Transport properties of strongly correlated metals: A dynamical mean-field approach

The temperature dependence of the transport properties of the metallic phase of a frustrated Hubbard model on the hypercubic lattice at half-filling is calculated. Dynamical mean-held theory, which maps the Hubbard model onto a single impurity,Anderson model that is solved self-consistently, and becomes exact in the limit of large dimensionality, is used. As the temperature increases there is a smooth crossover from coherent Fermi liquid excitations at low temperatures to incoherent excitations at high temperatures. This crossover leads to a nonmonotonic temperature dependence for the resistance, thermopower, and Hall coefficient, unlike in conventional metals. The resistance smoothly increases from a quadratic temperature dependence at low temperatures to large values which can exceed the Mott-Ioffe-Regel value ha/e(2) (where a is a lattice constant) associated with mean free paths less than a lattice constant. Further signatures of the thermal destruction of quasiparticle excitations are a peak in the thermopower and the absence of a Drude peak in the optical conductivity. The results presented here are relevant to a wide range of strongly correlated metals, including transition metal oxides, strontium ruthenates, and organic metals.

Ferromagnetism, paramagnetism, and a Curie-Weiss metal in an electron-doped Hubbard model on a triangular lattice

Motivated by the unconventional properties and rich phase diagram of NaxCoO2 we consider the electronic and magnetic properties of a two-dimensional Hubbard model on an isotropic triangular lattice doped with electrons away from half-filling. Dynamical mean-field theory (DMFT) calculations predict that for negative intersite hopping amplitudes (t < 0) and an on-site Coulomb repulsion, U, comparable to the bandwidth, the system displays properties typical of a weakly correlated metal. In contrast, for t > 0 a large enhancement of the effective mass, itinerant ferromagnetism, and a metallic phase with a Curie-Weiss magnetic susceptibility are found in a broad electron doping range. The different behavior encountered is a consequence of the larger noninteracting density of states (DOS) at the Fermi level for t > 0 than for t < 0, which effectively enhances the mass and the scattering amplitude of the quasiparticles. The shape of the DOS is crucial for the occurrence of ferromagnetism as for t > 0 the energy cost of polarizing the system is much smaller than for t < 0. Our observation of Nagaoka ferromagnetism is consistent with the A-type antiferromagnetism (i.e., ferromagnetic layers stacked antiferromagnetically) observed in neutron scattering experiments on NaxCoO2. The transport and magnetic properties measured in NaxCoO2 are consistent with DMFT predictions of a metal close to the Mott insulator and we discuss the role of Na ordering in driving the system towards the Mott transition. We propose that the Curie-Weiss metal phase observed in NaxCoO2 is a consequence of the crossover from a bad metal with incoherent quasiparticles at temperatures T > T-* and Fermi liquid behavior with enhanced parameters below T-*, where T-* is a low energy coherence scale induced by strong local Coulomb electron correlations. Our analysis also shows that the one band Hubbard model on a triangular lattice is not enough to describe the unusual properties of NaxCoO2 and is used to identify the simplest relevant model that captures the essential physics in NaxCoO2. We propose a model which allows for the Na ordering phenomena observed in the system which, we propose, drives the system close to the Mott insulating phase even at large dopings.

On the Hartman-Grobman Theorem with Parameters

The Hartman-Grobman Theorem of linearization is extended to families of dynamical systems in a Banach space X, depending continuously on parameters. We prove that the conjugacy also changes continuously. The cases of nonlinear maps and flows are considered, and both in global and local versions, but global in the parameters. To use a special version of the Banach-Caccioppoli Theorem we introduce equivalent norms on X depending on the parameters. The functional setting is suitable for applications to some nonlinear evolution partial differential equations like the nonlinear beam equation.

