Electronic structure of extended systems in strong magnetic fields

This project is devoted to the development of novel computational methods for materials in strong magnetic fields. Quantum mechanical systems often show peculiar and sometimes unexpected behavior in response to magnetic fields - consider, e.g., the integer and fractional quantum Hall effects, the Aharonov-Bohm effect, the so-called ``Hofstadter-butterfly`` spectrum or the striking conduction properties of graphene. Phenomena like these not only have elucidated many important aspects of quantum physics, but also offer interesting possibilities for future technological applications. Our proposal is additionally motivated by recent experimental prospects to reach field strengths of up to 100 Tesla. However, while electronic structure calculations in the absence of magnetic fields are fairly standard nowadays, simulations of materials in strong magnetic fields are much less established, and a number of theoretical and methodological problems remain to be solved. The key issues addressed in this project are: 1. When Schrödinger-type equations are represented on a finite-dimensional basis set or on a numerical grid, gauge invariance is usually destroyed. This leads to a pronounced gauge-dependent error of the eigenfunctions and eigenvalues. It is thus crucial to develop methods that guarantee manifestly gauge invariant solutions of the Schrödinger equation. 2. For an infinitely extended, periodic system, Bloch`s theorem no longer holds when a finite magnetic field is applied. The band structure in the presence of a magnetic field is thus fundamentally different from the zero-field band structure, and standard methods for band-structure calculations cannot be applied. Partly due to these difficulties, simulations involving magnetic fields are often carried out perturbatively, i.e., calculations are performed at zero field and magnetic response properties are then computed using linear response theory. In many situations, however, magnetic fields are not sufficiently small to allow for a perturbative treatment. This is the case, e.g., for the Quantum Hall and Aharonov-Bohm effects and the Hofstadter spectrum mentioned above. Calculations for extended systems commonly use effective one-band Hamiltonians and the tight-binding approximation, an approach that has been criticized recently. The primary goal of our project is to develop ab initio simulation methods for extended systems in strong magnetic fields that go beyond these approximations. A strong focus of the project is on applying the developed techniques to simulations of realistic materials. We have selected the following applications, based on their relevance to both fundamental research and technology: 1. Fullerenes: The question whether Fullerenes are spherical aromatic molecules has been debated since their discovery. In recent years, magnetic criteria, such as nuclear magnetic resonance shifts of encapsulated molecules, have been discussed as indicators to decide on this matter. 2. Graphene ribbons: The recent discovery of free-standing, isolated graphene has ignited intense theoretical and experimental research, motivated mainly by the peculiar band structure of this material that results in unusual, "relativistic" magnetoconduction properties. Theoretical investigations of graphene in magnetic fields have so far mostly been performed at the tight-binding level of theory. The goal of this project is to perform ab initio calculations of graphene in strong magnetic fields, and to study the influence of the edge states on the band structure. 3. Carbon nanotubes: Pronounced oscillations of the band gap as a function of the external magnetic field have been observed in carbon nanotubes due to the Aharonov-Bohm effect. Our goal is to investigate possible experimental signatures of the Hofstadter butterfly in such systems, in particular, of the anomalous Hall effect predicted recently. A second objective is the study of nuclear magnetic resonance shifts of small molecules encapsulated in infinitely long nanotubes. 4. Quantum dot arrays: Quantum dot lattices with a period comparable to the cyclotron radius have already been suggested by Hofstadter as a system to experimentally detect the fractal energy spectra predicted by Harper`s equation. The goal here is again to find and investigate possible experimental signatures of the Hofstadter butterfly.