;pAA;)(27rA; F A/c 2 ). Thus the crossing diagram (b) is suppressed relative to the non-crossing diagram (a) with a factor l/k p l.

We have now resummed most of the diagrams in the diagrammatic expansion of ({?k); mp with the exception of wigwam-diagrams with crossing lines. In Fig. 10.6 are shown two different types of irreducible diagrams of the same order in both n imp and u^. Also sketched is the phase space 0 available for the internal momenta k x and k 2 in the two cases. At zero temperature the energy broadening around the Fermi energy e F is given by |E| sa H/t which relaxes m|, |k 2 | = k F a bit. In k-space the broadening Ak is given by h 2 (k F + Ak) 2 /2m sa s F + H/t which gives Ak sa l/u F r = 1/1, i.e. the inverse scattering length. This means that k x and k 2 are both confined to a thin spherical shell of thickness l/l and radius k F .

In Fig. 10.6(a), where no crossing of scattering lines occurs, no further restrictions applies, so the volume of the available phase space is Q a = (4irk F /l) 2 . In Fig. 10.6(b), where the scattering lines crosses, the Feynman rules dictate that one further constraint, namely |k + k x — k 2 | sa k F . Thus only one of the two internal momenta are free to be anywhere on the Fermi shell, the other is bound to the intersection between two Fermi shells, i.e. on a ring with radius ~ k F and a cross section l/l 2 as indicated in Fig. 10.6(b). So Oft = (4:Trk F /l)(2Trk F /l 2 ). Thus by studying the phase space available for the non-crossed and the crossed processes we have found that the crossed ones are suppressed by a factor Oft/O a sa l/(k F l). Such a suppression factor enters the calculation for each crossing

10.6. SUMMARY AND OUTLOOK

179

of scattering lines in a diagram. Since for metals l/k F ~ 1 A we find that

< 1, for I > 1 A.

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