Abstract:
We apply a general method developed recently for the derivation of the diagonal representation of an arbitrary matrix valued quantum Hamiltonian to the particular case of Bloch electrons in an external electromagnetic field. We find the diagonal representation as a series expansion to the second order in $\hbar .$ This result is the basis for the determination of the effective in-band Hamiltonian of interacting Bloch electrons living in different energy bands. Indeed, the description of effects such as magnetic moment-moment interactions mediated by the magnetic part of the full electromagnetic interaction requires a computation to second order in $\hbar $. It is found that the electronic current is made of two contributions: the first one comes from the velocity and the second one is a magnetic moment current similar to the spin current for Dirac particles. This last contribution is responsible for the interaction between magnetic moments similarly to the spin-spin interaction in the Breit Hamiltonian for Dirac electrons in interaction.

Abstract:
The scattering diameters of Sgr A* and several nearby OH masers (~ 1" at 1 GHz) indicate that a region of enhanced scattering is along the line of sight to the Galactic center. We combine radio-wave scattering data and free-free emission and absorption measurements in a likelihood analysis that constrains the following parameters of the GC scattering region: The GC-scattering region separation, d; the angular extent of the region, \psi_l; the outer scale on which density fluctuations occur, l_0; and the gas temperature, T. The maximum likelihood estimates of these parameters are d = 133_{-80}^{+200} pc, 0.5 degrees <= \psi_l <~ 1 degrees, and (l_0/1 pc)^{2/3}T^{-1/2} = 10^{-7 +/- 0.8}. As host media for the scattering, we consider the photoionized surface layers of molecular clouds and the interfaces between molecular clouds and the 10^7 K ambient gas. We are unable to make an unambiguous determination, but we favor an interface model in which the scattering medium is hot (T ~ 10^6 K) and dense (n_e ~ 10 cm^{-3}). The GC scattering region produces a 1 GHz scattering diameter for an extragalactic source of 90", if the region is a single screen, or 180", if the region wraps around the GC, as appears probable. We modify the Taylor-Cordes model for the Galactic distribution of free electrons in order to include an explicit GC component. Pulsars seen through this region will have a dispersion measure of approximately 2000 pc cm^{-3}, of which 75% arises from the GC component. We stress the uniqueness of the GC scattering region, probably resulting from the high-pressure environment in the GC.

Abstract:
We study interacting electrons in a periodic potential and a uniform magnetic field ${\bf B}$ taking the spin-orbit interaction into account. We first establish a perturbation expansion for those electrons with respect to the Bloch states in zero field. It is shown that the expansion can be performed with the zero-field Feynman diagrams of satisfying the momentum and energy conservation laws. We thereby clarify the structures of the self-energy and the thermodynamic potential in a finite magnetic field. We also provide a prescription of calculating the electronic structure in a finite magnetic field within the density functional theory starting from the zero-field energy-band structure. On the basis of these formulations, we derive explicit expressions for the magnetic susceptibility of ${\bf B}\to{\bf 0}$ at various approximation levels on the interaction, particularly within the density functional theory, which include the result of Roth [J. Phys. Chem. Solids {\bf 23} (1962) 433] as the non-interacting limit. We finally study the de Haas-van Alphen oscillation in metals to show that quasiparticles at the Fermi level with the many-body effective mass are directly relevant to the phenomenon. The present argument may be more transparent than that by Luttinger [Phys. Rev. {\bf 121} (1961) 1251] of using the gauge invariance and has an advantage that the change of the band structure with the field may be incorporated.

Abstract:
A molecular description for magic-number configurations of interacting electrons in a quantum dot in high magnetic fields developed by one of the authors has been elaborated for four, five and six electron dots. For four electrons, the magic spin-singlet states are found to alternate between two different resonating valence bond (RVB)-like states. For the five-electron spin-polarized case, the molecular description is shown to work for the known phenomenon of magic-number sequences that correspond to both the N-fold symmetric ring configuration and a $(N-1)$-fold symmetric one with a center electron. A six-electron dot is shown here to have an additional feature in which inclusion of quantum mechanical mixing between classical configurations, which are deformed and degenerate, restores the N-fold symmetry and reproduces the ground-state energy accurately.

