Anderson Localization In One-Dimensional Quasi-Periodical Lattices

The realization of ultra-cold atoms and cooling atoms technology with laser have opened up a new field of researching on cold atom physics,providing sufficient technological preparation for the experimental realization of Bose-Einstein condensation.Experimentally,atoms are trapped in the optical lattices formed by interference of multiple laser beams to study the physical properties of ultra-cold atoms.With the introduction of the concept of "Anderson localization",one introduces the disordered random potential into the optical lattice,and finds that in the absence of the disorder,the particle exhibits an extended state,of which eigenfunction is Bloch function,and as the disorder increases gradually,the wave function exhibits exponential decay in space,which also means that Anderson localization occurs.Therefore,it becomes a focus of attention which will contribute to extensive investigation of Anderson localization in other disordered systems.Aubry and André predicted that Anderson localization also exists in one-dimensional quasi-periodic optical lattices.They proposed the famous Aubry-André(AA)model by introducing the incommensurate on-site potential which aroused great interest in researching on AA model.This paper shows the classical AA model and some properties of single particles in this model.The Anderson localization transition point of AA model is at ?/ J(28)1,when?/ J(27)1,all eigenstates are extended and when ?/ J(29)1,all the states are localized.At the transition point,the eigenstates exhibt multifractal features.Hence,there is no mobility edges in AA model.By changing the forms of the on-site potential and the hopping rules between sites,there emerges a series of generalized AA models with a simple expression of mobility edges.Mobility edge as one of the most important concepts in the disordered systems which draws great interest all the time.However,the existence of the many-body mobility edges in the interacting disordered systems is still an open question due to the dimension of the Hilbert space beyond the numerical capacity.In this paper,we obtain the effective Hamiltonian of particle pairs by using the perturbation theory and obtain the mobility edges of the boson pairs trapped in one dimensional quasi-periodical lattices subjected to strongly interactions.And we get numerical results by the exact diagonalization method to calculate the standard participation rate(NPR)value of each eigenstate.The numerical calculations finely coincide with the analytic results for b(28)0 and small b cases.Especially,for b(28)0 case,the mobility edges of the bosonic pairs are described as? ?-1/ E.The extended regime and the one with the mobility edges will vanish with the increase of the interaction U to infinity.We also study the scaling of the NPR with system size in both extended and localized regimes.The NPR of the extended states ?(E)?1/ L with the increase of L and L ? ?,?(E)tend to finite value,while for localized case,2?(E)?(1/ L)tend to zero when L ? ?.b ?1 limits are also considered.Since the modulated potential approaches singularity when b ?1,the analytic expression does not fit very well.We hope our theory will provide new insights for studying the many-body mobility edges.