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By Melia F.

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3] Un (q1 α1 . . +qn ,0 i β αP2 αPn ×Θ (k + q P2 ) · · · Θ n 1 n! n) k (k + q P2 + . . + q Pn ) ×G0 (k)G0 (k + qP2 ) · · · G0 (k + qP2 + . . + qPn ) . +qn ,0 denotes a Kronecker-δ in wave-vector and frequency space. We have used the invariance of Skin,n {φα } under relabeling of the fields to symmetrize the vertices Un with respect to the interchange of any two labels. n) is over the n! permutations of n integers, and Pi denotes the image of i under the permutation. Note that the vertices Un are uniquely determined by the energy dispersion ǫk − µ.

The proper choice depends on the shape of the Fermi surface and on the nature of the interaction. Although the variations in the direction of the local normal vector can always be reduced by choosing a sufficiently small patch cutoff Λ, this cutoff cannot be made arbitrarily small. The reason is that for practical calculations the sectorization turns out to be only useful if scattering processes that transfer momentum between different boxes (so called around-the-corner processes) can be neglected.

1) and truncating the expansion at some finite order. 56) should then also be performed perturbatively to this order. 1) are neglected, so that one sets ˆ 0 Vˆ + 1 Tr G ˆ 0 Vˆ Skin {φα } ≈ Tr G 2 2 . 2) 48 4. Bosonization of the Hamiltonian and . . 56) reduces to a trivial Gaussian integration. Evidently the effective action S˜eff {ρ˜α } of the collective density field is then also quadratic. Note that in the work by Houghton et al. 34] it is implicitly assumed that the Gaussian approximation is justified.

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