Numerical and analytical approaches to strongly correlated electron systems

In this thesis we consider three different models for strongly correlated electrons, namely a multi-band Hubbard model as well as the spinless Falicov-Kimball model, both with a semi-elliptical density of states in the limit of infinite dimensions d, and the attractive Hubbard model on a square lattice in d=2. In the first part, we study a two-band Hubbard model with unequal bandwidths and anisotropic Hund's rule coupling (J_z-model) in the limit of infinite dimensions within the dynamical mean-field theory (DMFT). Here, the DMFT impurity problem is solved with the use of quantum Monte Carlo (QMC) simulations. Our main result is that the J_z-model describes the occurrence of an orbital-selective Mott transition (OSMT), in contrast to earlier findings. We investigate the model with a high-precision DMFT algorithm, which was developed as part of this thesis and which supplements QMC with a high-frequency expansion of the self-energy. The main advantage of this scheme is the extraordinary accuracy of the numerical solutions, which can be obtained already with moderate computational effort, so that studies of multi-orbital systems within the DMFT+QMC are strongly improved. We also found that a suitably defined Falicov-Kimball (FK) model exhibits an OSMT, revealing the close connection of the Falicov-Kimball physics to the J_z-model in the OSM phase. In the second part of this thesis we study the attractive Hubbard model in two spatial dimensions within second-order self-consistent perturbation theory. This model is considered on a square lattice at finite doping and at low temperatures. Our main result is that the predictions of first-order perturbation theory (Hartree-Fock approximation) are renormalized by a factor of the order of unity even at arbitrarily weak interaction (U->0). The renormalization factor q can be evaluated as a function of the filling n for 00, the q-factor vanishes, signaling the divergence of self-consistent perturbation theory in this limit. Thus we present the first asymptotically exact results at weak-coupling for the negative-U Hubbard model in d=2 at finite doping.

Exact quantum Monte Carlo methods for correlated electron systems

Thema dieser Arbeit ist die Entwicklung und Kombination verschiedener numerischer Methoden, sowie deren Anwendung auf Probleme stark korrelierter Elektronensysteme. Solche Materialien zeigen viele interessante physikalische Eigenschaften, wie z.B. Supraleitung und magnetische Ordnung und spielen eine bedeutende Rolle in technischen Anwendungen. Es werden zwei verschiedene Modelle behandelt: das Hubbard-Modell und das Kondo-Gitter-Modell (KLM). In den letzten Jahrzehnten konnten bereits viele Erkenntnisse durch die numerische Lösung dieser Modelle gewonnen werden. Dennoch bleibt der physikalische Ursprung vieler Effekte verborgen. Grund dafür ist die Beschränkung aktueller Methoden auf bestimmte Parameterbereiche. Eine der stärksten Einschränkungen ist das Fehlen effizienter Algorithmen für tiefe Temperaturen.rnrnBasierend auf dem Blankenbecler-Scalapino-Sugar Quanten-Monte-Carlo (BSS-QMC) Algorithmus präsentieren wir eine numerisch exakte Methode, die das Hubbard-Modell und das KLM effizient bei sehr tiefen Temperaturen löst. Diese Methode wird auf den Mott-Übergang im zweidimensionalen Hubbard-Modell angewendet. Im Gegensatz zu früheren Studien können wir einen Mott-Übergang bei endlichen Temperaturen und endlichen Wechselwirkungen klar ausschließen.rnrnAuf der Basis dieses exakten BSS-QMC Algorithmus, haben wir einen Störstellenlöser für die dynamische Molekularfeld Theorie (DMFT) sowie ihre Cluster Erweiterungen (CDMFT) entwickelt. Die DMFT ist die vorherrschende Theorie stark korrelierter Systeme, bei denen übliche Bandstrukturrechnungen versagen. Eine Hauptlimitation ist dabei die Verfügbarkeit effizienter Störstellenlöser für das intrinsische Quantenproblem. Der in dieser Arbeit entwickelte Algorithmus hat das gleiche überlegene Skalierungsverhalten mit der inversen Temperatur wie BSS-QMC. Wir untersuchen den Mott-Übergang im Rahmen der DMFT und analysieren den Einfluss von systematischen Fehlern auf diesen Übergang.rnrnEin weiteres prominentes Thema ist die Vernachlässigung von nicht-lokalen Wechselwirkungen in der DMFT. Hierzu kombinieren wir direkte BSS-QMC Gitterrechnungen mit CDMFT für das halb gefüllte zweidimensionale anisotrope Hubbard Modell, das dotierte Hubbard Modell und das KLM. Die Ergebnisse für die verschiedenen Modelle unterscheiden sich stark: während nicht-lokale Korrelationen eine wichtige Rolle im zweidimensionalen (anisotropen) Modell spielen, ist in der paramagnetischen Phase die Impulsabhängigkeit der Selbstenergie für stark dotierte Systeme und für das KLM deutlich schwächer. Eine bemerkenswerte Erkenntnis ist, dass die Selbstenergie sich durch die nicht-wechselwirkende Dispersion parametrisieren lässt. Die spezielle Struktur der Selbstenergie im Impulsraum kann sehr nützlich für die Klassifizierung von elektronischen Korrelationseffekten sein und öffnet den Weg für die Entwicklung neuer Schemata über die Grenzen der DMFT hinaus.

