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By Peter Kopietz
The writer provides intimately a brand new non-perturbative method of the fermionic many-body challenge, bettering the bosonization process and generalizing it to dimensions d1 through practical integration and Hubbard--Stratonovich changes. partly I he in actual fact illustrates the approximations and boundaries inherent in higher-dimensional bosonization and derives definitely the right relation with diagrammatic perturbation conception. He indicates how the non-linear phrases within the strength dispersion will be systematically integrated into bosonization in arbitrary d, in order that in d1 the curvature of the Fermi floor should be taken into consideration. half II offers purposes to difficulties of actual curiosity. The booklet addresses researchers and graduate scholars in theoretical condensed subject physics.
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Extra resources for Bosonization of Interacting Fermions in Arbitrary Dimensions
Example text
The corresponding non-interacting Green’s function is then approximated by G0 (kα + q, i˜ ωn ) ≡ Gα q) ≈ 0 (˜ 1 . 8) according to k = kα + q, we obtain Unα (q1 . . +qn ,0 × i β n 1 n! n) q˜ α α q + qP2 G0 (˜ q )G0 (˜ q + qP2 ) · · · Gα 0 (˜ + . . + qPn ) . 11) Recall that we have introduced the convention that q˜ = [q, i˜ ωn ] labels fermionic Matsubara frequencies, while q = [q, iωm ] labels bosonic ones. Because the sum of a bosonic and a fermionic Matsubara frequency is a fermionic one, the external labels q1 , .
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 amazing fact is now that there exists a physically interesting limit where all higher order vertices Un with n ≥ 3 vanish.
1. 58) where : . . : denotes normal ordering, and it is assumed that the variations ′ of fqkk are negligible if k and k′ are restricted to given boxes, so that it is allowed to introduce coarse-grained interaction functions ′ ′ fqαα = kk′ Θα (k)Θα (k′ )fqkk kk′ Θα (k)Θα′ (k′ ) ′ . 32]. 58) is already the bosonized potential energy. It should be mentioned that the usefulness of the geometric construction described above is not restricted to higher-dimensional bosonization. A very similar construction has recently been used by Feldman et al.