Haldane fractional statistics in the fractional quantum Hall effect

We have tested Haldane's ``fractional-Pauli-principle'' description of excitations around the $\nu = 1/3$ state in the FQHE, using exact results for small systems of electrons. We find that Haldane's prediction $\beta = \pm 1/m$ for quasiholes and quasiparticles, respectively, describes our results well with the modification $\beta_{qp} = 2-1/3$ rather than $-1/3$. We also find that this approach enables us to better understand the {\it energetics\/} of the ``daughter'' states; in particular, we find good evidence, in terms of the effective interaction between quasiparticles, that the states $\nu = 4/11$ and 4/13 should not be stable.Comment: 9 pages, 3 Postscript figures, RevTex 3.0. (UCF-CM-93-005

Inferring Pattern and Disorder in Close-Packed Structures from X-ray Diffraction Studies, Part I: epsilon-Machine Spectral Reconstruction Theory

In a recent publication [D. P. Varn, G. S. Canright, and J. P. Crutchfield, Phys. Rev. B {\bf 66}:17, 156 (2002)] we introduced a new technique for discovering and describing planar disorder in close-packed structures (CPSs) directly from their diffraction spectra. Here we provide the theoretical development behind those results, adapting computational mechanics to describe one-dimensional structure in materials. By way of contrast, we give a detailed analysis of the current alternative approach, the fault model (FM), and offer several criticisms. We then demonstrate that the computational mechanics description of the stacking sequence--in the form of an epsilon-machine--provides the minimal and unique description of the crystal, whether ordered, disordered, or some combination. We find that we can detect and describe any amount of disorder, as well as materials that are mixtures of various kinds of crystalline structure. Underlying this approach is a novel method for epsilon-machine reconstruction that uses correlation functions estimated from diffraction spectra, rather than sequences of microscopic configurations, as is typically used in other domains. The result is that the methods developed here can be adapted to a wide range of experimental systems in which spectroscopic data is available.Comment: 26 pages, 23 figures, 8 tables, 110 citations; http://www.santafe.edu/projects/CompMech/papers/ipdcpsi.htm

Inferring Pattern and Disorder in Close-Packed Structures from X-ray Diffraction Studies, Part II: Structure and Intrinsic Computation in Zinc Sulphide

In the previous paper of this series [D. P. Varn, G. S. Canright, and J. P. Crutchfield, Physical Review B, submitted] we detailed a procedure--epsilon-machine spectral reconstruction--to discover and analyze patterns and disorder in close-packed structures as revealed in x-ray diffraction spectra. We argued that this computational mechanics approach is more general than the current alternative theory, the fault model, and that it provides a unique characterization of the disorder present. We demonstrated the efficacy of computational mechanics on four prototype spectra, finding that it was able to recover a statistical description of the underlying modular-layer stacking using epsilon-machine representations. Here we use this procedure to analyze structure and disorder in four previously published zinc sulphide diffraction spectra. We selected zinc sulphide not only for the theoretical interest this material has attracted in an effort to develop an understanding of polytypism, but also because it displays solid-state phase transitions and experimental data is available.Comment: 15 pages, 14 figures, 4 tables, 57 citations; http://www.santafe.edu/projects/CompMech/papers/ipdcpsii.htm

Quantum Hall fractions for spinless Bosons

We study the Quantum Hall phases that appear in the fast rotation limit for Bose-Einstein condensates of spinless bosonic atoms. We use exact diagonalization in a spherical geometry to obtain low-lying states of a small number of bosons as a function of the angular momentum. This allows to understand or guess the physics at a given filling fraction nu, ratio of the number of bosons to the number of vortices. This is also the filling factor of the lowest Landau level. In addition to the well-known Bose Laughlin state at nu =1/2 we give evidence for the Jain principal sequence of incompressible states at nu =p/(p+- 1) for a few values of p. There is a collective mode in these states whose phenomenology is in agreement with standard arguments coming e.g. from the composite fermion picture. At filling factor one, the potential Fermi sea of composite fermions is replaced by a paired state, the Moore-Read state. This is most clearly seen from the half-flux nature of elementary excitations. We find that the hierarchy picture does not extend up to the point of transition towards a vortex lattice. While we cannot conclude, we investigate the clustered Read-Rezayi states and show evidence for incompressible states at the expected ratio of flux vs number of Bose particles.Comment: RevTeX 4, 11 pages, 13 figure

Analytical solution for the Fermi-sea energy of two-dimensional electrons in a magnetic field: lattice path-integral approach and quantum interference

We derive an exact solution for the total kinetic energy of noninteracting spinless electrons at half-filling in two-dimensional bipartite lattices. We employ a conceptually novel approach that maps this problem exactly into a Feynman-Vdovichenko lattice walker. The problem is then reduced to the analytic study of the sum of magnetic phase factors on closed paths. We compare our results with the ones obtained through numerical calculations.Comment: 5 pages, RevTe

"Spin-Disentangled" Exact Diagonalization of Repulsive Hubbard Systems: Superconducting Pair Propagation

By a novel exact diagonalization technique we show that bound pairs propagate between repulsive Hubbard clusters in a superconducting fashion. The size of the matrices that must be handled depends on the number of fermion configurations {\em per spin}, which is of the order of the square root of the overall size of the Hilbert space. We use CuO$_{4}$ units connected by weak O-O links to model interplanar coupling and c-axis superconductivity in Cuprates. The numerical evidence on Cu$_{2}$O$_{8}$ and Cu$_{3}$O$_{12}$ prompts a new analytic scheme describing the propagation of bound pairs and also the superconducting flux quantization in a 3-d geometry.Comment: 5 pages, 3 figure

Exclusion statistics for fractional quantum Hall states on a sphere

We discuss exclusion statistics parameters for quasiholes and quasielectrons excited above the fractional quantum Hall states near $\nu=p/(2np+1)$. We derive the diagonal statistics parameters from the (``unprojected'') composite fermion (CF) picture. We propose values for the off-diagonal (mutual) statistics parameters as a simple modification of those obtained from the unprojected CF picture, by analyzing finite system numerical spectra in the spherical geometry.Comment: 9 pages, Revtex, 4 Postscript figures. Universality of the statistics parameters is stressed, 2 figs adde

The Energy Density in the Maxwell-Chern-Simons Theory

A two-dimensional nonrelativistic fermion system coupled to both electromagnetic gauge fields and Chern-Simons gauge fields is analysed. Polarization tensors relevant in the quantum Hall effect and anyon superconductivity are obtained as simple closed integrals and are evaluated numerically for all momenta and frequencies. The correction to the energy density is evaluated in the random phase approximation (RPA), by summing an infinite series of ring diagrams. It is found that the correction has significant dependence on the particle number density. In the context of anyon superconductivity, the energy density relative to the mean field value is minimized at a hole concentration per lattice plaquette (0.05 \sim 0.06) (p_c a/\hbar)^2 where p_c and a are the momentum cutoff and lattice constant, respectively. At the minimum the correction is about -5 % \sim -25 %, depending on the ratio (2m \omega_c)/(p_c^2) where \omega_c is the frequency cutoff. In the Jain-Fradkin-Lopez picture of the fractional quantum Hall effect the RPA correction to the energy density is very large. It diverges logarithmically as the cutoff is removed, implying that corrections beyond RPA become important at large momentum and frequency.Comment: 19 pages (plain Tex), 12 figures not included, UMN-TH-1246/9