Abstract:
A metallic disk with strong spin orbit interaction is investigated. The finite disk geometry introduces a confining potential. Due to the strong spin-orbit interaction and confining potential the metal disk is described by an effective one-dimensional model with a harmonic potential. The harmonic potential gives rise to classical turning points. As a result, open boundary conditions must be used. We bosonize the model and obtain chiral Bosons for each spin on the edge of the disk. When the filling fraction is reduced to the electron-electron interactions are studied by using the Jordan Wigner phase for composite fermions which give rise to a Luttinger liquid. When the metallic disk is in the proximity with a superconductor, a Fractional Topological Insulator is obtained. An experimental realization is proposed. We show that by tunning the chemical potential we control the classical turning points for which a Fractional Topological Insulator is realized.

Abstract:
Using two piezoelectric transducers, one measures the stress tensor response from the strain field generated by the second transducer. The ratio between the stress response and strain velocity determines the dissipative response. In the first part, we show that the dissipative stress response can be used for studying excitations in a topological superconductor. We investigate a topological superconductor for the case when an Abrikosov vortex lattice is formed. In this case, the Majorana fermions are dispersive, a fact that is used to compute the dissipative stress response. In the second part, we analyse the dissipative superfluid flow through solid 4He discoused recently. We identify low energy, an excitation which plays the role of the Majorana mode which is free to move in a direction perpendicular to the two dimensional plane spaces of the dislocations.

Abstract:
We present a new method for computing the wave function in the presence of constraints. As an explicit example we compute the wave function for the many electrons problem in coupled metallic rings in the presence of external magnetic fluxes. For equal fluxes and an even number of electrons the constraints enforce a wave function with a vanishing total momentum and a large persistent current and magnetization in contrast to the odd number of electrons where at finite temperatures the current is suppressed. We propose that the even-odd property can be verified by measuring the magnetization as a function of a varying gate voltage coupled to the rings. By reversing the flux in one of the ring the current and magnetization vanish in both rings; this can be used as a non-local control device.

Abstract:
Dirac's method for constraints is used for solving the problem of exclusion of double occupancy for Correlated Electrons. The constraints are enforced by the pair operator $Q(\vec{x})=\psi_{\downarrow}(\vec{x})\psi_{\uparrow}(\vec{x})$ which annihilates the ground state $|\Psi^0>$. Away from half fillings the operator $Q(\vec{x})$ is replaced by a set of $first$ $class$ Non-Abelian constraints $Q^{(-)}_{\alpha}(\vec{x})$ restricted to negative energies. The propagator for a single hole away from half fillings is determined by modified measure which is a function of the time duration of the hole propagator. As a result: a) The imaginary part of the self energy - is linear in the frequency. At large hole concentrations a Fermi Liquid self energy is obtained. b) For the Superconducting state the constraints generate an asymmetric spectrum excitations between electrons and holes giving rise to an asymmetry tunneling density of states.

Abstract:
We report the first calculation of persistent current in two coupled rings which form a character ``8'' genus g=2 structure. We obtain an exact solution for the persistent current and investigated the exact solution numerically. For two large coupled rings with equal fluxes, we find that the persistent current in the two coupled rings is equal to that in a single ring. For opposite fluxes the energy has a chaotic structure. For both cases the periodicity is $h/e$. This results are obtained within an extension of Dirac's second class method to fermionic constraints. This theory can be tested in the ballistic regime.

Abstract:
We investigate the Hubbard model in the limit $U=\infty$, which is equivalent to the statistical condition of exclusion of double occupancy. We solve this problem using Dirac's method for constraints. The constraints are solved within the Bosonization method. We find that the constraints modify the anomalous commutator. We apply this theory to quantum wires at finite temperatures where the Hubbard interaction is $U=\infty$. We find that the anomalous commutator induced by the constraints gives rise to the 0.7 anomalous conductance.

Abstract:
The two dimensional Rashba Hamiltonian is investigated using the momentum representation.One finds that the SU(2) transformation which diagonalizes the Hamiltonian gives rise to non commuting Cartesian coordinates for K=0 and zero otherwise.This result corresponds to the Aharonov -Bohm phase in the momentum space. The Spin -Hall conductivity is given by $\frac{e}{4\pi}$ which disagree with the result $\frac{e}{8\pi}$ given in the literature.Using Stokes theorem we find that the Spin -Hall current is carried equally by the up and down electrons on the Fermi surface. We identify the Magnetic current and find that for an electric field with a finite Fourier component in the momentum space a non zero Spin -Hall current is obtained.For the electric field which is constant in space, the orbital magnetic current cancels the spin current. In order to measure the Spin-Hall conductance we propose to apply a magnetic field gradient $\Delta H_{2}$ in the $i=2$ direction and to measure a Charge- Hall current $\frac{e}{h}(\frac{g}{2})\mu_{B}\Delta H_{2}$ in the $i=1$ direction.

Abstract:
We show that for a \textbf{multiple-connected} space the low energy strain fields excitations are given by instantons. Dirac fermions with a chiral mass and a pairing field propagates effectively in a multiple conected space. When the elastic strain field response is probed one finds that it is given by the \textbf{Pointriagin} characteristic. As a result the space time metric is modified. Applying an external stress field we observe that the phonon path bends in the transverse direction to the initial direction.

Abstract:
Using a $d_{x^2 - y^2}$ superconductor in 2+1 dimensions we show that the Nambu Goldstone fluctuations are replaced by dissipative excitations. We find that the nodal quasi-particles damping is caused by the strong dissipative excitations near the nodal points. As a result we find that the scattering rates are linear in frequency and not cubic as predicted in the literature for the ``d'' wave superconductors. Our results explain the recent angle resolved photoemission spectroscopy and optical conductivity in the BSCCO high $T_c$ compounds.

Abstract:
A Hamiltonian renormalization group is presented. Such a formulation is relevant for chiralic systems and more appropriate than the Lagrangian formalism. An application to 1D system is presented.