Abstract:
The distribution function for a system of interacting electrons in metals is multivalent in a certain region of wave vectors. One solution among many is isotropic. For other solutions the distribution of electrons over the wave vectors is anisotropic. In the simplest case, the anisotropy arises as a result of the repulsion between electrons in states with the wave vectors $\bf k$ and $-\hh\bf k$.

Abstract:
The transmission of two electrons through a region where they interact is found to be enhanced by a renormalization of the repulsive interaction. For a specific example of the single-particle Hamiltonian, which includes a strongly attractive potential, the renormalized interaction becomes attractive, and the transmission has a pronounced maximum as function of the depth of the single-electron attractive potential. The results apply directly to a simple model of scattering of two interacting electrons by a quantum dot.

Abstract:
Motivated by the recent finding of superconductivity in layered CoO_2 compounds, we investigate superconducting and magnetic instabilities of interacting electrons on the two-dimensional triangular lattice. Using a one-loop renormalization group scheme for weak to moderate coupling strengths, we find that for purely local interactions U>0 and small Fermi surfaces the renormalization group flow remains bounded down to very low scales and no superconducting or other instabilities can be detected. Antiferromagnetic exchange interactions J generate a wide density region with a d_{x^2-y^2}+id_{xy}-wave superconducting instability similar to recent proposals for the strongly correlated t-J model. For larger Fermi surface volumes the interactions flow to strong coupling also for purely local interactions U>0. We find a singlet pairing instability in the vicinity of strong magnetic ordering tendencies at three wavevectors for the van Hove filling.

Abstract:
Quantum entanglement is a concept commonly used with reference to the existence of certain correlations in quantum systems that have no classical interpretation. It is a useful resource to enhance the mutual information of memory channels or to accelerate some quantum processes as, for example, the factorization in Shor's Algorithm. Moreover, entanglement is a physical observable directly measured by the von Neumann entropy of the system. We have used this concept in order to give a physical meaning to the electron correlation energy in systems of interacting electrons. The electronic correlation is not directly observable, since it is defined as the difference between the exact ground state energy of the many--electrons Schroedinger equation and the Hartree--Fock energy. We have calculated the correlation energy and compared with the entanglement, as functions of the nucleus--nucleus separation using, for the hydrogen molecule, the Configuration Interaction method. Then, in the same spirit, we have analyzed a dimer of ethylene, which represents the simplest organic conjugate system, changing the relative orientation and distance of the molecules, in order to obtain the configuration corresponding to maximum entanglement.

Abstract:
The eigenstates and the scattering transmission for two interacting electrons are found exactly for I quantum dots, including the hybridization with the states on the leads. The results imply limitations on the validity of the Coulomb blockade picture. The ground states for I=1, and 2 on a one--dimensional chain (modeling single and double quantum dots) exhibit quantum delocalization and magnetic transitions. The effective transmission T of two interacting electrons through one impurity (I=1) is enhanced by a renormalization of the repulsive interaction, when one of the electrons is captured in a strongly localized state.

Abstract:
We construct a model for interacting electrons with strong spin orbit coupling in the pyrochlore iridates. We establish the importance of the direct hopping process between the Ir atoms and use the relative strength of the direct and indirect hopping as a generic tuning parameter to study the correlation effects across the iridates family. We predict novel quantum phase transitions between conventional and/or topologically non-trivial phases. At weak coupling, we find topological insulator and metallic phases. As one increases the interaction strength, various magnetic orders emerge. The topological Weyl semi-metal phase is found to be realized in these different orders, one of them being the all-in/all-out pattern. Our findings establish the possible magnetic ground states for the iridates and suggest the generic presence of the Weyl semi-metal phase in correlated magnetic insulators on the pyrochlore lattice. We discus the implications for existing and future experiments.