Numerical and analytical approaches to strongly correlated electron systems

In this thesis we consider three different models for strongly correlated electrons, namely a multi-band Hubbard model as well as the spinless Falicov-Kimball model, both with a semi-elliptical density of states in the limit of infinite dimensions d, and the attractive Hubbard model on a square lattice in d=2. In the first part, we study a two-band Hubbard model with unequal bandwidths and anisotropic Hund's rule coupling (J_z-model) in the limit of infinite dimensions within the dynamical mean-field theory (DMFT). Here, the DMFT impurity problem is solved with the use of quantum Monte Carlo (QMC) simulations. Our main result is that the J_z-model describes the occurrence of an orbital-selective Mott transition (OSMT), in contrast to earlier findings. We investigate the model with a high-precision DMFT algorithm, which was developed as part of this thesis and which supplements QMC with a high-frequency expansion of the self-energy. The main advantage of this scheme is the extraordinary accuracy of the numerical solutions, which can be obtained already with moderate computational effort, so that studies of multi-orbital systems within the DMFT+QMC are strongly improved. We also found that a suitably defined Falicov-Kimball (FK) model exhibits an OSMT, revealing the close connection of the Falicov-Kimball physics to the J_z-model in the OSM phase. In the second part of this thesis we study the attractive Hubbard model in two spatial dimensions within second-order self-consistent perturbation theory. This model is considered on a square lattice at finite doping and at low temperatures. Our main result is that the predictions of first-order perturbation theory (Hartree-Fock approximation) are renormalized by a factor of the order of unity even at arbitrarily weak interaction (U->0). The renormalization factor q can be evaluated as a function of the filling n for 00, the q-factor vanishes, signaling the divergence of self-consistent perturbation theory in this limit. Thus we present the first asymptotically exact results at weak-coupling for the negative-U Hubbard model in d=2 at finite doping.

Strategies to identify and further characterize novel and known genes involved in regulation of skeletogenesis

372 osteochondrodysplasias and genetically determined dysostoses were reported in 2007 [Superti-Furga and Unger, 2007]. For 215 of these conditions, an association with one or more genes can be stated, while the molecular changes for the remaining syndromes remain illusive to date. Thus, the present dissertation aims at the identification of novel genes involved in processes regarding cartilage/ bone formation, growth, differentiation and homeostasis, which may serve as candidate genes for the above mentioned conditions. Two different approaches were undertaken. Firstly, a high throughput EST sequencing project from a human fetal cartilage library was performed to identify novel genes in early skeletal development (20th week of gestation until 2nd year of life) that could be investigated as potential candidate genes. 5000 EST sequences were generated and analyzed representing 1573 individual transcripts, corresponding to known (1400) and to novel, yet uncharacterized genes (173). About 7% of the proteins were already described in cartilage/ bone development or homeostasis, showing that the generated library is tissue specific. The remaining profile of this library was compared to previously published libraries from different time points (8th–12th, 18th–20th week and adult human cartilage) that also showed a similar distribution, reflecting the quality of the presented library analyzed. Furthermore, three potential candidate genes (LRRC59, CRELD2, ZNF577) were further investigated and their potential involvement in skeletogenesis was discussed. Secondly, a disease-orientated approach was undertaken to identify downstream targets of LMX1B, the gene causing Nail-Patella syndrome (NPS), and to investigate similar conditions. Like NPS, Genitopatellar syndrome (GPS) is characterized by aplasia or hypoplasia of the patella and renal anomalies. Therefore, six GPS patients were enrolled in a study to investigate the molecular changes responsible for this relatively rare disease. A 3.07 Mb deletion including LMX1B and NR5A1 (SF1) was found in one female patient that showed features of both NPS and GPS and investigations revealed a 46,XY karyotype and ovotestes indicating true hermaphroditism. The microdeletion was not seen in any of the five other patients with GPS features only, but a potential regulatory element between the two genes cannot be ruled out yet. Since Lmx1b is expressed in the dorsal limb bud and in podocytes, proteomic approaches and expression profiling were performed with murine material of the limbs and the kidneys to identify its downstream targets. After 2D-gel electrophoresis with protein extracts from E13.5 fore limb buds and newborn kidneys of Lmx1b wild type and knock-out mice and mass spectrometry analysis, only two proteins, agrin and carbonic anhydrase 2, remained of interest, but further analysis of the two genes did not show a transcriptional down regulation by Lmx1b. The focus was switched to expression profiles and RNA from newborn Lmx1b wild type and knock-out kidneys was compared by microarray analysis. Potential Lmx1b targets were almost impossible to study, because of the early death of Lmx1b deficient mice, when the glomeruli, containing podocytes, are still immature. Because Lmx1b is also expressed during limb development, RNA from wild type and knock-out Lmx1b E11.5 fore limb buds was investigated by microarray, revealing four potential Lmx1b downstream targets: neuropilin 2, single-stranded DNA binding protein 2, peroxisome proliferative activated receptor, gamma, co-activator 1 alpha, and short stature homeobox 2. Whole mount in situ hybridization strengthened a potential down regulation of neuropilin 2 by Lmx1b, but further investigations including in situ hybridization and protein-protein interaction studies will be needed.