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Modern Physics Letters B

LOW DENSITY HUBBARD MODEL IN TWO AND THREE DIMENSIONS

Sergei Khlebnikov

Published May 20, 1992• 0 citations
Publisher

Abstract

We consider two- and three-dimensional repulsive Hubbard models at low density, using the gas approximation. The large-U three-dimensional model is shown to behave as a Fermi liquid and the corresponding two-particle scattering length is found. In the two-dimensional case we find that the gas approximation is inconsistent, as indicated by an isolated pole of the two-particle Green function below the occupied states in the upper half-plane. This behaviour is specific to the lattice system as opposed to the continuum two-dimensional Fermi gas with short-range repulsion.

DOI
10.1142/s0217984992000831

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A central problem in the theory of strongly correlated fermion systems, concerns the nature of their low-lying excitations. It is generally accepted that, except for special cases with symmetry breaking, three dimensional (3D) interacting Fermi systems, are well described in terms of weakly correlated quasi-particles with quantum numbers in one-toone correspondence with those identifying the excitations of a non-interacting Fermi gas. This is the basic assumption underlying the Landau theory of Fermi liquids (FL), which was originally proposed as a phenomenological description of strongly interacting fermions by Landau in early 1956[1), an then established microscopically by Nozieres and Luttinger in 1962[2]. Quite a different scenario occurs in lD interacting Fermi systems. Apart from specific models developing gaps in the excitation spectrum, most fermion systems in one dimension with repulsive interactions, do have low-lying gapless excitations, like ordinary Fermi liquids, but totally different from the quasi-particles predicted by Landau theory. The low-energy behavior of most of these lD interacting fermion systems with gapless linear excitations, can be understood in terms of few model dependent constants, which parametrize all the long-wavelength properties...

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We consider two- and three-dimensional repulsive Hubbard models at low density, using the gas approximation. The large-U three-dimensional model is shown to behave as a Fermi liquid and the corresponding two-particle scattering length is found. In the two-dimensional case we find that the gas approximation is inconsistent, as indicated by an isolated pole of the two-particle Green function below the occupied states in the upper half-plane. This behaviour is specific to the lattice system as opposed to the continuum two-dimensional Fermi gas with short-range repulsion.
The two-dimensional Hubbard model has been studied in the region of low electron density based on the t -matrix approximation. This can take proper account of the singular behaviors of the forward scattering whose importance was indicated by Anderson. By performing first the angular integrations in the irreducible particle-particle correlation function, we can prove analytically that all of the poles of the t -matrix lie on the real axis. Hence the t -matrix and then the self-energy of the retarded Green function, Σ( k , z ), are analytic in the upper half-plane of the energy variable, z . This, together with Im Σ( k F , ε+i δ)∝ε 2 ln |ε|, leads to the conclusion that the electrons described by the two-dimensional Hubbard model are Fermi liquid, at least in the region of low electron density.
The two dimensional Hubbard model with repulsive interaction is studied in the t -matrix approximation, which is the valid approximation in low electron density. The particle-particle correlation function and the t -matrix are obtained analytically. By use of these analytical results it is shown that the t -matrix and hence the self-energy have no singularities in the upper half plane of the complex energy variable. The phase angle δ(2 k F + q , ω), with k F being the Fermi wave vector, in the limit of q =0 and ω=0 is found to depend strongly on the ratio ω/ q and is not always zero. The case of the attractive interaction is also briefly studied.
We study the dilute-gas expansion for a two-dimensional Fermi system with arbitrary short-range repulsive interactions. In contrast with the three-dimensional case, we find an unusual pole in the vertex part in the particle-particle channel, for all center-of-mass momenta q2kF{\mathit{k}}_{\mathit{F}}kF​. We show that this represents an excitation consisting of bound hole pairs, which disperses down to zero energy at q=2kF{\mathit{k}}_{\mathit{F}}kF​. We study the effect of these bound states on the single-particle self-energy and find that the quasiparticles are well defined. Thus in the low-density regime, there is no breakdown of Fermi-liquid theory. In the Appendix we discuss the two-particle phase shift in the dilute Fermi gas, its connection with bound states, and the analogy with the potential scattering phase shift.
We consider the 2D Hubbard model in the strong-coupling case (U ≫ W) and at low electron density (nd2 ≪ 1). We find an antibound state as a pole in the two-particle T-matrix. The contribution of this pole in the self-energy reproduces a two-pole structure in the dressed one-particle Green-function similar to the Hubbard-I approximation. We also discuss briefly the Engelbrecht-Randeria mode which corresponds to the pairing of two holes below the bottom of the band for U ≫ W and low electron density. Both poles produce nontrivial corrections to Landau Fermi-liquid picture already at low electron density but do not destroy it in 2D.
A central problem in the theory of strongly correlated fermion systems, concerns the nature of their low-lying excitations. It is generally accepted that, except for special cases with symmetry breaking, three dimensional (3D) interacting Fermi systems, are well described in terms of weakly correlated quasi-particles with quantum numbers in one-toone correspondence with those identifying the excitations of a non-interacting Fermi gas. This is the basic assumption underlying the Landau theory of Fermi liquids (FL), which was originally proposed as a phenomenological description of strongly interacting fermions by Landau in early 1956[1), an then established microscopically by Nozieres and Luttinger in 1962[2]. Quite a different scenario occurs in lD interacting Fermi systems. Apart from specific models developing gaps in the excitation spectrum, most fermion systems in one dimension with repulsive interactions, do have low-lying gapless excitations, like ordinary Fermi liquids, but totally different from the quasi-particles predicted by Landau theory. The low-energy behavior of most of these lD interacting fermion systems with gapless linear excitations, can be understood in terms of few model dependent constants, which parametrize all the long-wavelength properties...