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Abstrak :
The book is based on the lectures given at the CIME school "Quantum many body systems" held in the summer of 2010. It provides a tutorial introduction to recent advances in the mathematics of interacting systems, written by four leading experts in the field, V. Rivasseau illustrates the applications of constructive Quantum Field Theory to 2D interacting electrons and their relation to quantum gravity, R. Seiringer describes a proof of Bose-Einstein condensation in the Gross-Pitaevski limit and explains the effects of rotating traps and the emergence of lattices of quantized vortices, J.-P. Solovej gives an introduction to the theory of quantum Coulomb systems and to the functional analytic methods used to prove their thermodynamic stability, finally, T. Spencer explains the supersymmetric approach to Anderson localization and its relation to the theory of random matrices. All the lectures are characterized by their mathematical rigor combined with physical insights.

Abstrak :
The dependence on the choice of these projections is studied in detail here and a method to overcome this ambiguity in DFT+U, by self-consistently determining the projections, is introduced. The author shows how nonorthogonal representations for electronic states may be used to construct these projections and, furthermore, how DFT+U may be implemented with a linearly increasing cost with respect to system size. The use of nonorthogonal functions in the context of electronic structure calculations is extensively discussed and clarified, with new interpretations and results, and, on this topic.

Abstrak :
This thesis presents a novel method for fabricating the smallest superconducting quantum interference device (SQUID) that resides on the apex of a very sharp tip. The nanoSQUID-on-tip displays a characteristic size down to 100 nm and a field sensitivity of 10^-3 Gauss/Hz^(1/2). A scanning SQUID microsope was constructed by gluing the nanoSQUID-on-tip to a quartz tuning-fork. This enabled the nanoSQUID to be scanned within nanometers of the sample surface, providing simultaneous images of sample topography and the magnetic field distribution. This microscope represents a significant improvement over the existing scanning SQUID techniques and is expected to be able to image the spin of a single electron.

Abstrak :
The present volume, covering such topics as quantum phase transitions of antiferromagnets and cuprate-based high-temperature superconductors, electronic liquid crystal phases, graphene physics, dynamical mean field theory applied to strongly correlated systems, transport through quantum dots, quantum information perspectives on many-body physics, frustrated magnetism, statistical mechanics of classical and quantum computational complexity, and integrable methods in statistical field theory.