Luttinger Liquid in a Solvable 2-dimensional Model. * Typeset Using Revt E X

We consider spinless electrons in two dimensions with the bare spectrum ǫ(p) = |px| + |py|. In momentum space, the interactions among electrons have a finite range q0, which is small compared to the Fermi momentum. A golden rule calculation of the electron lifetime indicates a breakdown of the Landau Fermi liquid in the model. At the one-loop level of perturbation theory, we show that the density wave and the superconducting instabilities cancel each other and there is no symmetry breaking. We solve the model via bosonization; the excitation spectrum is found to consist of gapless bosonic modes as in a one-dimensional Luttinger liquid. Typeset using REVTEX 1 Recently, the question about validity of Landau’s Fermi liquid (LFL) theory in two dimensions has attracted much interest. This is because there are strong indications that in the normal state of the high temperature superconductors (HTSC’s), some kind of a non-LFL state is realized. This idea is due to Anderson [1], who realized that there are striking analogies between the phenomenology of the cuprates and of one-dimensional metals. The latter are known to be Fermi liquids of a very special type (Luttinger liquids): the Fermi surface (defined as a set of points, where the momentum distribution function has a singularity) still exists and has the same volume as for non-interacting electrons in agreement with Luttinger’s theorem, but elementary excitations are completely different from those of a usual metal in D = 3: there are no quasiparticle excitations and the spectrum consists of collective modes only. However, in a D = 2 low density Fermi gas with short ranged interactions, the LFL phase was shown (within perturbation theory) to be stable [2]. It is believed, that LFL-behavior can break down in perturbation theory only in specific situations: assuming particular bandstructure (like in the nested Fermi liquid theory [3]) and/or singular long ranged interactions. The latter suggestion is due to Haldane [4] (see also [5]), who proposed a generalization of bosonization as a tool to study non-LFL behavior in higher dimensions. Haldane’s method was applied recently to study fermions interacting with long-ranged current-current interactions [6]. Here we study a model with nesting. We consider spinless fermions in D = 2 described by the Hamiltonian