Random Aharonov-Bohm vortices and some exact families of integrals: Part III

As a sequel to [1] and [2], I present some recent progress on Bessel integrals \int_0^{\infty}{\rmd u}\; uK_0(u)^{n}, \int_0^{\infty}{\rmd u}\; u^{3}K_0(u)^{n}, ... where the power of the integration variable is odd and where $n$, the Bessel weight, is a positive integer. Some of these integrals for weights n=3 and n=4 are known to be intimately related to the zeta numbers zeta(2) and zeta(3). Starting from a Feynman diagram inspired representation in terms of n dimensional multiple integrals on an infinite domain, one shows how to partially integrate to n-2 dimensional multiple integrals on a finite domain. In this process the Bessel integrals are shown to be periods. Interestingly enough, these "reduced" multiple integrals can be considered in parallel with some simple integral representations of zeta numbers. One also generalizes the construction of [2] on a particular sum of double nested Bessel integrals to a whole family of double nested integrals. Finally a strong PSLQ numerical evidence is shown to support a surprisingly simple expression of zeta(5) as a linear combination with rational coefficients of Bessel integrals of weight n= 8.Comment: 13 pages. arXiv admin note: substantial text overlap with arXiv:1209.103

On Thouless bandwidth formula in the Hofstadter model

We generalize Thouless bandwidth formula to its n-th moment. We obtain a closed expression in terms of polygamma, zeta and Euler numbers.Comment: 8 pages, 2 figure

Area distribution of two-dimensional random walks and non Hermitian Hofstadter quantum mechanics

When random walks on a square lattice are biased horizontally to move solely to the right, the probability distribution of their algebraic area can be exactly obtained. We explicitly map this biased classical random system on a non hermitian Hofstadter-like quantum model where a charged particle on a square lattice coupled to a perpendicular magnetic field hopps only to the right. In the commensurate case when the magnetic flux per unit cell is rational, an exact solution of the quantum model is obtained. Periodicity on the lattice allows to relate traces of the Nth power of the Hamiltonian to probability distribution generating functions of biased walks of length N.Comment: 14 pages, 7 figure

Fractal energy carpets in non-Hermitian Hofstadter quantum mechanics

We study the non-Hermitian Hofstadter dynamics of a quantum particle with biased motion on a square lattice in the background of a magnetic field. We show that in quasi-momentum space the energy spectrum is an overlap of infinitely many inequivalent fractals. The energy levels in each fractal are space-filling curves with Hausdorff dimension 2. The band structure of the spectrum is similar to a fractal spider net in contrast to the Hofstadter butterfly for unbiased motion.Comment: 12 pages, 18 figures. Fractal properties of the energy levels are visualised in the supplementary video material https://www.youtube.com/watch?v=ODS3QVkPTP

Topological 2-Dimensional Quantum Mechanics

We define a Chern- Simons Lagrangian for a system of planar particles topologically interacting at a distance. The anyon model appears as a particular case where all the particles are identical. We propose exact N-body eigenstates, set up a perturbative algorithm, discuss the case where some particles are fixed on a lattice, and also consider curved manifolds. PACS numbers: 05.30.-d, 11.10.-zComment: 18 pages, Orsay Report IPNO/TH 92-10

Persistent Currents and Magnetization in two-dimensional Magnetic Quantum Systems

Persistent currents and magnetization are considered for a two-dimensional electron (or gas of electrons) coupled to various magnetic fields. Thermodynamic formulae for the magnetization and the persistent current are established and the ``classical'' relationship between current and magnetization is shown to hold for systems invariant both by translation and rotation. Applications are given, including the point vortex superposed to an homogeneous magnetic field, the quantum Hall geometry (an electric field and an homogeneous magnetic field) and the random magnetic impurity problem (a random distribution of point vortices).Comment: 27 pages latex, 1 figur