Hamiltonian Theory of Fractionally Filled Chern Bands, and Disorder in Quantum Hall Ferromagnets

University Of Kentucky Research Foundation, Lexington KY

Investigators

Abstract

NONTECHNICAL SUMMARY This award supports theoretical research and education aimed to investigate novel states of electrons. Electrons are generally thought to be indivisible which is true at room temperature and under the conditions under which solid state devices normally operate. Most important to this research are the quantum Hall states, which occur in a two-dimensional sheet of electrons, usually at an artificially engineered interface between semiconductors, cooled to temperatures less than one degree from absolute zero, and placed in an a very strong magnetic field perpendicular to the sheet. In the simplest such state the electron can be thought of as "split" into three objects known as composite fermions. Each composite fermion has a charge one-third that of the electron along with other exotic properties. The charge and other properties of the excitation depend on the particular state. Some of these states offer possible platforms for quantum computing. Conventional, semiconductor quantum Hall states need extreme conditions. In the past decade a new way of potentially realizing such states has been proposed where the environment inside some materials can generate the equivalent of very strong internal magnetic fields. The PI aims to study both conventional quantum Hall states and these newly discovered possibilities, known as topological band materials. The PI will investigate whether topological band materials can support states that have not been discovered before, even in traditional quantum Hall systems. A main goal of the research is to understand the properties of such states in an approximate analytical way where the underlying physics is clear. This will enable understanding the similarities and the differences between conventional quantum Hall states and those in topological band materials. The PI will also utilize this approach to investigate conventional quantum Hall materials subjected to elastic strain. Any real material contains lattice imperfections, substituted atoms, and defects, collectively known as disorder. The PI will develop a controlled approach to investigate the role of disorder where experiments on some quantum Hall states suggest that the effect of disorder is important. The PI will also contribute to the organization of Winter Schools in India and will participate in the reorganization of the University Honors Program which provides a mechanism for students to learn about many disciplines and benefit from experimental learning. TECHNICAL SUMMARY This award supports theoretical research and education aimed to investigate quantum Hall states and topological states in materials. It has recently been established that materials with strong spin-orbit coupling can form new types of insulators, known as topological insulators, because of the topological properties of the band structure. When such bands are full, they have a quantized Hall conductance. With partial filling and strong electron-electron interactions fractional quantum Hall-like states form. The research has two major thrusts: 1. Investigating novel states in fractionally filled topological bands: The PI will use an analytical approach to investigate Composite Fermion states in topological bands. (a) Ground state energies for gapped states at the principal fractions and collective excitations will be computed in the Hamiltonian approach developed for the fractional quantum Hall effect. (b) Transitions between principal fraction states of different spin will be investigated using ground state energy crossings. (c) Two different possibilities for the half-filled state, an electron fluid and a Composite Fermion fluid, will be investigated. The nature of the phase transition and low-energy excitations near the phase transition will be studied. (d) Edge states of fractionally filled topological bands will be studied using a conserving approximation. This is relevant for determining whether the topological band materials have excitations other than those of conventional fractional quantum Hall states. (e) Two time-reversed copies of topological bands are a model of a time-reversal invariant topological insulator. Fractionally filled states of strongly interacting electrons will be studied in this model. (f) Tilted fields or strain in the conventional fractional quantum Hall effects produces an anisotropy which has been measured. The PI will develop an analytical theory of such anisotopic states, and potential phase transitions into nematic-like states. 2. Elucidating the role of quenched disorder in quantum Hall ferromagnets: The prototypical system is the filling 1 bilayer, where experimental observations pose numerous challenges to theory, and where disorder seems to be essential. (a) The PI and collaborators will, at the first stage, mimic the nonperturbative effects of disorder by imposing a strong periodic potential on the quantum Hall system. Hartree-Fock and effective low-energy theories will be used to determine the generation of topological charges in response to the potential. (b) Collective excitations will be computed to derive an experimental signature in light scattering of such states. (c) A low-energy field theory will be constructed near the phase transitions between different arrangements of topological charge. (d) Weak disorder will be put in at this stage and renormalization group techniques will be used to determine the low-energy long-wavelength behavior near the transitions. The PI will also contribute to the organization of Winter Schools in India and will participate in the reorganization of the University Honors Program which provides a mechanism for students to learn about many disciplines and benefit from experimental learning.