Two-particle states in the Hubbard model

Abstract

We consider a pair of bosonic particles in a one-dimensional tight-binding periodic potential described by the Hubbard model with attractive or repulsive on-site interaction. We derive explicit analytic expressions for the two-particle states, which can be classified as (i) scattering states of asymptotically free particles, and (ii) interaction-bound dimer states. Our results provide a very transparent framework to understand the properties of interacting pairs of particles in a lattice.

Bound states and expansion dynamics of interacting bosons on a one-dimensional lattice

“…In this case, despite the non-integrability of the model [35], it is possible to solve the Schrödinger equation exactly, by the Bethe ansatz technique, thanks to the separation of centre-of-mass and relative coordinates. A similar analysis was exploited by Valiente and Petrosyan in [29] in the thermodynamic limit; instead, we work at finite size L+1, with L even. Moreover, in the rest of the paper (unless otherwise stated), we will choose = a = 1.…”

“…Very recently, bound states have also entered the debate about equilibration and thermalization [28]. For bosonic systems, the existence of bound states in lattice models has been investigated theoretically and experimentally, both from a static [29][30][31] and a dynamic [23,24] point of view. They can, in principle, play an important role in the expanding dynamics of a lattice Bose gas: it is natural to expect that if the wave function of the system possesses a large projection over the set of the bound states, the dynamics should be quasistationary, and the expansion very different from a ballistic one.…”

NOON states via a quantum walk of bound particles

“…The key point here is that, provided that the onsite interaction strength U is large enough, the resulting state composed of M > 1 bounded particles on the same site is stable against dissociation during the time evolution [32, 34-36, 48, 49], and behaves like an effective single particle. Indeed, as explicitly discussed in [35,36] for a few values of M, the bounded-particle states lie in an energy band which is well-separated (by an energy separation ∝ U) from other states, provided that U is suitably large. In the following we introduce a general theory to model the effective interactions between stable bounded particles.…”

“…One characteristic feature of the Bose-Hubbard model is that the onsite interaction enables the creation of "bound" states when several particles are in the same site [2,[32][33][34][35][36]. Here we are interested in a non-equilibrium configuration, in the low filling regime, where M particles are initially located on a single site.…”

“…From the theoretical point of view, optical lattice systems in a low filling limit are modeled by the Bose-Hubbard Hamiltonian, which contains a hopping term between neighboring sites and an onsite interaction. If more than one particles are initially located in the same site and the onsite interaction is sufficiently strong, this model favors the creation of bound states, which are stable against dissociation [2, [32][33][34][35][36]. A natural question is then whether bound states have some advantages for non-classical state production tasks.…”