Excited states of spherium

We report analytic solutions of a recently discovered quasi-exactly solvable model consisting of two electrons, interacting {\em via} a Coulomb potential, but restricted to remain on the surface of a $\mathcal{D}$-dimensional sphere. Polynomial solutions are found for the ground state, and for some higher ($L\le3$) states. Kato cusp conditions and interdimensional degeneracies are discussed.Comment: 6 pages, 2 figures, to appear in Mol. Phy

Algebraic Approach to Interacting Quantum Systems

We present an algebraic framework for interacting extended quantum systems to study complex phenomena characterized by the coexistence and competition of different states of matter. We start by showing how to connect different (spin-particle-gauge) {\it languages} by means of exact mappings (isomorphisms) that we name {\it dictionaries} and prove a fundamental theorem establishing when two arbitrary languages can be connected. These mappings serve to unravel symmetries which are hidden in one representation but become manifest in another. In addition, we establish a formal link between seemingly unrelated physical phenomena by changing the language of our model description. This link leads to the idea of {\it universality} or equivalence. Moreover, we introduce the novel concept of {\it emergent symmetry} as another symmetry guiding principle. By introducing the notion of {\it hierarchical languages}, we determine the quantum phase diagram of lattice models (previously unsolved) and unveil hidden order parameters to explore new states of matter. Hierarchical languages also constitute an essential tool to provide a unified description of phases which compete and coexist. Overall, our framework provides a simple and systematic methodology to predict and discover new kinds of orders. Another aspect exploited by the present formalism is the relation between condensed matter and lattice gauge theories through quantum link models. We conclude discussing applications of these dictionaries to the area of quantum information and computation with emphasis in building new models of computation and quantum programming languages.Comment: 44 pages, 14 psfigures. Advances in Physics 53, 1 (2004

Stringy Membranes in AdS/CFT

We study membrane configurations in AdS_{7/4}xS^{4/7}. The membranes are wrapped around the compact manifold S^{4/7} and are dynamically equivalent to bosonic strings in AdS_5. We thus conveniently identify them as "stringy membranes". For the case of AdS_7xS^4, their construction is carried out by embedding the Polyakov action for classical bosonic strings in AdS_5, into the corresponding membrane action. Therefore, every string configuration in AdS_5 can be realized by an appropriately chosen stringy membrane in AdS_7xS^4. We discuss the possibility of this being also the case for stringy membranes in AdS_4xS^7/Z^k (k > 1 or k = 1). By performing a stability analysis to the constructed solutions, we find that the (membrane) fluctuations along their transverse directions are organized in multiple Lam\'{e} stability bands and gaps in the space of parameters of the configurations. In this membrane picture, strings exhibit a single band/gap structure. Since the spectrum of quadratic fluctuations is important for the quantization of membranes, our analysis suggests the idea that quantum membranes behave as collective excitations of a multitude of interacting quantum strings