We employ exact diagonalization with strong coupling expansion to the massless and massive Schwinger model. New results are presented for the ground state energy and scalar mass gap in the massless model, which improve the precision to nearly $10^{-9
} %$. We also investigate the chiral condensate and compare our calculations to previous results available in the literature. Oscillations of the chiral condensate which are present while increasing the expansion order are also studied and are shown to be directly linked to the presence of flux loops in the system.
We develop a systematic strong coupling approach for studying an extended t-V model with interactions of a finite range. Our technique is not based on the Bethe ansatz and is applicable to both integrable and non-integrable models. We illustrate our
technique by presenting analytic results for the ground state energy (up to order 7 in t/V), the current density and density-density correlations for integrable and non-integrable models with commensurate filling factors. We further present preliminary numerical results for incommensurate non-integrable models.
Numerical investigations of the XY model, the Heisenberg model and the J-J Heisenberg model are conducted, using the exact diagonalisation, the numerical renormalisation and the density matrix renormalisation group approach. The low-lying energy leve
ls are obtained and finite size scaling is performed to estimate the bulk limit values. The results are found to be consistent with the exact values. The DMRG results are found to be most promising. The Schwinger model is also studied using the exact diagonalisation and the strong coupling expansion. The massless, the massive model and the model with a background electric field are explored. Ground state energy, scalar and vector particle masses and order parameters are examined. The achieved values are observed to be consistent with previous results and theoretical predictions. Path to the future studies is outlined.
The two-dimensional Ising model is studied by performing computer simulations with one of the Monte Carlo algorithms - the worm algorithm. The critical temperature T_C of the phase transition is calculated by the usage of the critical exponents and t
he results are compared to the analytical result, giving a very high accuracy. We also show that the magnetic ordering of impurities distributed on a graphene sheet is possible, by simulating the properly constructed model using the worm algorithm. The value of T_C is estimated. Furthermore, the dependence of T_C on the interaction constants is explored. We outline how one can proceed in investigating this relation in the future.
