This thesis addresses several challenging problems in low-dimensional systems, which have rarely or never been studied using quantum Monte Carlo methods. It begins with an investigation into weak van der Waals-like interactions in bilayer graphene and extends to graphene placed on top of boron nitride at four different stacking configurations. The in-plane optical phonon frequencies for the latter heterostructure as well as the out-of-plane phonon frequencies for both structures are calculated. We find that the binding energies (BEs) of these structures are almost within the same range and are less than 20 meV/atom. Although the phonon vibrations are comparable within both the diffusion quantum Monte Carlo (DMC) method and density functional theory (DFT), DFT gives quantitatively wrong BEs for vdW structures. Next, the BEs of 2D biexcitons are studied at different mass ratios and a variety of screening lengths. Our exact DMC results show that the BEs of biexcitons in different kinds of transition-metal dichalcogenides are in the range 15 − 30 meV bound at room temperature.
Besides 2D systems, the electronic properties of 1D hydrogen-terminated oligoynes and polyyne are studied by calculating their DMC quasiparticle and excitonic gaps. By minimising the DMC energy of free-standing polyyne with respect to the lattice constant and the bond-length alternation, DMC predicts geometry in agreement with that obtained by accurate quantum chemistry methods. The DMC longitudinal optical phonon is within the range of experimental values. Our results confirm that DMC is capable of accurately describing Peierls-distorted materials.