A Mean Value Theorem for non Differentiable Mappings in Banach Spaces

We prove that if f is a real valued lower semicontinuous function on a Banach space X and if there exists a C^1, real valued Lipschitz continuous function on X with bounded support and which is not identically equal to zero, then f is Lipschitz continuous of constant K provided all lower subgradients of f are bounded by K. As an application, we give a regularity result of viscosity supersolutions (or subsolutions) of Hamilton-Jacobi equations in infinite dimensions which satisfy a coercive condition. This last result slightly improves some earlier work by G. Barles and H. Ishii.

Generalized Fractional Evolution Equation

2000 Mathematics Subject Classification: Primary 46F25, 26A33; Secondary: 46G20

Reduction of infinite dimensional systems to finite dimensions: Compact convergence approach

We consider parameter dependent semilinear evolution problems for which, at the limit value of the parameter, the problem is finite dimensional. We introduce an abstract functional analytic framework that applies to many problems in the existing literature for which the study of asymptotic dynamics can be reduced to finite dimensions via the invariant manifolds technique. Some practical models are considered to show wide applicability of the theory. © 2013 Society for Industrial and Applied Mathematics.

Cat(0) and Cat (-1) dimensions of torsion free

We show that a particular free-by-cyclic group has CAT(0) dimension equal to 2, but CAT(-1) dimension equal to 3. We also classify the minimal proper 2-dimensional CAT(0) actions of this group; they correspond, up to scaling, to a 1-parameter family of locally CAT(0) piecewise Euclidean metrics on a fixed presentation complex for the group. This information is used to produce an infinite family of 2-dimensional hyperbolic groups, which do not act properly by isometries on any proper CAT(0) metric space of dimension 2. This family includes a free-by-cyclic group with free kernel of rank 6.

Étude des transitions de phases dans le modèle de Higgs abélien en (2+1) dimensions et l'effet du terme de Chern-Simons

Nous avons investigué, via les simulations de Monte Carlo, les propriétés non-perturbatives du modèle de Higgs abélien en 2+1 dimensions sans et avec le terme de Chern-Simons dans la phase de symétrie brisée, en termes de ses excitations topologiques: vortex et anti-vortex. Le but du présent travail est de rechercher les phases possibles du système dans ce secteur et d'étudier l'effet du terme de Chern-Simons sur le potentiel de confinement induit par les charges externes trouvé par Samuel. Nous avons formulé une description sur réseau du modèle effectif en utilisant une tesselation tétraédrique de l'espace tridimensionnel Euclidien pour générer des boucles de vortex fermées. En présence du terme de Chern-Simons, dans une configuration donnée, nous avons formulé et calculé le nombre d'enlacement entre les différentes boucles de vortex fermées. Nous avons analysé les propriétés du vide et calculé les valeurs moyennes de la boucle de Wilson, de la boucle de Polyakov à différentes températures et de la boucle de 't Hooft en présence du terme de Chern-Simons. En absence du terme de Chern-Simons, en variant la masse des boucles de vortex, nous avons trouvé deux phases distinctes dans le secteur de la symétrie brisée, la phase de Higgs habituelle et une autre phase caractérisée par l'apparition de boucles infinies. D'autre part, nous avons trouvé que la force entre les charges externes est écrantée correpondant à la loi périmètre pour la boucle de Wilson impliquant qu'il n'y a pas de confinement. Cependant, après la transition, nous avons trouvé qu'il existe toujours une portion de charges externes écrantée, mais qu'après une charge critique, l'énergie libre diverge. En présence du terme de Chern-Simons, et dans la limite de constante de couplage faible de Chern-Simons nous avons trouvé que les comportements de la boucle de Wilson et de la boucle de 't Hooft ne changent pas correspondants à une loi périmètre, impliquant qu'il n'y a pas de confinement. De plus, le terme de Chern-Simons ne contribue pas à la boucle de Wilson.

