Bosonization of the low energy excitations of Fermi liquids.

We bosonize the low energy excitations of Fermi Liquids in any number of dimensions in the limit of long wavelengths. The bosons are coherent superposition of electron-hole pairs and are related with the displacements of the Fermi Surface in some arbitrary direction. A coherent-state path integral for the bosonized theory is derived and it is shown to represent histories of the shape of the Fermi Surface. The Landau equation for the sound waves is shown to be exact in the semiclassical approximation for the bosons. PACS numbers: 05.30.Fk, 05.30.Jp, 11.10.Ef, 11.40.Fy, 71.27.+a, 71.45.-d Typeset using REVTEX 1 The attempts of description of fermionic systems by bosons date to the early days of second quantization. In the early 50’s Tomonaga [1], generalizing earlier work by Bloch [2] on sound waves in dense fermi systems, gave an explicit construction of the Bloch waves for systems in one spacial dimension. His work was subsequently generalized by many authors [3] who derived an explicit fermi-bose transmutation in one-dimensional systems. These works uncovered deep connections in relativistic field theories (both fermionic and bosonic) and with condensed matter systems. The success of the bosonization approach in one dimension is related to phase space considerations. Even for non-interacting fermions two excitations with arbitrarily low energies, moving in the same direction move at the same speed (the Fermi velocity) and, hence, are almost a bound state. Consequently, even the weakest interactions can induce dramatic changes in the nature of the low lying states. These effects are detected even in perturbation theory and result in the presence of marginal operators. In dimensions higher than one, phase space considerations change the physics of the low-lying states and there are no marginal operators left. This observation is at the root of the stability of the Landau theory of the Fermi liquid [4]. It is, thus, hardly surprising that very few attempts have been made to generalize the bosonization approach to dimensions higher than one. The earliest serious attempt at bosonization in higher dimensions was carried out by Luther [5]. Luther constructed a generalized bosonization formula in terms of the fluctuations of the Fermi Sea along radial directions in momentum space. Interest in the construction of bosonized versions of Fermi liquids has been revived recently in the context of strongly correlated systems [6]. That particle-hole excitations have bosonic character is well known since the early days of the Landau Theory, e. g. the sound waves (zero sound collective modes) of the Fermi Surface of neutral liquids or plasmons in charged Fermi Liquids [7]. Haldane [8] has recently derived an algebra for the densities for a Fermi Liquid in a form of a generalized Kac-Moody algebra (which is central to the construction in one dimension) (see also ref. [9]). The main point of this article is to derive a description of the Fermi Liquids as the physics of the dynamics of the Fermi Surface. The main reason to believe that the Fermi 2 Surface is a real dynamical entity is based on the following observation: the total energy of an interacting electronic system can be written as an integral in momentum space of the form [10] E = ∑ ~k E~k. For the case of Fermi Liquids, E~k has a discontinuity at the Fermi Surface (exactly as for the case of the occupation number [11]). We can think of this discontinuity as due to the difference of energy density across the Fermi Surface [12]. Hence, we can define an energy per unit area of the Fermi Surface, i. e. a surface tension. It readily follows that this surface tension is proportional to the quasiparticle residue and therefore it vanishes for Non-Fermi Liquid behavior. From this point of view we see the Fermi Surface as a drumhead where the elementary excitations are the sound waves (and in particular the zero sound) which propagate on it. In this paper we will derive an effective bosonized theory for the dynamics of these excitations. Our starting point to approach this problem resembles the microscopic approaches for the foundations of Fermi Liquid Theory [7]. However, instead of working with the dynamics of the response functions, we will work directly with the dynamics of operators as in the standard procedures of bosonization in one dimension. Our main result is the derivation of a bosonized theory of the fluctuations of the shape of the Fermi Surface. The main ingredients of our construction are an effective algebra for the local (in momentum space ) of particle-hole operators valid in a Hilbert space restricted to the vicinity of the Fermi Surface and a boson coherent state path-integral constructed from these states. In particular, we get a bosonized version of Landau’s theory of the Fermi Liquid in all dimensions, at zero temperature and at very low energies. The coherent-state path integral can be viewed as a sum over the histories of the shape of the Fermi Surface, a bosonic shape field. For the case of the Landau hamiltonian we find that resulting bosonic action is quadratic in the shape fields. We find that the semiclassical approximation to our path-integral yields the Landau equation for the sound modes. For simplicity we will consider a system of interacting spinless fermions (the spin index can be introduced without problems in the formalism). The density of fermions at some point ~r in the d dimensional space at some time t, is given by 3 ρ(~r, t) = ψ(~r, t)ψ(~r, t) = ∑ ~k,~ q c † ~k− ~q 2 (t)c~k+ ~q 2 (t) e q.~r (1) where c ~k and c~k are the creation and annihilation operator of an electron at some momentum ~k which obey the Fermionic algebra, {