Coulomb repulsion versus Hubbard repulsion in a disordered chain

We study the difference between on site Hubbard and long range Coulomb repulsions for two interacting particles in a disordered chain. While Hubbard repulsion can only yield weak critical chaos with intermediate spectral statistics, Coulomb repulsion can drive the two particle system to quantum chaos with Wigner-Dyson spectral statistics. For intermediate strengths U of the two repulsions in one dimension, there is a crossover regime where delocalization and spectral rigidity are maximum, whereas the limits of weak and strong U are characterized by a stronger localization and uncorrelated energy levels.Comment: 8 pages, 10 figure

Level curvatures, spectral statistics and scaling for interacting particles

The mobility of two interacting particles in a random potential is studied, using the sensitivity of their levels to a change of boundary conditions. The delocalization in Hilbert space induced by the interaction of the two particle Fock states is shown to decrease the mobility in metals and to increase it in insulators. In contrast to the single particle case, the spectral rigidity is not directly related to the level curvature. Therefore, another curvature of topological origin is introduced, which defines the energy scale below which the spectrum has the universal Wigner-Dyson rigidity.Comment: 4 pages, RevTe

Magnetoconductance of ballistic chaotic quantum dots: A Brownian motion approach for the SS-matrix

Using the Fokker-Planck equation describing the evolution of the transmission eigenvalues for Dyson's Brownian motion ensemble, we calculate the magnetoconductance of a ballistic chaotic dot in in the crossover regime from the orthogonal to the unitary symmetry. The correlation functions of the transmission eigenvalues are expressed in terms of quaternion determinants for arbitrary number $N$ of scattering channels. The corresponding average, variance and autocorrelation function of the magnetoconductance are given as a function of the Brownian motion time $t$. A microscopic derivation of this $S$-Brownian motion approach is discussed and $t$ is related to the applied flux. This exactly solvable random matrix model yields the right expression for the suppression of the weak localization corrections in the large $N$-limit and for small applied fluxes. An appropriate rescaling of $t$ could extend its validity to larger magnetic fluxes for the averages, but not for the correlation functions.Comment: 33 pages, 7 postscript figure

From independent particle towards collective motion in two electron square lattices

The two dimensional crossover from independent particle towards collective motion is studied using 2 spinless fermions interacting via a U/r Coulomb repulsion in a LxL square lattice with periodic boundary conditions and nearest neighbor hopping t. Three regimes characterize the ground state when U/t increases. Firstly, when the fluctuation $\Delta r$ of the spacing r between the two particles is larger than the lattice spacing, there is a scaling length $Lo=\sqrt{8}\pi^2(t/U)$ such that the relative fluctuation $\Delta r/$ is a universal function of the dimensionless ratio L/Lo, up to finite size corrections of order $L^{-2}$. LLo are respectively the limits of the free particle Fermi motion and of the correlated motion of a Wigner molecule. Secondly, when U/t exceeds a threshold U*(L)/t, $\Delta r$ becomes smalller than the lattice spacing, giving rise to a correlated lattice regime where the previous scaling breaks down and analytical expansions in powers of (t/U) become valid. A weak random potential reduces the scaling length and favors the correlated motion.Comment: 8 pages, 11 figures, to be published in EP

Universal Scaling of the Quantum Conductance of an Inversion-Symmetric Interacting Model

We consider quantum transport of spinless fermions in a 1D lattice embedding an interacting region (two sites with inter-site repulsion U and inter-site hopping td, coupled to leads by hopping terms tc). Using the numerical renormalization group for the particle-hole symmetric case, we study the quantum conductance g as a function of the inter-site hopping td. The interacting region, which is perfectly reflecting when td -> 0 or td -> infinity, becomes perfectly transmitting if td takes an intermediate value \tau(U,tc) which defines the characteristic energy of this interacting model. When td < tc sqrt(U), g is given by a universal function of the dimensionless ratio X=td/\tau. This universality characterizes the non-interacting regime where \tau=tc^2, the perturbative regime (U < tc^2) where \tau can be obtained using Hartree-Fock theory, and the non-perturbative regime (U > tc^2) where \tau is twice the characteristic temperature TK of an orbital Kondo effect induced by the inversion symmetry. When td < \tau, the expression g(X)=4/(X+1/X)^2 valid without interaction describes also the conductance in the presence of the interaction. To obtain those results, we map this spinless model onto an Anderson model with spins, where the quantum impurity is at the end point of a semi-infinite 1D lead and where td plays the role of a magnetic field h. This allows us to describe g(td) using exact results obtained for the magnetization m(h) of the Anderson model at zero temperature. We expect this universal scaling to be valid also in models with 2D leads, and observable using 2D semi-conductor heterostructures and an interacting region made of two identical quantum dots with strong capacitive inter-dot coupling and connected via a tunable quantum point contact.Comment: 14 pages, 18 figure

Brownian motion ensembles and parametric correlations of the transmission eigenvalues: Application to coupled quantum billiards and to disordered wires

The parametric correlations of the transmission eigenvalues $T_i$ of a $N$-channel quantum scatterer are calculated assuming two different Brownian motion ensembles. The first one is the original ensemble introduced by Dyson and assumes an isotropic diffusion for the $S$-matrix. The second Brownian motion ensemble assumes for the transfer matrix $M$ an isotropic diffusion yielded by a multiplicative combination law. We review the qualitative differences between transmission through two weakly coupled quantum dots and through a disordered line and we discuss the mathematical analogies between the Fokker-Planck equations of the two Brownian motion models.Comment: 33 pages, 7 postscript figures, the presented abstract is shortened in comparison to the abstract of the pape

Two interacting particles in a disordered chain I: Multifractality of the interaction matrix elements

For $N$ interacting particles in a one dimensional random potential, we study the structure of the corresponding network in Hilbert space. The states without interaction play the role of the ``sites''. The hopping terms are induced by the interaction. When the one body states are localized, we numerically find that the set of directly connected ``sites'' is multifractal. For the case of two interacting particles, the fractal dimension associated to the second moment of the hopping term is shown to characterize the Golden rule decay of the non interacting states and the enhancement factor of the localization length.Comment: First paper of a serie of four, to appear in Eur. Phys.

Effect of measurement probes upon the conductance of an interacting nanosystem: Detection of an attached ring by non local many body effects

We consider a nanosystem connected to measurement probes via leads. When a magnetic flux is varied through a ring attached to one lead at a distance Lc from the nanosystem, the effective nanosystem transmission |ts|^2 exhibits Aharonov-Bohm oscillations if the electrons interact inside the nanosystem. These oscillations can be very large if Lc is small and if the nanosystem has almost degenerate levels which are put near the Fermi energy by a local gate

Andreev-Lifshitz supersolid revisited for a few electrons on a square lattice II

In this second paper, using N=3 polarized electrons (spinless fermions) interacting via a U/r Coulomb repulsion on a two dimensional L * L square lattice with periodic boundary conditions and nearest neighbor hopping, we show that a single unpaired fermion can co-exist with a correlated two particle Wigner molecule for intermediate values of the Coulomb energy to kinetic energy ratio r_s.Comment: 15 pages, 28 postscript figure