ESI workshop December 1-4, 2004: Abstracts ----------------------------------------------------------------- Bruno Nachtergaele A Ferromagnetic Lieb-Mattis Theorem We prove ferromagnetic ordering of energy levels for XXX Heisenberg chains of any spin and XXZ spin chains with all spins equal to $\frac{1}{2}$. Ferromagnetic ordering means that the minimum energies in the invariant subspaces of fixed total spin are monotone decreasing as a function of the total spin. This result provides a ferromagnetic analogue of the well-known theorem by Lieb and Mattis about ordering of energy levels in antiferromagnetic and ferrimagnetic systems on bipartite graphs. Some applications will be discussed. ----------------------------------------------------------------- Giosi Benfatto Generalized Luttinger liquid construction by Ward Identities: Part 1 There are up to now two different ways to prove the key property on which our Luttinger liquid rigorous construction rests, the vanishing of the leading part of the Beta function. The first one was developed in the last years and is based in an essential way on the exact Mattis-Lieb solution of the Luttinger model. More recently, we found a new proof, based on the Ward identities obtained by a chiral local gauge transformation, applied to a Luttinger model with ultraviolet and infrared cutoffs. This is an old approach in the physical literature, but its implementation in an RG scheme is not trivial at all, because the ultraviolet and infrared cutoffs destroy local Gauge invariance and produce "correction terms'' with respect to the formal Ward identities. We discover however a new set of identities, called "Correction Identities", relating the corrections to the Schwinger functions. By combining Ward and Correction identities with a Dyson equation, the vanishing of the Beta function follows, so that the infrared cutoff can be removed. As a byproduct, even the ultraviolet cutoff can be removed, after a suitable ultraviolet renormalization, so that a Quantum Field Theory corresponding to the Thirring model is constructed, showing the phenomenon of Chiral anomaly. ----------------------------------------------------------------- Vieri Matropietro Generalized Luttinger liquid construction by Renormalization Group and Ward Identities: Part 2 We describe a general and rigorous method for constructing systems behaving as Luttinger liquids or generalized Luttinger liquids (that is including for instance mass terms). Our method can be applied to a large class of systems, where exact solutions are lacking and bosonization methods fail. We explain the general strategy in two important examples, the Ashkin-Teller model (isotropic or anisotropic) , describing two 2d Ising models with different parameters and coupled by a quartic interaction, and the 1d Hubbard model (including next to nearest neigbor interaction, external potentials, magnetic fields and so on). The method is based on multiscale analysis implemented with Ward identities; the Grassmann integrals are expressed as analytic series in a set of running coupling constants, and the running coupling constants verify a recursive equation called Beta function, which can be written as sum of two terms, one (the principal part) which is common to all the models we can treat, and the other which is model dependent. The running coupling constants remain in the analiticity radius only if on can prove that the principal part of the Beta function is vanishing; such property is a consequence of approximate Ward identities (consequence of an hidden and approximate local Gauge invariance) described in the first part of the talk. ----------------------------------------------------------------- Ruedi Seiler Q-Shannon-MacMillan Theorem for the Heisenberg Chain is the equivalence of ensembles ----------------------------------------------------------------- Volker Bach Ferromagnetism in the Hartree-Fock-Z Approximation for the Hubbard Model In a joint work with E. Lieb and M. Travaglia, we study the Hartree-Fock approximation for the Hubbard model, additionally requiring that all spin components are parallel or antiparallel to the Z-axis. We prove that the ground state for this model is the saturated ferromagnet, provided the filling is small and the coupling U>0 is sufficiently large. ----------------------------------------------------------------- Harald Grosse Renormalization of scalar quantum field theory on noncommutative R^4 After a short introduction into the formulation of noncommutative field theory and the discussion of the IR/UV mixing, I review the main ideas and techniques of the proof with Raimar Wulkenhaar that the duality-covariant four-dimensional noncommutative scalar model is renormalizable to all orders. This includes the reformulation as a dynamical matrix model, the solution of the free theory by orthogonal polynomials as well as the renormalization by flow equations involving power-counting theorems for ribbon graphs drawn on Riemann surfaces. Finally the behavior of the beta function is discussed. ----------------------------------------------------------------- Heide Narnhofer Mesoscopic observables in mean field theories In mean field theories equilibrium states are typically product states.Their fluctuation algebra in the sense of Verbeure et.al can be calculated and coincides with the Weyl algebra over R. The time evolution differs from the local effective time evolution. However stability properties are related on the microscopic and mesoscopic level. The BCS model below the critical temperature is discussed. The time evolution for the mesoscopic algebra of the factor states is unstable. If the limiting of the mesoscopic algebra and of the groundstate is coupled, a different scaling becomes necessary. As mesoscopic algebra we obtain the Weylalgebra on the Torus with trivial time evolution. This algebra reflects the Meissner effect and the Josephson effect. ----------------------------------------------------------------- Walter Thirring A mean-field mechanism for high-T_c superconductivity ----------------------------------------------------------------- Stephane Afchain Some results on the two-dimensional Hubbard model at half-filling We present a study of the 2D-Hubbard model, whose Fermi surface at half-filling is exactly a square. The control of the perturbative series defining the correlation functions is carried out using constructive methods, which allow to prove that the model is analytic in the domain $| \lambda | \leq \frac{K}{\log^2 T}$. But we will emphasize a more interesting result : some second derivative of the self-energy diverges sufficiently quickly as $T \rightarrow 0^+$ to show that Salmhofer's criterion for a Fermi liquid behaviour is violated, and therefore that the model exhibits some features similar to a Luttinger liquid. ----------------------------------------------------------------- Jakob Yngvason Bosons in disc-shaped traps ----------------------------------------------------------------- Valentin Zagrebnov Bose Gas in Random Potential The Lifshitz-tail behaviour of the one-particle spectrum reduces the lower critical dimensionality of the Bose-Einstein Condensation (BEC) for the perfect Bose-gas to $d_c =1$. The following questions will be considered in the talk: - One-Particle Integrated Density of States (IDS) - Self-Averaging of the IDS and BEC - BEC in 1D-Random Poisson Potential - BEC Localization and ODLRO ----------------------------------------------------------------- -----------------------------------------------------------------