615 research outputs found

    On Some Open Problems in Many-Electron Theory

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    Mel Levy and Elliott Lieb are two of the most prominent researchers who have dedicated their efforts to the investigation of fundamental questions in many-electron theory. Their results have not only revolutionized the theoretical approach of the field, but, directly or indirectly, allowed for a quantum jump in the computational treatment of realistic systems as well. For this reason, at the conclusion of our book where the subject is treated across different disciplines, we have asked Mel Levy and Elliott Lieb to provide us with some open problems, which they believe will be a worth challenge for the future also in the perspective of a synergy among the various disciplines.Comment: "Epilogue" chapter in "Many-Electron Approaches in Physics, Chemistry and Mathematics: A Multidisciplinary View", Volker Bach and Luigi Delle Site Eds. pages 411-416; Book Series: Mathematical Physics Studies, Springer International Publishing Switzerland, 2014. The original title has been modified in order to clarify the subject of the chapter out of the context of the boo

    Including a phase in the Bethe equations of the Hubbard model

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    We compute the Bethe equations of generalized Hubbard models, and study their thermodynamical limit. We argue how they can be connected to the ones found in the context of AdS/CFT correspondence, in particular with the so-called dressing phase problem. We also show how the models can be interpreted, in condensed matter physics, as integrable multi-leg Hubbard models.Comment: 30 page

    Mean-field evolution of fermions with singular interaction

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    We consider a system of N fermions in the mean-field regime interacting though an inverse power law potential V(x)=1/xαV(x)=1/|x|^{\alpha}, for α(0,1]\alpha\in(0,1]. We prove the convergence of a solution of the many-body Schr\"{o}dinger equation to a solution of the time-dependent Hartree-Fock equation in the sense of reduced density matrices. We stress the dependence on the singularity of the potential in the regularity of the initial data. The proof is an adaptation of [22], where the case α=1\alpha=1 is treated.Comment: 16 page

    Field evolution of the magnetic structures in Er2_2Ti2_2O7_7 through the critical point

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    We have measured neutron diffraction patterns in a single crystal sample of the pyrochlore compound Er2_2Ti2_2O7_7 in the antiferromagnetic phase (T=0.3\,K), as a function of the magnetic field, up to 6\,T, applied along the [110] direction. We determine all the characteristics of the magnetic structure throughout the quantum critical point at HcH_c=2\,T. As a main result, all Er moments align along the field at HcH_c and their values reach a minimum. Using a four-sublattice self-consistent calculation, we show that the evolution of the magnetic structure and the value of the critical field are rather well reproduced using the same anisotropic exchange tensor as that accounting for the local paramagnetic susceptibility. In contrast, an isotropic exchange tensor does not match the moment variations through the critical point. The model also accounts semi-quantitatively for other experimental data previously measured, such as the field dependence of the heat capacity, energy of the dispersionless inelastic modes and transition temperature.Comment: 7 pages; 8 figure

    Pauli's Principle in Probe Microscopy

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    Exceptionally clear images of intramolecular structure can be attained in dynamic force microscopy through the combination of a passivated tip apex and operation in what has become known as the "Pauli exclusion regime" of the tip-sample interaction. We discuss, from an experimentalist's perspective, a number of aspects of the exclusion principle which underpin this ability to achieve submolecular resolution. Our particular focus is on the origins, history, and interpretation of Pauli's principle in the context of interatomic and intermolecular interactions.Comment: This is a chapter from "Imaging and Manipulation of Adsorbates using Dynamic Force Microscopy", a book which is part of the "Advances in Atom and Single Molecule Machines" series published by Springer [http://www.springer.com/series/10425]. To be published late 201

    Smooth Entropy in Axiomatic Thermodynamics

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    Thermodynamics can be formulated in either of two approaches, the phenomenological approach, which refers to the macroscopic properties of systems, and the statistical approach, which describes systems in terms of their microscopic constituents. We establish a connection between these two approaches by means of a new axiomatic framework that can take errors and imprecisions into account. This link extends to systems of arbitrary sizes including very small systems, for which the treatment of imprecisions is pertinent to any realistic situation. Based on this, we identify the quantities that characterise whether certain thermodynamic processes are possible with entropy measures from information theory. In the error-tolerant case, these entropies are so-called smooth min and max entropies. Our considerations further show that in an appropriate macroscopic limit there is a single entropy measure that characterises which state transformations are possible. In the case of many independent copies of a system (the so-called i.i.d. regime), the relevant quantity is the von Neumann entropy. Transformations among microcanonical states are characterised by the Boltzmann entropy

    Blow-up profile of rotating 2D focusing Bose gases

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    We consider the Gross-Pitaevskii equation describing an attractive Bose gas trapped to a quasi 2D layer by means of a purely harmonic potential, and which rotates at a fixed speed of rotation Ω\Omega. First we study the behavior of the ground state when the coupling constant approaches a_a\_* , the critical strength of the cubic nonlinearity for the focusing nonlinear Schr{\"o}dinger equation. We prove that blow-up always happens at the center of the trap, with the blow-up profile given by the Gagliardo-Nirenberg solution. In particular, the blow-up scenario is independent of Ω\Omega, to leading order. This generalizes results obtained by Guo and Seiringer (Lett. Math. Phys., 2014, vol. 104, p. 141--156) in the non-rotating case. In a second part we consider the many-particle Hamiltonian for NN bosons, interacting with a potential rescaled in the mean-field manner a_NN2β1w(Nβx),with--a\_N N^{2\beta--1} w(N^{\beta} x), with wapositivefunctionsuchthat a positive function such that \int\_{\mathbb{R}^2} w(x) dx = 1.Assumingthat. Assuming that \beta < 1/2andthat and that a\_N \to a\_*sufficientlyslowly,weprovethatthemanybodysystemisfullycondensedontheGrossPitaevskiigroundstateinthelimit sufficiently slowly, we prove that the many-body system is fully condensed on the Gross-Pitaevskii ground state in the limit N \to \infty$

    Degenerate Stars and Gravitational Collapse in AdS/CFT

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    We construct composite CFT operators from a large number of fermionic primary fields corresponding to states that are holographically dual to a zero temperature Fermi gas in AdS space. We identify a large N regime in which the fermions behave as free particles. In the hydrodynamic limit the Fermi gas forms a degenerate star with a radius determined by the Fermi level, and a mass and angular momentum that exactly matches the boundary calculations. Next we consider an interacting regime, and calculate the effect of the gravitational back-reaction on the radius and the mass of the star using the Tolman-Oppenheimer-Volkoff equations. Ignoring other interactions, we determine the "Chandrasekhar limit" beyond which the degenerate star (presumably) undergoes gravitational collapse towards a black hole. This is interpreted on the boundary as a high density phase transition from a cold baryonic phase to a hot deconfined phase.Comment: 75 page

    Two remarks on generalized entropy power inequalities

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    This note contributes to the understanding of generalized entropy power inequalities. Our main goal is to construct a counter-example regarding monotonicity and entropy comparison of weighted sums of independent identically distributed log-concave random variables. We also present a complex analogue of a recent dependent entropy power inequality of Hao and Jog, and give a very simple proof.Comment: arXiv:1811.00345 is split into 2 papers, with this being on
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