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Renyi entropies are calculated for some multiparticle systems. Arguments are presented that measurements of Renyi entropies as functions of the average number of particles produced in high energy collisions carry important information on the produced system.

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Bose–Einstein correlations are being exploited to obtain information about the structure of the sources of hadrons in multiple particle production processes. In this paper the principles of this approach are described and some of the controversies about their implementation are discussed.

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The discovery of the superfluid phases of Helium 3 in 1971 opened the door to one of the most fascinating systems known in condensed matter physics. Superfluidity of Helium 3, originating from pair condensation of Helium 3 atoms, turned out to be the ideal test ground for many fundamental concepts of modern physics, such as macroscopic quantum phenomena, (gauge-)symmetries and their spontaneous breakdown, topological defects, etc. Thereby the superfluid phases of Helium 3 enriched condensed matter physics enormously. In particular, they contributed significantly — and continue to do so — to our understanding of various other physical systems, from heavy fermion and high-\(T_{\rm c}\) superconductors all the way to neutron stars, particle physics, gravity and the early universe. A simple introduction into the basic concepts and questions is presented.

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In this article we discuss two manifestations of electron correlation effects, namely the electrical characteristics and quantum critical phenomena observed in the NiS\(_{2-x}\)Se\(_x\) system, and the electrical properties and orbital ordering effects encountered in the V\(_2\)O\(_3\) system. Considerable emphasis will be placed on recent developments.

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We describe briefly a new method of approach to the interacting fermion and boson systems. Namely, we determine the explicit form of the single-particle wave functions \(\{w_{i}({r})\}\) appearing in the microscopic parameters of models in the second-quantization representation. The method is illustrated on the examples of H\(_{2}\) molecule and He atom, the Hubbard chain, and a ring of \(N\leq 10\) atoms.

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In these lectures, superconductivity in impure thin films close to the absolute zero of temperature is discussed. The behavior as function of the applied magnetic field and the amount of impurities suggests the presence of a superconductor–insulator transition at zero temperature. The theory of superconductivity in the limit where all the electrons become tightly bound in pairs is used to explain the main characteristics of the transition. In that limit, where the theory becomes equivalent to a phase-only theory, electron pairs exist on either side of the transition. The presentation is pedagogical in nature and includes exercises as a learning aid for those new to the field.

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We discuss the origin of topological defects in phase transitions and analyze their role as a “diagnostic tool” in the study of the non-equilibrium dynamics of symmetry breaking. Homogeneous second order phase transitions are the focus of our attention, but the same paradigm is applied to the cross-over and inhomogeneous transitions. The discrepancy between the experimental results in \(^3\)He and \(^4\)He is discussed in the light of recent numerical studies. The possible role of the Ginzburg regime in determining the vortex line density for the case of a quench in \(^4\)He is raised and tentatively dismissed. The difference in the anticipated origin of the dominant signal in the two (\(^3\)He and \(^4\)He) cases is pointed out and the resulting consequences for the subsequent decay of vorticity are noted. The possibility of a significant discrepancy between the effective field theory and (quantum) kinetic theory descriptions of the order parameter is briefly touched upon, using atomic Bose–Einstein condensates as an example.

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We review recent results obtained for the stripe phases in the Hubbard model. The experimentally observed half-filled (01) stripes with the filling of one hole per two domain wall atoms are stabilized by electron correlation effects. We show that the metallic stripe phases obtained using the dynamical mean-field approximation are stabilized by a pseudogap and are qualitatively different from insulating stripes derived from the one-particle (Hartree–Fock) simulations. They reproduce the doping dependence of the size of magnetic domains in (01) stripe phases and agree with the experimental data of angle resolved photoemission for La\(_{2-x}\)Sr\(_x\)CuO\(_4\).

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The theory of a Coulomb blockade phenomenon in carbon nanotubes is briefly reviewed and its experimental consequences are discussed. This review is based on the joint paper Y. Oreg, K. Byczuk, B.I. Halperin, Phys. Rev. Lett. 85, 365 (2000).

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Low-dimensional antiferromagnets in an external magnetic field provide an ideal illustration of the physics of quantum phase transitions. This theoretical analysis is motivated by the two-leg spin ladder geometry, which has been the subject of much experimental study in the material CuHpCl. The non-linear sigma model is used to characterise the quantum phases of the system, and the bond-operator description to discuss excitation spectra and quantum phase transitions between ground states.

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I review arguments for the existence of a critical point \(E\) in the QCD phase diagram as a function of temperature \(T\) and baryon chemical potential \(\mu \). I describe how heavy ion collision experiments at the SPS and RHIC can discover the tell-tale signatures of such a critical point, thus mapping this region of the QCD phase diagram. I then review the phenomena expected in cold dense quark matter: color superconductivity and color-flavor locking. I close with a snapshot of ongoing explorations of the implications of recent developments in our understanding of cold dense quark matter for the physics of compact stars.

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The formation and the structure of a mixed quark–nucleon phase in neutron star cores are studied, for different models of the nuclear symmetry energy. Simple parametrizations of the nuclear matter equation of state and the bag model for the quark phase are used. For lower values of the bag constant \(B\) the properties of the mixed phase do not depend strongly on the symmetry energy. For larger \(B\) we find that a critical pressure for the first quark droplets to form in the nucleon medium is strongly dependent on the nuclear symmetry energy, but the pressure at which last nucleons disappear is independent of it.

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We consider quantum superposition states in Bose–Einstein condensates. A decoherence rate for the Schrödinger cat is calculated and shown to be a significant threat to this macroscopic quantum superposition of BEC’s. An experimental scenario is outlined where the decoherence rate due to the thermal cloud is dramatically reduced thanks to trap engineering and “symmetrization” of the environment. We show that under the proposed scenario the Schrödinger cat belongs to an approximate decoherence-free pointer subspace.

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A consistent physical understanding of electronic, magnetic and spectroscopic properties of PrNi\(_{5}\) and ErNi\(_{5}\) is obtained with treating the \(f\) electrons as highly-correlated. The fine electronic structure, related with the atomic-like states and determined by the crystal-field and spin-orbit interactions, has been evaluated by means of different experimental techniques. The importance of the higher-order charge multipolar interactions and the local symmetry of the crystal field for the realized fine electronic structure and for the ground state are pointed out. We point out that a significant success of the crystal-field theory indicates on the substantial preservation of the atomic-like structure of the open-shell atoms even when they become part of solid.

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A modified form of the polar model of crystals is proposed. A peculiarity of the model is the dependence of the hopping integral on the site occupation. In the cases of weak and strong interactions the effective model Hamiltonian, which generalises the known forms of the effective Hamiltonian, is derived. It is shown that the model has the electron-hole asymmetry, in contrast to the Hubbard model. The metal-insulator transition within the model is also studied. The obtained results are compared with experimental data for narrow-band materials. Some specific narrow-band effects are discussed.