ar X iv : c on d - m at / 9 50 50 59 v 2 3 1 O ct 1 99 5 Why normal Fermi systems with sufficiently singular interactions do not have a sharp Fermi surface

We use a bosonization approach to show that the momentum distribution nk of normal Fermi systems with sufficiently singular interactions is analytic in the vicinity of the non-interacting Fermi surface. These include singular density-density interactions that diverge in d dimensions stronger than |q|−2(d−1) for vanishing momentum transfer q, but also fermions that are coupled to transverse gauge fields in d < 3. PACS. 05.30.Fk Fermi systems and electron gas. PACS. 11.15.-q Gauge field theories. Typeset using REVTEX 1 As first noticed by Bares and Wen [1], singular density-density interactions with Fourier transform fq ∝ |q|−η destroy in d dimensions the Fermi liquid state for η ≥ 2(d − 1). The case η = 2(d − 1) is marginal and corresponds to a Luttinger liquid, while for η > 2(d− 1) one obtains normal metals which are neither Fermi liquids nor Luttinger liquids. We shall call these metals strongly correlated quantum liquids. The properties of these systems are not very well understood. Certainly strongly correlated quantum liquids cannot be studied by means of conventional many-body perturbation theory, because the perturbative calculation of the self-energy leads to power-law divergencies. In the present work we shall use higher dimensional bosonization to calculate the momentum distribution nk in these systems. We find that for η > 2(d− 1) the momentum distribution nk does not have any singularities, so that a sharp Fermi surface cannot be defined. We then argue that below three dimensions the momentum distribution of electrons that are coupled to transverse gauge fields has also this property. Bosonization in arbitrary dimensions has recently been discussed by a number of authors [2–8]. The fundamental geometric construction is the subdivision of the Fermi surface into patches of area Λd−1. With each patch one then associates a “squat box” K [4] of radial hight λ ≪ kF and volume λΛd−1, and partitions the degrees of freedom close to the Fermi surface into the boxes K. Here kF is the Fermi wave-vector, and α labels the boxes in some convenient ordering. If the size of the patches is chosen small enough, then the curvature of the Fermi surface can be locally neglected. The essential motivation for this construction is that it opens the way for a local linearization of the non-interacting energy dispersion ǫk: If k points to the suitably defined center of box K, then for k ∈ K we may shift k = k+q, and expand ǫkα+q − μ ≈ v · q, where μ is the chemical potential and v is the local Fermi velocity. At high densities and for interactions that are dominated by small momentum transfers the linearization implies in arbitrary dimension a large-scale cancellation between self-energy and vertex-corrections (generalized closed loop theorem [8]), so that the entire perturbation series can be summed in a controlled way. The final result for the MatsubaraGreen’s function G(k, iω̃n) of the interacting many-body system is then 2