arXiv subjects
Publications and source records attributed to Wellington da Cruz.
We prove that the Farey sequences can be express into equivalence classes labeled by a fractal parameter which looks like a Hausdorff dimension $h$ defined within the interval 1 < h < 2. The classes $h$ satisfy the same properties of the Farey series and for each value of $h$ there exists an algebraic equation.
We obtain the distribution functions for anyonic excitations classified into equivalence classes labeled by Hausdorff dimension $h$ and as an example of such anyonic systems, we consider the collective excitations of the Fractional Quantum Hall Effect (FQHE). We also introduce the concept of duality between such classes, defined by $\tilde{h}=3-h$. In this way, we confirm that the filling factors for which the FQHE were observed just appears into these classes and the internal duality for a given class $h$ or $\tilde{h}$ is between quasihole and quasiparticle excitations for these FQHE systems. Exchanges of dual pairs $(ν,\tildeν)$, suggests conformal invariance. A connection between equivalence classes $h$ and the modular group for the quantum phase transitions of the FQHE is also obtained. A $β-$function is also defined for the complex conductivity which embodies the $h$ classes.
We consider for an anyon gas its termodynamics properties taking into account the fractal statistics obtained by us recently. This approach describes the anyonic excitations in terms of equivalence classes labeled by fractal parameter or Hausdorff dimension $h$. An exact equation of state is obtained in the high-temperature and low-temperature limits, for gases with a constant density of states.
We consider anyonic excitations classified into equivalence classes labeled by Hausdorff dimension, $h$ and introduce the concept of duality between such classes, defined by $\tilde{h}=3-h$. In this way, we confirm that the filling factors for which the Fractional Quantum Hall Effect (FQHE) were observed just appears into these classes and the duality in case is between quasihole and quaasiparticle excitations for these FQHE systems.Exchanges of dual pairs $(ν,\tildeν)$, suggests conformal invariance.
We obtain the distribution functions for anyonic excitations classified into equivalence classes labeled by Hausdorff dimension, $h$ and as an example of such anyonic systems, we consider the collective excitations of the Fractional Quantum Hall Effect (FQHE).
We obtain for an anyon gas in the high temperature limit a relation between the exclusion statistics parameter $g$ and the Hausdorff dimension $h$, given by $g=h(2-h)$. The anyonic excitations are classified into equivalence classes labeled by Hausdorff dimension, $h$, and in that limit, the parameter $g$ give us the second virial coefficient for any statistics, $ν$. The anyonic excitations into the same class $h$ get the same value of this virial coefficient.
We obtain the Hausdorff dimension, $h=2-2s$, for particles with fractional spins in the interval, $0\leq s \leq 0.5$, such that the manifold is characterized by a topological invariant given by, ${\cal W}=h+2s-2p$. This object is related to fractal properties of the path swept out by fractional spin particles, the spin of these particles, and the genus (number of anyons) of the manifold. We prove that the anyonic propagator can be put into a path integral representation which gives us a continuous family of Lagrangians in a convenient gauge. The formulas for, $h$ and ${\cal W}$, were obtained taking into account the anyon model as a particle-flux system and by a qualitative inference of the topology.
We obtain for any spin, $s$, the Hausdorff dimension, $h_{i}$, for fractional spin particles and we discuss the connection between this number, $h_{i}$, and the Chern-Simons potential. We also define the topological invariants, $W_s$, in terms of the statistics of these particles.
The technique of functional integration over velocities is applied to the calculation of the propagator of a spinning particle with and without anomalous magnetic moment. A representation for the spin factor is obtained in this context for the particle in a constant electromagnetic field. As a by-product, we also obtain a Schwinger representation for the first case.