# Bosonization of Interacting Fermions in Arbitrary Dimensions by Peter Kopietz

By Peter Kopietz

The writer offers intimately a brand new non-perturbative method of the fermionic many-body challenge, bettering the bosonization approach and generalizing it to dimensions *d*1 through useful integration and Hubbard--Stratonovich differences. partially I he truly illustrates the approximations and barriers inherent in higher-dimensional bosonization and derives the appropriate relation with diagrammatic perturbation concept. He exhibits how the non-linear phrases within the power dispersion may be systematically incorporated into bosonization in arbitrary *d*, in order that in *d*1 the curvature of the Fermi floor may be taken into consideration. half II supplies purposes to difficulties of actual curiosity. The publication addresses researchers and graduate scholars in theoretical condensed subject physics.

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Fermions and the Fermi surface term in the expansion of the energy dispersion close to the Fermi surface is quadratic. Obviously, the effect of Van Hove singularities on the low-energy behavior of the Green’s function cannot be studied within an approximation that relies on the linearization of the energy dispersion close to the Fermi surface. However, our more refined functional bosonization approach developed in Chaps. 2 retains the quadratic term in the energy dispersion, so that our method might shed some new light on the problem of Van Hove singularities in strongly correlated Fermi systems.

14) reduces to ′ Π0αα (q) = − 1 V ′ Θα (k)Θα (k + q) k f (ξk+q ) − f (ξk ) ξk+q − ξk − iωm By relabeling k + q → k it is easy to see that . 15) 36 3. Hubbard-Stratonovich transformations ′ ′ Π0αα (q) = Π0α α (−q) . 41). We would like to emphasize that in the above functional integral representations of the correlation functions the precise normalization for the integration measure D{ψ} is irrelevant, because the measure appears always in the numerator as well as in the denominator. 2 The first Hubbard-Stratonovich transformation We decouple the two-body interaction between the fermions with the help of a Hubbard-Stratonovich field φα .

Note that in the work by Houghton et al. 34] it is implicitly assumed that the Gaussian approximation is justified. However, in none of these works the corrections to the Gaussian approximation have been considered, so that the small parameter which actually controls the accuracy of the Gaussian approximation has not been determined. 1) truncates at the second order. In this case we have exactly ˆ 0 Vˆ ˆ 0 Vˆ ] = Tr G ˆ 0 Vˆ + 1 Tr G −Tr ln[1 − G 2 2 . 1]. A few years later T. 2], and formulated it as a theorem, which he called the closed loop theorem.