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Abstract:

We present the results of an exact diagonalization study of the spin-1/2 Heisenberg antiferromagnet on a two-dimensional version of the pyrochlore lattice, also known as the square lattice with crossings or the checkerboard lattice. Examining the low energy spectra for systems of up to 24 spins, we find that all clusters studied have non-degenerate ground states with total spin zero, and big energy gaps to states with higher total spin. We also find a large number of non-magnetic excitations at energies within this spin gap. Spin-spin and spin-Peierls correlation functions appear to be short-ranged, and we suggest that the ground state is a spin liquid.

Abstract:

We study the universality class for localization which arises from models of non-interacting quasiparticles in disordered superconductors that have neither time-reversal nor spin-rotation symmetries. Two-dimensional systems in this category, which is known as class D, can display phases with three different types of quasiparticle dynamics: metallic, localized, or with a quantized (thermal) Hall conductance. Correspondingly, they can show a variety of delocalization transitions. We illustrate this behavior by investigating numerically the phase diagrams of network models with the appropriate symmetry, and for the first time show the appearance of the metallic phase.

Abstract:

We study the two-dimensional chiral metal, which is formed at the surface of a layered three-dimensional system exhibiting the integer quantum Hall effect by hybridization of the edge states associated with each layer of the sample. We investigate mesoscopic fluctuations, dynamical screening and inelastic scattering in the chiral metal, focussing particularly on fluctuations of conductance, $δg(B)$, with magnetic field, $B$. The correlation function $<δg(B) δg(B+δB)>$ provides information on the inelastic scattering rate, $τ_{in}^{-1}$, through both the variance of fluctuations and the range of correlations in $δB$. We calculate this correlation function for samples which are not fully phase coherent. Two regimes of behaviour exist, according to whether $τ_{in}^{-1}$ is smaller or larger than $τ_{\perp}^{-1}$, the rate for inter-edge tunneling, and we give results in both regimes. We also investigate dynamical screening of Coulomb interactions in the chiral metal and calculate the contribution to $τ_{in}^{-1}$ from electron-electron scattering, finding $τ_{in}^{-1} \propto T^{3/2}$ for $τ_{in}^{-1} \ll τ_{\perp}^{-1}$ at temperature $T$.

Abstract:

We study spectral properties of the Fokker-Planck operator that represents particles moving via a combination of diffusion and advection in a time-independent random velocity field, presenting in detail work outlined elsewhere [J. T. Chalker and Z. J. Wang, Phys. Rev. Lett. 79, 1797 (1997)]. We calculate analytically the ensemble-averaged one-particle Green function and the eigenvalue density for this Fokker-Planck operator, using a diagrammatic expansion developed for resolvents of non-Hermitian random operators, together with a mean-field approximation (the self-consistent Born approximation) which is well controlled in the weak-disorder regime for dimension d>2. The eigenvalue density in the complex plane is nonzero within a wedge that encloses the negative real axis. Particle motion is diffusive at long times, but for short times we find a novel time dependence of the mean-square displacement, approximately t(2/d) in dimension d>2, associated with the imaginary parts of eigenvalues.