Mapping Formalism Research Articles

SDG Fermion-Pair Algebraic SO(12) and Sp(10) Models and Their Boson Realizations

It is shown how the boson mapping formalism may be applied as a useful many-body tool to solve a fermion problem. This is done in the context of generalized Ginocchio models for which we introduce S-, D-, and G-pairs of fermions and subsequently construct the sdg-boson realizations of the generalized Dyson type. The constructed SO(12) and Sp(10) fermion models are solved beyond the explicit symmetry limits. Phase transitions to rotational structures are obtained also in situations where there is no underlying SU(3) symmetry.

Erratum: Suppression of localization in Kronig-Penney models with correlated disorder

We consider the electron dynamics and transport properties of one-dimensional continuous models with random, short-range correlated impurities. We develop a generalized Poincar\'e map formalism to cast the Schr\"odinger equation for any potential into a discrete set of equations, illustrating its application by means of a specific example. We then concentrate on the case of a Kronig-Penney model with dimer impurities. The previous technique allows us to show that this model presents infinitely many resonances (zeroes of the reflection coefficient at a single dimer) that give rise to a band of extended states, in contradiction with the general viewpoint that all one-dimensional models with random potentials support only localized states. We report on exact transfer-matrix numerical calculations of the transmission coefficient, density of states, and localization length for various strengths of disorder. The most important conclusion so obtained is that this kind of system has a very large number of extended states. Multifractal analysis of very long systems clearly demonstrates the extended character of such states in the thermodynamic limit. In closing, we briefly discuss the relevance of these results in several physical contexts.

Concept mapping and other formalisms as mindtools for representing knowledge

Concept mapping and other formalisms as mindtools for representing knowledge

Suppression of localization in Kronig-Penney models with correlated disorder.

We consider the electron dynamics and transport properties of one-dimensional continuous models with random, short-range correlated impurities. We develop a generalized Poincar\'e map formalism to cast the Schr\odinger equation for any potential into a discrete set of equations, illustrating its application by means of a specific example. We then concentrate on the case of a Kronig-Penney model with dimer impurities. The previous technique allows us to show that this model presents infinitely many resonances (zeroes of the reflection coefficient at a single dimer) that give rise to a band of extended states, in contradiction with the general viewpoint that all one-dimensional models with random potentials support only localized states. We report on exact transfer-matrix numerical calculations of the transmission coefficient, density of states, and localization length for various strengths of disorder. The most important conclusion so obtained is that this kind of system has a very large number of extended states. Multifractal analysis of very long systems clearly demonstrates the extended character of such states in the thermodynamic limit. In closing, we briefly discuss the relevance of these results in several physical contexts.

Quantum mechanics in the gaussian wave-packet phase space representation III: From phase space probability functions to wave-functions

Using a mapping formalism, the Wave-Packet Phase Space Representation, we show that for phase space probability distributions that do not violate the Uncertainty Principle one can associate a density matrix, ϱ( x, x'), when some conditions are fulfilled. Furthermore if condition Tr ∧ϱ 2 = 1 is satisfied it becomes possible to find a wave function associated to the probability distribution.

Analysis of the truncation schemes for the physical boson states with Dyson's description.

We analyze various truncation schemes for the physical boson basis states which are required for the calculations in the boson space, in the Dyson boson mapping formalism. The available approximate schemes violate the Pauli principle and are found to be inadequate. The necessity to evolve yet a reliable and useful truncation scheme is stressed.

A semiclassical theory of underdamped nonlinear Brownian processes

In the limit of weak noise, the conditional probabilities of a damped, noise-driven anharmonic oscillator can be expressed (at least formally) in terms of solutions to a nonlinear ordinary differential equation (“Onsager-Machlup” equation) of fourth order in the time derivative. In the case of a one-dimensional oscillator, this equation is recast in the form of a mapping; and in the limit of long time and weak time-independent friction, it is shown how to reduce the iteration of this map to quadratures. This procedure is checked by using it to reproduce a number of standard results usually derived in other ways. In connection with the statistical mechanics of electron-positron colliding-beam storage rings, an attempt is also made to apply this mapping formalism to the case of weak, periodically time-dependent friction. When no frequency in the Fourier decomposition of the friction coefficient is close to an (even, for symmetric potentials) integral multiple of the (amplitude-dependent) frequency of the oscillator, then results can be derived that parallel those obtained in the case of constant friction. Otherwise (“thermal resonance”) the situation is not clear.

Microscopic structure of an interacting boson model in terms of the Dyson boson mapping

In an application of the generalized Dyson boson mapping to a shell model Hamiltonian acting in a single $j$ shell, a clear distinction emerges between pair bosons and kinematically determined seniority bosons. As in the Otsuka-Arima-Iachello method it is found that the latter type of boson determines the structure of an interactive boson-model-like Hamiltonian for the single $j$-shell model. It is furthermore shown that the Dyson boson mapping formalism is equally well suited for investigating possible interactive boson-model-like structures in a multishell case, where dynamical considerations are expected to play a much more important role in determining the structure of physical bosons.NUCLEAR STRUCTURE Boson mapping, interacting boson model, shell model.

Universality for the critical lines of the eight-vertex, Ashkin-Teller, and Gaussian models

A perturbation expansion has been performed for the critical lines of the eight-vertex and Ashkin-Teller models around the decoupling point. The critical temperature and the details of the mapping of these lines onto the Gaussian-model critical line is calculated to second order in the marginal parameters. We find agreement with the exactly known results for these models and with the mapping formalism of Kadanoff and Brown.