An important, but also difficult, research field in condensed matter physics is that of strongly correlated systems. This thesis considers two topics in this field. The first topic is disorder and frustration in spin models. The introduction of disorder into quantum spin chains creates a complex problem. The ground state of the random-bond spin-1 Heisenberg chain is studied by means of stochastic series expansion quantum Monte Carlo simulation, applying the concept of directed loops. It is found that this system undergoes a phase transition to the random-singlet phase if the bond disorder is strong enough. Further a frustrated spin system is investigated. The frustration is introduced by having spins positioned on a triangular lattice. Performing a quantum Monte Carlo simulation for such a frustrated lattice leads to the occurrence of the infamous sign problem. This problem is investigated and it is shown that it is possible to use a meron cluster approach to reduce its effect for some specific models. The second topic concerns atomic condensates in optical lattices. A system of trapped bosonic atoms in such a lattice is described by a Bose-Hubbard model with an external confining potential. Using quantum Monte Carlo simulations it is demonstrated that the local density approximation that relates the observables of the unconfined and the confined models yields quantitatively correct results in most of the interesting parameter range of the model. Further, the same model with the addition that the atoms carry spin-1 is analyzed using density matrix renormalization group calculations. The anticipated phase diagram, with Mott insulating regions of dimerized spin-1 chains for odd particle density, and on-site singlets for even density is confirmed. Also an ultracold gas of bosonic atoms in an anisotropic two dimensional optical lattice is studied. It is found that if the system is finite in one direction it exhibits a quantum phase transition. The Monte Carlo simulations performed show that the transition is of Kosterlitz-Thouless type. / QC 20100628

The experimentally obtained value of the critical exponent ν is presently in significant disagreement with current theoretical predictions for the λ-universality class. We suggest two novel approaches of determining the exponents ν and η by utilizing the effects of finite size scaling. The numerical computations are performed using Monte Carlo simulations of a 3D XY model, realized on a bc-lattice. Different sizes of systems are then either compared in pairs (pairwise fit) or all together (joint fit), in order to fit the correct value of the critical exponents to our sampled data. We find for the pairwise fitting procedure that ν = 0.6731(36) and η = 0.0351(39). Likewise, the joint fitting procedure yields ν = 0.6727(58) and η = 0.0349(49). The predictions for ν are very consistent with existing works, while the values for η are somewhat lower than expected from existing literature results. / Det experimentella värdet av den kritiska exponenten, ν, skiljer sig för närvarande från de teoretiska uppskattningarna som gjorts för λ-universalitetsklassen. Vi föreslår två nya metoder för att bestämma exponenterna ν och η, genom att använda effekterna från ändlig storleksskalning. De numeriska beräkningarna utförs med hjälp av Monte Carlo simulering av en tredimensionell XY-modell, realiserad på ett kubiskt gitter. Olika sys- temstorlekar jämförs sedan antingen parvis eller gemensamt, för att kunna anpassa värdet på de kritiska exponenterna till data. Vi erhåller ν = 0.6731(36) och η = 0.0351(39), för parvis anpassning, samt ν = 0.6727(58) and η = 0.0349(49) för gemensam anpassning. Det beräknade värdet på ν stämmer bra överens med andra arbeten, medan värdena på η är något lägre än förväntat från tidigare uppskattningar.