Two electrons in an external oscillator potential: hidden algebraic structure

It is shown that the Coulomb correlation problem for a system of two electrons (two charged particles) in an external oscillator potential possesses a hidden $sl_2$-algebraic structure being one of recently-discovered quasi-exactly-solvable problems. The origin of existing exact solutions to this problem, recently discovered by several authors, is explained. A degeneracy of energies in electron-electron and electron-positron correlation problems is found. It manifests the first appearence of hidden $sl_2$-algebraic structure in atomic physics.Comment: 7 pages (plus one figure avaliable via direct request), LaTeX, Preprint IFUNAM FT 94-4

Physical realization of the parity anomaly and quantum hall effect

We present the lattice version of the anomalous Dirac operator in (2+1) dimensions. This version is associated with a group of magnetic translations rather than translations. This group naturally provided homologically nontrivial fiber bundles, the first Chern class of which is directly related with the number of zero modes and spectral asymmetry of the Dirac operator. The results are in agreement with the lattice fermions doubling phenomenon. The relation of the parity anomaly in (2+1) and the Hall effect is elucidated

Raman scattering and anomalous current algebra in Mott insulators

We present a theory of large shift Raman scattering in Mott insulators and show that inelastic light scattering provides information about electronic current algebra. We argue that the recent experiment where a new excitation below the optical absorption threshold was observed in crossed polarizations gives evidence of anomalous terms in the current alegebra. We show that it suggests there exists an exciton bound state with a topological magnetic excitation with odd parity with respect to a spatial reflection

Hierarchical Structure of Azbel-Hofstader Problem: Strings and loose ends of Bethe Ansatz

We present numerical evidence that solutions of the Bethe Ansatz equations for a Bloch particle in an incommensurate magnetic field (Azbel-Hofstadter or AH model), consist of complexes-"strings". String solutions are well-known from integrable field theories. They become asymptotically exact in the thermodynamic limit. The string solutions for the AH model are exact in the incommensurate limit, where the flux through the unit cell is an irrational number in units of the elementary flux quantum. We introduce the notion of the integral spectral flow and conjecture a hierarchical tree for the problem. The hierarchical tree describes the topology of the singular continuous spectrum of the problem. We show that the string content of a state is determined uniquely by the rate of the spectral flow (Hall conductance) along the tree. We identify the Hall conductances with the set of Takahashi-Suzuki numbers (the set of dimensions of the irreducible representations of $U_q(sl_2)$ with definite parity). In this paper we consider the approximation of noninteracting strings. It provides the gap distribution function, the mean scaling dimension for the bandwidths and gives a very good approximation for some wave functions which even captures their multifractal properties. However, it misses the multifractal character of the spectrum.Comment: revtex, 30 pages, 6 Figs, 8 postscript files are enclosed, important references are adde

Theta-terms in nonlinear sigma-models

We trace the origin of theta-terms in non-linear sigma-models as a nonperturbative anomaly of current algebras. The non-linear sigma-models emerge as a low energy limit of fermionic sigma-models. The latter describe Dirac fermions coupled to chiral bosonic fields. We discuss the geometric phases in three hierarchies of fermionic sigma-models in spacetime dimension (d+1) with chiral bosonic fields taking values on d-, d+1-, and d+2-dimensional spheres. The geometric phases in the first two hierarchies are theta-terms. We emphasize a relation between theta-terms and quantum numbers of solitons.Comment: 10 pages, no figures, revtex, typos correcte

Quantum Group and Magnetic Translations. Bethe-Ansatz Solution for Azbel-Hofstadter Problem

We present a new approach to the problem of Bloch electrons in magnetic ( sometimes called Azbel-Hofstadter problem) field, by making explicit a natural relation between the group of magnetic translations and the quantum group $U_{q}(sl_2)$. The approach allows us to express the "mid" band spectrum of the model and the Bloch wave function as solutions of the Bethe-Ansatz equations typical for completely integrable quantum systems. The zero mode wave functions are found explicitly in terms of $q$-deformed classical orthogonal polynomials.In this paper we present solution for the isotropic problem. We also present a class of solvable quasiperiodic equations related to $U_{q}(sl_2)$.Comment: 19 pages, Revte

Energy Reflection Symmetry of Lie-Algebraic Problems: Where the Quasiclassical and Weak Coupling Expansions Meet

We construct a class of one-dimensional Lie-algebraic problems based on sl(2) where the spectrum in the algebraic sector has a dynamical symmetry E -> - E. All 2j+1 eigenfunctions in the algebraic sector are paired, and inside each pair are related to each other by simple analytic continuation x -> ix, except the zero mode appearing if j is integer. At j-> infinity the energy of the highest level in the algebraic sector can be calculated by virtue of the quasiclassical expansion, while the energy of the ground state can be calculated as a weak coupling expansion. The both series coincide identically.Comment: Latex, 16 pages, 3 figures. Minor styllistic changes made, typos corrected, a remark on the energy-reflection symmetry in the quantum-algebraic Hamiltonians emerging in finite-difference problems added. Final version, to be published in Physical Review

Topological Mechanism of Superconductivity

We outline the basic ideas of the topological mechanisms of superconductivity. A gauged model of correlated electronic system where a topological fluid is formed as a result of a strong interaction is discussed.Comment: 38 pages, latex, no figure

Ground state energy and quasiparticle gaps in ν=N2N±1\nu={N\over{2N\pm 1}} FQHE states

Applying the transformation of fermion operators to new fermion quasiparticles introduced by Halperin, Lee, and Read we estimate a scaling behavior of the ground state energy and quasiparticle gaps as a function of filling fraction for a "principal sequence" of FQHE $\nu={N\over{2N\pm 1}}$ states converging towards the gapless state at half filling. The exponent describing the shape of the cusp $\delta E(\nu)\sim |\delta\nu|^{\eta}$ is found to be greater than one and to depend nontrivially on the interaction potential. The dependence of quasiparticle gaps agrees with the results of recent measurements by R.R.Du et al.Comment: 15 pages, TeX, C Version 3.0, preprint ETH-TH/93-3

Anderson-Yuval approach to the multichannel Kondo problem

We analyze the structure of the perturbation expansion of the general multichannel Kondo model with channel anisotropic exchange couplings and in the presence of an external magnetic field, generalizing to this case the Anderson-Yuval technique. For two channels, we are able to map the Kondo model onto a generalized resonant level model. Limiting cases in which the equivalent resonant level model is solvable are identified. The solution correctly captures the properties of the two channel Kondo model, and also allows an analytic description of the cross-over from the non Fermi liquid to the Fermi liquid behavior caused by the channel anisotropy.Comment: 23 pages, ReVTeX, 4 figures av. on reques