Diagonalization- and Numerical Renormalization-Group-Based Methods for Interacting Quantum Systems

In these lecture notes, we present a pedagogical review of a number of related {\it numerically exact} approaches to quantum many-body problems. In particular, we focus on methods based on the exact diagonalization of the Hamiltonian matrix and on methods extending exact diagonalization using renormalization group ideas, i.e., Wilson's Numerical Renormalization Group (NRG) and White's Density Matrix Renormalization Group (DMRG). These methods are standard tools for the investigation of a variety of interacting quantum systems, especially low-dimensional quantum lattice models. We also survey extensions to the methods to calculate properties such as dynamical quantities and behavior at finite temperature, and discuss generalizations of the DMRG method to a wider variety of systems, such as classical models and quantum chemical problems. Finally, we briefly review some recent developments for obtaining a more general formulation of the DMRG in the context of matrix product states as well as recent progress in calculating the time evolution of quantum systems using the DMRG and the relationship of the foundations of the method with quantum information theory.Comment: 51 pages; lecture notes on numerically exact methods. Pedagogical review appearing in the proceedings of the "IX. Training Course in the Physics of Correlated Electron Systems and High-Tc Superconductors", Vietri sul Mare (Salerno, Italy, October 2004

Modal interaction in postbuckled plates. Theory

Plates can have more than one buckled solution for a fixed set of boundary conditions. The theory for the identification and the computation of multiple solutions in buckled plates is examined. The theory predicts modal interaction (which is also called change in buckle pattern or secondary buckling) in experiments on certain plates with multiple theoretical solutions. A set of coordinate functions is defined for Galerkin's method so that the von Karman plate equations are reduced to a coupled set of cubic equations in generalized coordinates that are uncoupled in the linear terms. An iterative procedure for solving modal interaction problems is suggested based on this cubic form

Some General Aspects of Coset Models and Topological Kazama-Suzuki Models

We study global aspects of N=2 Kazama-Suzuki coset models by investigating topological G/H Kazama-Suzuki models in a Lagrangian framework based on gauged Wess-Zumino-Witten models. We first generalize Witten's analysis of the holomorphic factorization of bosonic G/H models to models with N=1 and N=2 supersymmetry. We also find some new anomaly-free and supersymmetric models based on non-diagonal embeddings of the gauge group. We then explain the basic properties (action, symmetries, metric independence, ...) of the topologically twisted G/H Kazama-Suzuki models. We explain how all of the above generalizes to non-trivial gauge bundles. We employ the path integral methods of localization and abelianization (shown to be valid also for non-trivial bundles) to establish that the twisted G/H models can be localized to bosonic H/H models (with certain quantum corrections), and can hence be reduced to an Abelian bosonic T/T model, T a maximal torus of H. We also present the action and the symmetries of the coupling of these models to topological gravity. We determine the bosonic observables for all the models based on classical flag manifolds and the bosonic observables and their fermionic descendants for models based on complex Grassmannians.Comment: expanded version to appear in NPB: construction of wave functions, proof of holomorphic factorization and localization extended to non-trivial gauge bundles; 73 pages, LaTeX fil

Euler systems for Rankin--Selberg convolutions of modular forms

We construct an Euler system in the cohomology of the tensor product of the Galois representations attached to two modular forms, using elements in the higher Chow groups of products of modular curves. We use this Euler system to prove a finiteness theorem for the strict Selmer group of the Galois representation when the associated p-adic Rankin--Selberg L-function is non-vanishing at s = 1.Comment: Revised version with many updates and correction

Brane Partons and Singleton Strings

We examine p-branes in AdS(D) in two limits where they exhibit partonic behavior: rotating branes with energy concentrated to cusp-like solitons; tensionless branes with energy distributed over singletonic partons on the Dirac hypercone. Evidence for a smooth transition from cusps to partons is found. First, each cusp yields D-2 normal-coordinate bound states with protected frequencies (for p>2 there are additional bound states); and can moreover be related to a short open p-brane whose tension diverges at the AdS boundary leading to a decoupled singular CFT at the ``brane at the end-of-the-universe''. Second, discretizing the closed p-brane and keeping the number N of discrete partons finite yields an sp(2N)-gauged phase-space sigma model giving rise to symmetrized N-tupletons of the minimal higher-spin algebra ho_0(D-1,2)\supset so(D-1,2). The continuum limit leads to a 2d chiral sp(2)-gauged sigma model which is critical in D=7; equivalent a la Bars-Vasiliev to an su(2)-gauged spinor string; and furthermore dual to a WZW model in turn containing a topological \hat{so}(6,2)_{-2}/(\hat{so}(6)\oplus \hat\so(2))_{-2} coset model with a chiral ring generated by singleton-valued weight-0 spin fields. Moreover, the two-parton truncation can be linked via a reformulation a la Cattaneo-Felder-Kontsevich to a topological open string on the phase space of the D-dimensional Dirac hypercone. We present evidence that a suitable deformation of the open string leads to the Vasiliev equations based on vector oscillators and weak sp(2)-projection. Geometrically, the bi-locality reflects broken boundary-singleton worldlines, while Vasiliev's intertwiner kappa can be seen to relate T and R-ordered deformations of the boundary and the bulk of the worldsheet, respectively.Comment: Latex, 83 pages, 4 figures. v2: Analysis of intertwiner and bi-local operators modified, some minor changes, references added, typos corrected. v3: final versio