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This thesis contains three theoretical works about certain aspects of the interplay of electronic correlations and topology in the Hubbard model.
In the first part of this thesis, the applicability of elementary band representations (EBRs) to diagnose interacting topological phases, that are protected by spatial symmetries and time-reversal-symmetry, in terms of their single-particle Matsubara Green’s functions is investigated. EBRs for the Matsubara Green’s function in the zero-temperature limit can be defined via the topological Hamiltonian. It is found that the Green’s function EBR classification can only change by (i) a gap closing in the spectral function at zero frequency, (ii) the Green’s function becoming singular i.e. having a zero eigenvalue at zero frequency or (iii) the Green’s function breaking a protecting symmetry. As an example, the use of the EBRs for Matsubara Green’s functions is demonstrated on the Su-Schriefer-Heeger model with exact diagonalization.
In the second part the Two-Particle Self-Consistent approach (TPSC) is extended to include spin-orbit coupling (SOC). Time-reversal symmetry, that is preserved in the presence of SOC, is used to derive new TPSC self-consistency equations including SOC. SOC breaks spin rotation symmetry which leads to a coupling of spin and charge channel. The local and constant TPSC vertex then consists of three spin vertices and one charge vertex. As a test case to study the interplay of Hubbard interaction and SOC, the Kane-Mele-Hubbard model is studied. The antiferromagnetic spin fluctuations are the leading instability which confirms that the Kane-Mele-Hubbard model is an XY antiferromagnet at zero temperature. Mixed spin-charge fluctuations are found to be small. Moreover, it is found that the transversal spin vertices are more strongly renormalized than the longitudinal spin vertex, SOC leads to a decrease of antiferromagnetic spin fluctuations and the self-energy shows dispersion and sharp features in momentum space close to the phase transition.
In the third part TPSC with SOC is used to calculate the spin Hall conductivity in the Kane-Mele-Hubbard model at finite temperature. The spin Hall conductivity is calculated once using the conductivity bubble and once including vertex corrections. Vertex corrections for the spin Hall conductivity within TPSC corresponds to the analogues of the Maki-Thompson contributions which physically correspond to the excitation and reabsorption of a spin, a charge or a mixed spin-charge excitation by an electron. At all temperatures, the vertex corrections show a large contribution in the vicinity of the phase transition to the XY antiferromagnet where antiferromagnetic spin fluctuations are large. It is found that vertex corrections are crucial to recover the quantized value of −2e^2/h in the zero-temperature limit. Further, at non-zero temperature, increasing the Hubbard interaction leads to a decrease of the spin Hall conductivity. The results indicate that scattering of electrons off antiferromagnetic spin fluctuations renormalize the band gap. Decreasing the gap can be interpreted as an effective increase of temperature leading to a decrease of the spin Hall conductivity.
Folding of G-protein coupled receptors (GPCRs) according to the two-stage model (Popot, J. L., and Engelman, D. M. (1990) Biochemistry 29, 4031–4037) is postulated to proceed in 2 steps: partitioning of the polypeptide into the membrane followed by diffusion until native contacts are formed. Herein we investigate conformational preferences of fragments of the yeast Ste2p receptor using NMR. Constructs comprising the first, the first two, and the first three transmembrane (TM) segments, as well as a construct comprising TM1–TM2 covalently linked to TM7 were examined. We observed that the isolated TM1 does not form a stable helix nor does it integrate well into the micelle. TM1 is significantly stabilized upon interaction with TM2, forming a helical hairpin reported previously (Neumoin, A., Cohen, L. S., Arshava, B., Tantry, S., Becker, J. M., Zerbe, O., and Naider, F. (2009) Biophys. J. 96, 3187–3196), and in this case the protein integrates into the hydrophobic interior of the micelle. TM123 displays a strong tendency to oligomerize, but hydrogen exchange data reveal that the center of TM3 is solvent exposed. In all GPCRs so-far structurally characterized TM7 forms many contacts with TM1 and TM2. In our study TM127 integrates well into the hydrophobic environment, but TM7 does not stably pack against the remaining helices. Topology mapping in microsomal membranes also indicates that TM1 does not integrate in a membrane-spanning fashion, but that TM12, TM123, and TM127 adopt predominantly native-like topologies. The data from our study would be consistent with the retention of individual helices of incompletely synthesized GPCRs in the vicinity of the translocon until the complete receptor is released into the membrane interior.