Two-body problem in graphene

Abstract

We study the problem of two Dirac particles interacting through non-relativistic potentials and confined to a two-dimensional sheet, which is the relevant case for graphene layers. The two-body problem cannot be mapped into that of a single particle, due to the non-trivial coupling between the center-of-mass and the relative coordinates, even in the presence of central potentials. We focus on the case of zero total momentum, which is equivalent to that of a single particle in a Sutherland lattice. We show that zero-energy states induce striking new features such as discontinuities in the relative wave function, for particles interacting through a step potential, and a concentration of relative density near the classical turning point, if particles interact via a Coulomb potential. In the latter case we also find that the two-body system becomes unstable above a critical coupling. These phenomena may have bearing on the nature of strong coupling phases in graphene.Comment: 13 pages, 7 figures, final versio

Exciton states in narrow-gap carbon nanotubes

“…Indeed for negative energies one must take special care and attention to include all possible solutions to obtain the correct negative energy spectrum and this shall be a topic of future study. An analogous case was found in the two dimensional problem [35] which was later resolved in [39]. However in our case only positive solutions are needed since we are considering a system containing a single electron above the Dirac point and a single hole below it.…”

“…Pair formation has also been studied in gapped graphene [33,34,35,24] and the two body problem was considered in bilayer graphene quantum dots [36]. Excitonic effects in Dirac materials have been studied using a variety of approaches such as the Bethe-Salpeter method [37] as well as the twobody tight-binding matrix Hamiltonian [35,38,39,29], we shall employ the later method.…”

Quantum collision theory in flat bands

“…In fact, a new subfield has emerged studying these so-called Dirac materials [33]. These systems are very interesting also from a few-body point of view due to their non-trivial scattering properties [34][35][36][37]. We note that truly flat bands are also of great interest in the context of graphene [38,39].…”

Pair states in one-dimensional Dirac systems

“…Restricting ourselves to the modes of relevant parity, we find that the same conditions hold true for Eqs. (32) and (33) at zero energy.…”

“…Excitonic effects in Dirac materials have been studied using a variety of approaches such as the Bethe-Salpeter method [31] as well as the two-body matrix Hamiltonian, based on the low-energy expansion of the tight binding [29,[32][33][34][35], which will be the method employed in this study. Pair formation has been studied in Dirac materials with effective mass, such as gapped graphene [32,[36][37][38][39], bilayer graphene [40], and graphene in the trigonal warping regime [41][42][43]. However, there is still much debate concerning the existence of coupled pairs in intrinsic graphene [34,44,45].…”