Development of efficient electron correlation methods for one- and two-dimensional extended systems and their applications

Oracio Navarro, C. Wang

Edoardo Fertitta

The clear distinction between metals and insulators is a well established and appartently simple concept which can be clarified by studying the behavior of the electrical conductivity. However, a theoretical description of the processes leading to one of the two extremes is no trivial task. The widely accepted band structure theory relies on an one-electron picture to characterize the metallic/insulating behavior of a material. However, in spite of the fact that it can be used to make successful predictions, this description does not capture one of the essential elements characterizing the nature of an electronic quantum system, i.e. the electron correlation. The key to the different behaviors is clearly contained in the correlated many- electron wave function, but the analysis of such a high-dimensional object in configuration space can be quite cumbersome. Therefore the development of powerful tools to simplify this task is necessary. The total position-spread (TPS) tensor, which was derived by Walter Kohn’s theory of the insulating state, plays a key role in the field of metal-insulator transitions (MITs). This tool provides information about the electron localization, which is strongly connected to the electrical conductivity. Therefore, in this work particular attention was paid to the calculation of the TPS tensor of small and extended systems. Moreover, part of this thesis focuses on a new formalism, which allows the study of spin mobility and spin entanglement and provides therefore deeper insight into the effects of electron correlation. In the framework of quantum information theory (QIT), the quantum entanglement can be quantified by means of measures such as the von Neumann entropy. Also, the analysis of the elements of the two-orbital reduced density matrix provides an overview of the complexity of the many-electron wave function. These insights were used in this work for the analysis of the metallic/insulating character of a system. However, in order to perform such investigastions, the problem of achieving an accurate description of the many- electron wave function has to be dealt with. Despite the tremedous progress in the efficiency of quantum-chemical methods, the treatement of extended systems using standard approaches is still a difficult (if not unfeasible) task. Therefore the development and testing of sophisticated, low-scaling methods is currently the subject of many investigations. In this work, two approaches were employed: 1) the ab-initio density matrix renormalization group (DMRG) approach which, based on a tensor product ansatz, provides very precise results and exploits QIT to improve its performance; 2) the method of increments (MoI) which, using a many-body expansion in terms of localized orbitals, yields accurate correlation energies. By using both methods on strongly correlated one-dimesional model systems, we worked on a new formalism of the method of increments which can be used to describe the whole dissociation curve without inconvenient discontinuities. Finally, aiming to calculate the total position-spread tensor of extended systems, we worked on the application of the MoI for quantities other than the correlation energy.

K. W. Becker, Peter Fulde

Eero Tölö

The quantum Hall effect observed in a two-dimensional electron gas exposed to a perpendicular magnetic field is one of the most important modern discoveries in condensed matter physics. Advances in technology have allowed both experimental and computer-based theoretical study of miniature-size counterparts of conventional quantum Hall devices. Due to their versatile tunable electronic and magnetic properties, these systems show great promise for future technological applications. This work investigates two-dimensional semiconductor quantum dots and extended quantum rings in the quantum Hall regime. Besides being interesting on purely theoretical grounds as experimentally reachable extreme quantum-mechanical interacting many-body systems, the systems considered have major applicational interest in the realm of quantum information processing. Our emphasis lies in the strongly correlated regime where the enhanced electron-electron interactions lead to a computationally hard problem. For the systems with a few electrons, we employ the in principle exact configuration interaction method, which allows accurate study of the effects of the confinement potential and the effective form of the electron-electron interaction. Larger systems are modeled by Monte Carlo and density functional based methods. Moreover, we develop a computational method based on the reduced density-matrix functional theory to study the physics at the strongly correlated fractional quantum Hall regime with large particle numbers. The main results are related to the interplay of electron correlation and magnetism as well as the effects of tuning the electron correlation through sample engineering. The results on magnetism may find applications in future spintronics devices and spin qubits. Meanwhile, the results on electron correlation are relevant for the attempts to utilize fractional quantum Hall states in topologically protected quantum computing.

Xin Zhao, Jun Liu, Yongxin Yao +2 more

Developing accurate and computationally efficient methods to calculate the electronic structure and total energy of correlated-electron materials has been a very challenging task in condensed matter physics and materials science. Recently, we have developed a correlation matrix renormalization (CMR) method which does not assume any empirical Coulomb interaction $U$ parameters and does not have double counting problems in the ground-state total energy calculation. The CMR method has been demonstrated to be accurate in describing both the bonding and bond breaking behaviors of molecules. In this study, we extend the CMR method to the treatment of electron correlations in periodic solid systems. Using a linear hydrogen chain as a benchmark system, we show that the results from the CMR method compare very well with those obtained recently by accurate quantum Monte Carlo (QMC) calculations. We also study the equation of states of three-dimensional crystalline phases of atomic hydrogen. We show that the results from the CMR method agree much better with the available QMC data in comparison with those from density functional theory and Hartree-Fock calculations.

Yu. P. Monarkha