Integer and fractional quantum Hall phases are prototypical examples of topologically ordered phases of matter. A standard setting for studying these phases is in continuum two-dimensional electron fluids with broken time-reversal symmetry, where the single particle bands are familiar Landau levels. In this thesis, we explore aspects of quantum Hall fluids in the presence of periodic lattice potential. We focus mostly on the perturbative, Harper-Hofstadter regime of low magnetic flux density, where the single-particle bands are effectively mixtures of Landau levels. We study the stability of strongly-correlated fractional quantum Hall fluids in relation to the geometry of single-particle Chern bands in tight-binding models. We first treat the case of the Hofstadter model and show the convergence of the geometry of Hofstadter bands to that of Landau levels in the limit of small flux density. We then introduce and study a new series of tight-binding models which do not necessarily converge to Landau levels but still host fractional phases. In each case, we observe correlations between band geometry and many-body gaps consistent with the "geometric stability hypothesis." Finally, we give a perturbative calculation of the current response of Chern bands to inhomogeneous electric fields. In continuum quantum Hall fluids with galilean symmetry, the O(q2) contribution to this response is related to the Hall viscosity, which measures the linear response to variations in spatial geometry in time-reversal odd two-dimensional fluids.

My thesis focuses on the effect of temperatures, the strengths of an external magnetic field, and the presence of impurities on Integer Quantum Hall Effect. The results show that the ratios of impurity strength over hopping strength less than 3 preserves gaps between Landau Levels (LLs) on a Hall system, a 2-dimensional 10x10-square lattice with an external magnetic field applied perpendicular to the plane. On the same Hall system, the transverse Hall conductance simulated using Kubo formula is integer multiples of e2/h between the LLs. As the magnetic flux decreased from 1/4 to 1/16, the transverse Hall conductance increased from e2/h to 4*e2/h on a 16x16 square lattice. The simulation on a 16x16-honeycomb lattice shows the increase of temperature causes Hall conductance to make smooth transitions from quantum phenomena toward classical characteristics as the slope decreases and the stepwise plot starts to disappear. On a 32x32- honeycomb lattice, the effect of increasing magnetic flux on the Hall conductance is the same as before, decreasing the Hall conductance. Finally, I explored the difference between LLs of toroidal geometry and the LLs of cylindrical geometry on a 10x10-square lattice. The cylindrical geometry shows continuous flow up of eigenenergies, each associated with its instantaneous eigenstates.

This graduate-level textbook is the first pedagogical synthesis of the field of topological insulators and superconductors, one of the most exciting areas of research in condensed matter physics. Presenting the latest developments, while providing all the calculations necessary for a self-contained and complete description of the discipline, it is ideal for graduate students and researchers preparing to work in this area, and it will be an essential reference both within and outside the classroom. The book begins with simple concepts such as Berry phases, Dirac fermions, Hall conductance and its link to topology, and the Hofstadter problem of lattice electrons in a magnetic field. It moves on to explain topological phases of matter such as Chern insulators, two- and three-dimensional topological insulators, and Majorana p-wave wires. Additionally, the book covers zero modes on vortices in topological superconductors, time-reversal topological superconductors, and topological responses/field theory and topological indices. The book also analyzes recent topics in condensed matter theory and concludes by surveying active subfields of research such as insulators with point-group symmetries and the stability of topological semimetals. Problems at the end of each chapter offer opportunities to test knowledge and engage with frontier research issues. Topological Insulators and Topological Superconductors will provide graduate students and researchers with the physical understanding and mathematical tools needed to embark on research in this rapidly evolving field.

Presenting the physics of the most challenging problems in condensed matter using the conceptual framework of quantum field theory, this book is of great interest to physicists in condensed matter and high energy and string theorists, as well as mathematicians. Revised and updated, this second edition features new chapters on the renormalization group, the Luttinger liquid, gauge theory, topological fluids, topological insulators and quantum entanglement. The book begins with the basic concepts and tools, developing them gradually to bring readers to the issues currently faced at the frontiers of research, such as topological phases of matter, quantum and classical critical phenomena, quantum Hall effects and superconductors. Other topics covered include one-dimensional strongly correlated systems, quantum ordered and disordered phases, topological structures in condensed matter and in field theory and fractional statistics.

After a foreword by Klaus von Klitzing, the first chapters of this book discuss the prehistory and the theoretical basis as well as the implications of the discovery of the Quantum Hall effect on superconductivity, superfluidity, and metrology, including experimentation. The second half of this volume is concerned with the theory of and experiments on the many body problem posed by fractional effect. Specific unsolved problems are mentioned throughout the book and a summary is made in the final chapter. The quantum Hall effect was discovered on about the hundredth anniversary of Hall's original work, and the finding was announced in 1980 by von Klitzing, Dorda and Pepper. Klaus von KIitzing was awarded the 1985 Nobel prize in physics for this discovery.