On Some Infinite Convex Invariants

The present study on some infinite convex invariants. The origin of convexity can be traced back to the period of Archimedes and Euclid. At the turn of the nineteenth centaury , convexicity became an independent branch of mathematics with its own problems, methods and theories. The convexity can be sorted out into two kinds, the first type deals with generalization of particular problems such as separation of convex sets[EL], extremality[FA], [DAV] or continuous selection Michael[M1] and the second type involved with a multi- purpose system of axioms. The theory of convex invariants has grown out of the classical results of Helly, Radon and Caratheodory in Euclidean spaces. Levi gave the first general definition of the invariants Helly number and Radon number. The notation of a convex structure was introduced by Jamison[JA4] and that of generating degree was introduced by Van de Vel[VAD8]. We also prove that for a non-coarse convex structure, rank is less than or equal to the generating degree, and also generalize Tverberg’s theorem using infinite partition numbers. Compare the transfinite topological and transfinite convex dimensions

Effective action for QED with fermion self-interaction in D=2 and D=3 dimensions

In this work we discuss the effect of the quartic fermion self-interaction of Thirring type in QED in D=2 and D=3 dimensions. This is done through the computation of the effective action up to quadratic terms in the photon field. We analyze the corresponding nonlocal photon propagators nonperturbatively in k/m, where k is the photon momentum and m the fermion mass. The poles of the propagators were determined numerically by using the MATHEMATICA software. In D=2 there is always a massless pole whereas for strong enough Thirring coupling a massive pole may appear. For D=3 there are three regions in parameter space. We may have one or two massive poles or even no pole at all. The interquark static potential is computed analytically in D=2. We notice that the Thirring interaction contributes with a screening term to the confining linear potential of massive two-dimensional QED (QED(2)). In D=3 the static potential must be calculated numerically. The screening nature of the massive QED(3) prevails at any distance, indicating that this is a universal feature of D=3 electromagnetic interaction. Our results become exact for an infinite number of fermion flavors.

Bound states by a pseudoscalar Coulomb potential in one-plus-one dimensions

The Dirac equation is solved for a pseudoscalar Coulomb potential in a two-dimensional world. An infinite sequence of bounded solutions are obtained. These results are in sharp contrast with those ones obtained in 3 + 1 dimensions where no bound-state solutions are found. Next the general two-dimensional problem for pseudoscalar power-law potentials is addressed consenting us to conclude that a nonsingular potential leads to bounded solutions. The behaviour of the upper and lower components of the Dirac spinor for a confining linear potential nonconserving- as well as conserving-parity, even if the potential is unbounded from below, is discussed in some detail. (C) 2003 Elsevier B.V. All rights reserved.

Integrability and conformal symmetry in higher dimensions: A model with exact hopfion solutions

We use ideas on integrability in higher dimensions to define Lorentz invariant field theories with an infinite number of local conserved currents. The models considered have a two-dimensional target space. Requiring the existence of lagrangean and the stability of static solutions singles out a class of models which have an additional conformal symmetry. That is used to explain the existence of an ansatz leading to solutions with non-trivial Hopf charges. © SISSA/ISAS 2002.