Dirac spectra, summation formulae, and the spectral action

Noncommutative geometry is a source of particle physics models with matter Lagrangians coupled to gravity. One may associate to any noncommutative space (A, H, D) its spectral action, which is defined in terms of the Dirac spectrum of its Dirac operator D. When viewing a spin manifold as a noncommutative space, D is the usual Dirac operator. In this paper, we give nonperturbative computations of the spectral action for quotients of SU(2), Bieberbach manifolds, and SU(3) equipped with a variety of geometries. Along the way we will compute several Dirac spectra and refer to applications of this computation.

Scanning tunneling spectroscopic studies on high-temperature superconductors and Dirac materials

This thesis details the investigations of the unconventional low-energy quasiparticle excitations in electron-type cuprate superconductors and electron-type ferrous superconductors as well as the electronic properties of Dirac fermions in graphene and three-dimensional strong topological insulators through experimental studies using spatially resolved scanning tunneling spectroscopy (STS) experiments.

Magnetic-field- and temperature-dependent evolution of the spatially resolved quasiparticle spectra in the electron-type cuprate La_0.1Sr_0.9CuO₂ (La-112) T_C = 43 K, are investigated experimentally. For temperature (T) less than the superconducting transition temperature (TC), and in zero field, the quasiparticle spectra of La-112 exhibits gapped behavior with two coherence peaks and no satellite features. For magnetic field measurements at T < TC, first ever observation of vortices in La-112 are reported. Moreover, pseudogap-like spectra are revealed inside the core of vortices, where superconductivity is suppressed. The intra-vortex pseudogap-like spectra are characterized by an energy gap of V_PG = 8.5 ± 0.6 meV, while the inter-vortex quasiparticle spectra shows larger peak-to-peak gap values characterized by Δ_pk-pk(H) >V_PG, and Δ_pk-pk (0)=12.2 ± 0.8 meV > Δ_pk-pk (H > 0). The quasiparticle spectra are found to be gapped at all locations up to the highest magnetic field examined (H = 6T) and reveal an apparent low-energy cutoff at the V_PG energy scale.

Magnetic-field- and temperature-dependent evolution of the spatially resolved quasiparticle spectra in the electron-type "122" iron-based Ba(Fe_1-xCo_x)₂A_s2 are investigated for multiple doping levels (x = 0.06, 0.08, 0.12 with T_C= 14 K, 24 K, and 20 K). For all doping levels and the T < T_C, two-gap superconductivity is observed. Both superconducting gaps decrease monotonically in size with increasing temperature and disappear for temperatures above the superconducting transition temperature, T_C. Magnetic resonant modes that follow the temperature dependence of the superconducting gaps have been identified in the tunneling quasiparticle spectra. Together with quasiparticle interference (QPI) analysis and magnetic field studies, this provides strong evidence for two-gap sign-changing s-wave superconductivity.

Additionally spatial scanning tunneling spectroscopic studies are performed on mechanically exfoliated graphene and chemical vapor deposition grown graphene. In all cases lattice strain exerts a strong influence on the electronic properties of the sample. In particular topological defects give rise to pseudomagnetic fields (B ~ 50 Tesla) and charging effects resulting in quantized conductance peaks associated with the integer and fractional Quantum Hall States.

Finally, spectroscopic studies on the 3D-STI, Bi₂Se₃ found evidence of impurity resonance in the surface state. The impurities are in the unitary limit and the spectral resonances are localized spatially to within ~ 0.2 nm of the impurity. The spectral weight of the impurity resonance diverges as the Fermi energy approaches the Dirac point and the rapid recovery of the surface state suggests robust topological protection against perturbations that preserve time reversal symmetry.

