Few-Particle Green’s Functions for Strongly Correlated Systems on Infinite Lattices

Abstract

We show how few-particle Green's functions can be calculated efficiently for models with nearestneighbor hopping, for infinite lattices in any dimension. As an example, for one dimensional spinless fermions with both nearest-neighbor and second nearest-neighbor interactions, we investigate the ground states for up to 5 fermions. This allows us not only to find the stability region of various bound complexes, but also to infer the phase diagram at small but finite concentrations.PACS numbers: 71.10. Li, 31.15.ac, 71.35.Pq Recently, there has been considerable interest in fewparticle solutions of interacting Hamiltonians. For example, in Ref. [1] it was shown that knowledge of the twoand three-body solutions allows for quantitatively accurate predictions of finite-temperature thermodynamic quantities for many-body systems. As another example, in the context of atomic and molecular physics, the predicted universal three-body Efimov structures [2] have now been seen experimentally [3], giving new impetus to their study and work on various generalizations [4].While the above work is for free space where particles have parabolic dispersions, there is equally strong interest in the lattice version of such few-body problems. For example, while stable excitons -bound pairs comprised of an electron and a hole -appear in many materials, it is less clear when a so-called charged exciton or trion, consisting of two holes and one electron or viceversa, is stable. That this can happen has been recently demonstrated in GaAs quantum wells [5] and in carbon nanotubes [6]. (Note that trion theory is still mostly based on continuous models and variational solutions, e.g. see Ref. [7]). Studying bigger bound complexes, for example bi-exciton pairs, is the next logical step.Few-particle bound states are relevant not only for the materials where they appear, but also in the interpretation of certain spectroscopic data. For instance, the role played by bound two-particle states, leading to atomic-like multiplet structures in the Auger spectra of narrow-band insulating oxides, is well established [8]. At low dopings, more complicated complexes may form and leave their fingerprints in various spectroscopic features. It is therefore useful to be able to study relatively easily few-particle solutions on an infinite lattice.In this Letter we show that few-particle Green's functions can be calculated efficiently for strongly correlated lattice Hamiltonians in the thermodynamic limit, at least so long as the hopping involves only nearest neighbor sites. For simplicity and to illustrate the technique and its usefulness, we focus here on a one-dimensional (1D) model of spinless fermions with nearest-neighbor (nn) and next-nearest-neighbor (nnn) interactions. However, the method generalizes straightforwardly to higher dimensions, longer (but finite) range interactions, mixtures of fermions (including spinful fermions) and/or bosons, etc. Such problems are of direct interest either in solid state physics, or for cold atoms in optical lattic...

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