Strongly interacting systems
Ultracold atoms in the optical lattice potential initially considered a "experimental toy model" for condensed matter physics, have recently become an independent and important field of quantum complex system physics. They are efficiently described by a Bose-Hubbard model, which shows similarity to the Hubbard model first used to model gas of electrons in a solid body. Atoms forming the ultracold gas interact through collisions, which typically may not be neglected : the Mott insulator phase, present in low-mobility regime, originates directly as a result of interparticle interactions. Ability to change systems parameters, by adjusting strength of lasers, enables to realize different regimes and phase transisiton and crossovers between them. Therefore description of a many-body problem must be done exactly, in a true many-body way.
Such systems pose a serious challenge from the computational point of view, due to large entanglement present in the system. However restriction of problem geometry to 1D enables to preserve many-bodiness and complex nature of such systems, providing at the same time a very powerful computational tool : the DMRG and related techniques. This is due to area laws satisfied by 1D systems, which limit the entanglement, and enable efficient analysys of systems consistings of hundreds of particles.
We have used it to solve problem of numerical calculation of many-body eigenvectors for a strongly coupled system [1,2], analyze effects caused by including higher Bloch bands on the dynamics .
 Mateusz Łącki, Dominique Delande, Jakub Zakrzewski Numerical computation of dynamically important excited states of many-body systems, Phys. Rev. A 86, 013602 (2012) arxiv http://pra.aps.org/abstract/PRA/v86... http://arxiv.org/abs/1106.4906
 Mateusz Łącki, Dominique Delande, Jakub Zakrzewski Extracting Information from Non Adiabatic Dynamics : Excited Symmetric States of the BosHubbard Model Acta Physica Polonica 12/2011 (2012) http://przyrbwn.icm.edu.pl/APP/PDF/...
 Mateusz Łącki, Dominique Delande, Jakub Zakrzewski Higher optical lattice bands in real-time dynamics : eﬀective Hamiltonian approach http://arxiv.org/pdf/1206.6740v2.pdf (2012) http://arxiv.org/pdf/1206.6740v2.pdf