Studying the physics of low dimensional systems has been a fruitful branch of condensed matter physics. A high level of control in experiments and the existence of numerous powerful analytical and numerical techniques for investigating these systems have made experimental and theoretical study of this subject accessible and rich.
A large portion of this thesis is devoted to the theoretical study of a class of two dimensional systems, the so-called moiré structures: these are structures made by stacking incommensurate layered materials, where e.g. a twist or a lattice mismatch between layers results in the formation of a large scale spatial pattern called the moiré pattern.
Different aspects of moiré systems are discussed; starting with the case of twisted bilayer graphene (TBG), first, the effects of an external magnetic field on TBG are studied. Hofstadter spectra and semiclassical analyses are carried out and it is shown that the rich band structure of TBG near its magic angle results in a nontrivial Landau level structure even at the noninteracting level that is different from naive expectations. Next, the quantized anomalous Hall effect (QAHE) that is observed experimentally in TBG is considered; through extensive Hartree-Fock computations, the regimes in which the QAHE is expected are obtained.
Another moiré platform which is considered in this thesis is moiré structures made of intrinsic Van der Waals magnets. A general methodology for studying them theoretically is introduced and it is utilized to analyze various cases of such moiré magnets; first, twisted bilayers of antiferromagnets and ferromagnets are considered and it is shown that a rich phase diagram exists for such systems when different parameters in the system are tuned. Considering next the case of heterobilayer moiré magnets made of ferromagnetic and antiferromagnetic layers, we show that more interesting magnetic textures such as skyrmion lattices could potentially be realized.
The last topic that is covered in this thesis is numerical computation in low dimensional many body systems. It has been known that many body computational methods in the continuum experience difficulty compared to lattice models. Having this in mind, we propose a method using the wavelet basis for many body computations in the continuum; in order to tackle the difficulty mentioned above, a fine graining procedure is introduced which is general and can be used in combination with classical or quantum variational approaches; ultimately, we use tensor network computations to exhibit the usefulness of this procedure.