Synthesis and characterization of 1,1-di-tert-butyldiazene

The synthesis and direct observation of 1,1-di-tert-butyldiazene (16) at -127°C is described. The absorption spectrum of a red solution of 1,1-diazene 16 reveals a structured absorption band with λ max at 506 run (Me_2O, -125°C). The vibrational spacing in S_1 is about 1200 cm^(-1). The excited state of 16 emits weakly with a single maximum at 715 run observed in the fluorescence spectrum (Me_2O:CD_2Cl_2, -196°C). The proton NMR spectrum of 16 occurs as a singlet at 1.41 ppm. Monitoring this NMR absorption at -94^0 ± 2°C shows that 1,1-diazene 16 decomposes with a first-order rate of 1.8 x 10^(-3) sec(-1) to form isobutane, isobutylene and hexarnethylethane. This rate is 10^8 and 10^(34) times faster than the thermal decomposition of the corresponding cis and trans 1,2-di-tert-butyldiazene isomers. The free energy of activation for decomposition of 1,1-diazene 16 is found to be 12.5 ± 0.2 kcal/mol at -94°C which is much lower than the values of 19.1 and 19.4 kcal/lmole calculated at -94°C for N-(2,2,6,6- tetramethylpiperidyl)nitrene (3) and N-(2,2,5,5- tetrarnethylpyrrolidyl)nitrene (4), respectively. This difference between 16 and the cyclic-1,1-diazenes 3 and 4 can be attributed to a large steric interaction between the tert-butyl groups in 1,1-diazene 16.

In order to investigate the nature of the singlet-triplet gap in 1,1-diazenes, 2,5-di-tert-butyl-N-pyrrolynitrene (22) was generated but was found to be too reactive towards dimerization to be persistent. In the presence of dimethylsulfoxide, however, N-pyrrolynitrene (22) can be trapped as N-(2,5-di-tert-butyl- N'-pyrrolyl)dimethylsulfoxirnine (38). N-(2,5-di-tert-butyl-N'-pyrrolyl)dimethylsulfoximine (38-d^6) exchanges with free dimethylsulfoxide at 50°C in solution, presumably by generation and retrapping of pyrrolynitrene 22.

I. The effect of N,N,N',N' -tetramethylethylenediamine on the Schlenk Equilibrium. II. The nature of the Di-Grignard reagent

I. The influence of N,N,N’,N’-tetramethylethylenediamine on the Schlenk equilibrium

The equilibrium between ethylmagnesium bromide, diethylmagnesium, and magnesium bromide has been studied by nuclear magnetic resonance spectroscopy. The interconversion of the species is very fast on the nmr time scale, and only an averaged spectrum is observed for the ethyl species. When N,N,N’,N’-tetramethylethylenediamine is added to solutions of these reagents in tetrahydrofuran, the rate of interconversion is reduced. At temperatures near -50°, two ethylmagnesium species have been observed. These are attributed to the different ethyl groups in ethylmagnesium bromide and diethylmagnesium, two of the species involved in the Schlenk equilibrium of Grignard reagents.

II. The nature of di-Grignard reagents

Di-Grignard reagents have been examined by nuclear magnetic resonance spectroscopy in an attempt to prove that dialkylmagnesium reagents are in equilibrium with alkylmagnesium halides. The di-Grignard reagents of compounds such as 1,4-dibromobutane have been investigated. The dialkylmagnesium form of this di-Grignard reagent can exist as an intramolecular cyclic species, tetramethylene-magnesium. This cyclic form would give an nmr spectrum different from that of the classical alkylmagnesium halide di-Grignard reagent. In dimethyl ether-tetrahydrofuran solutions of di-Grignard reagents containing N N,N,N’,N’-Tetramethylethylenediamine, evidence has been found for the existence of an intramolecular dialkylmagnesium species. This species is rapidly equilibrating with other forms, but at low temperatures, the rates of interconversion are reduced. Two species can be seen in the nmr spectrum at -50°. One is the cyclic species; the other is an open form.

Inversion of the carbon at the carbon-magnesium bond in di-Grignard reagents has also been studied. This process is much faster than in corresponding monofunctional Grignard reagents.