Strongly-correlated lattice systems on a finite cylinder

In this thesis, I present my research on the numerical simulation of finite-size strongly-correlated lattice systems using the density matrix renormalisation group (DMRG). Ultracold gases in optical lattices have become the experimental setup of choice to simulate models from condensed-matter physics, because of their high degree of tunability and control. Their inherently finite size and alternative implementation call for a more in-detail study of how to define and characterise the phases of the models that they simulate. I use DMRG to implement lattice Hamiltonians of highly-correlated systems with long-range interactions and identify their ground states on a finite system geometry. I describe in detail the process of constructing long-range Hamiltonians for their use in DMRG. I then apply these numerical methods to two fundamental models. I study the ground state of a fractional quantum Hall system and identify it as the well-known Laughlin state in a still unexplored parameter regime, by calculating its topological entanglement entropy and by using a set of physical observables that are available in a finite cylindrical geometry. I then study a dipolar Bose-Hubbard model and perform a systematic study of the order parameters that can best be used to characterise its phases in an ultracold gas setting by comparing how different types of observables are sensitive to finite-size and boundary effects. My results are meant to provide guidance for future experimental realisations of bosonic lattice models of small sizes.