249 research outputs found

    The Local Semicircle Law for Random Matrices with a Fourfold Symmetry

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    We consider real symmetric and complex Hermitian random matrices with the additional symmetry hxy=hN−x,N−yh_{xy}=h_{N-x,N-y}. The matrix elements are independent (up to the fourfold symmetry) and not necessarily identically distributed. This ensemble naturally arises as the Fourier transform of a Gaussian orthogonal ensemble (GOE). It also occurs as the flip matrix model - an approximation of the two-dimensional Anderson model at small disorder. We show that the density of states converges to the Wigner semicircle law despite the new symmetry type. We also prove the local version of the semicircle law on the optimal scale.Comment: 20 pages, to appear in J. Math. Phy

    Semi-classical analysis of non self-adjoint transfer matrices in statistical mechanics. I

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    We propose a way to study one-dimensional statistical mechanics models with complex-valued action using transfer operators. The argument consists of two steps. First, the contour of integration is deformed so that the associated transfer operator is a perturbation of a normal one. Then the transfer operator is studied using methods of semi-classical analysis. In this paper we concentrate on the second step, the main technical result being a semi-classical estimate for powers of an integral operator which is approximately normal.Comment: 28 pp, improved the presentatio

    Interacting Fermi liquid at finite temperature: Part I: Convergent Attributions

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    Using the method of continuous renormalization group around the Fermi surface, we prove that a two-dimensional jellium interacting system of Fermions at low temperature T is a Fermi liquid (analytic in the coupling constant g) for g < const/|logT|, and satisfying uniform bounds on the first and second derivatives of the selfenergy. This bound is also a step in the program of rigorous (non-perturbative) study of the BCS phase transition for many Fermions systems; it proves in particular that in dimension two the transition temperature (if any) must be non-perturbative in the coupling constant. The proof is organized into two parts: the present paper deals with the convergent contributions, and a companion paper (Part II) deals with the renormalization of dangerous two point subgraphs and achieves the proof.Comment: Latex, 54 pages, 6 figures, minor change

    One-loop β\beta functions of a translation-invariant renormalizable noncommutative scalar model

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    Recently, a new type of renormalizable ϕ4⋆4\phi^{\star 4}_{4} scalar model on the Moyal space was proved to be perturbatively renormalizable. It is translation-invariant and introduces in the action a a/(θ2p2)a/(\theta^2p^2) term. We calculate here the β\beta and γ\gamma functions at one-loop level for this model. The coupling constant βλ\beta_\lambda function is proved to have the same behaviour as the one of the ϕ4\phi^4 model on the commutative R4\mathbb{R}^4. The βa\beta_a function of the new parameter aa is also calculated. Some interpretation of these results are done.Comment: 13 pages, 3 figure

    Anderson localization vs. Mott-Hubbard metal-insulator transition in disordered, interacting lattice fermion systems

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    We review recent progress in our theoretical understanding of strongly correlated fermion systems in the presence of disorder. Results were obtained by the application of a powerful nonperturbative approach, the Dynamical Mean-Field Theory (DMFT), to interacting disordered lattice fermions. In particular, we demonstrate that DMFT combined with geometric averaging over disorder can capture Anderson localization and Mott insulating phases on the level of one-particle correlation functions. Results are presented for the ground-state phase diagram of the Anderson-Hubbard model at half filling, both in the paramagnetic phase and in the presence of antiferromagnetic order. We find a new antiferromagnetic metal which is stabilized by disorder. Possible realizations of these quantum phases with ultracold fermions in optical lattices are discussed.Comment: 25 pages, 5 figures, typos corrected, references update
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