24,423 research outputs found
Generalized Kneser coloring theorems with combinatorial proofs
The Kneser conjecture (1955) was proved by Lov\'asz (1978) using the
Borsuk-Ulam theorem; all subsequent proofs, extensions and generalizations also
relied on Algebraic Topology results, namely the Borsuk-Ulam theorem and its
extensions. Only in 2000, Matou\v{s}ek provided the first combinatorial proof
of the Kneser conjecture.
Here we provide a hypergraph coloring theorem, with a combinatorial proof,
which has as special cases the Kneser conjecture as well as its extensions and
generalization by (hyper)graph coloring theorems of Dol'nikov,
Alon-Frankl-Lov\'asz, Sarkaria, and Kriz. We also give a combinatorial proof of
Schrijver's theorem.Comment: 19 pages, 4 figure
Two-Dimensional Electrons in a Strong Magnetic Field with Disorder: Divergence of the Localization Length
Electrons on a square lattice with half a flux quantum per plaquette are
considered. An effective description for the current loops is given by a
two-dimensional Dirac theory with random mass. It is shown that the
conductivity and the localization length can be calculated from a product of
Dirac Green's functions with the {\it same} frequency. This implies that the
delocalization of electrons in a magnetic field is due to a critical point in a
phase with a spontaneously broken discrete symmetry. The estimation of the
localization length is performed for a generalized model with fermion
levels using a --expansion and the Schwarz inequality. An argument for the
existence of two Hall transition points is given in terms of percolation
theory.Comment: 10 pages, RevTeX, no figure
Level statistics and localization in a 2D quantum percolation problem
A two dimensional model for quantum percolation with variable tunneling range
is studied. For this purpose the Lifshitz model is considered where the
disorder enters the Hamiltonian via the nondiagonal elements. We employ a
numerical method to analyze the level statistics of this model. It turns out
that the level repulsion is strongest around the percolation threshold. As we
go away from the maximum level repulsion a crossover from a GOE type behavior
to a Poisson like distribution is indicated. The localization properties are
calculated by using the sensitivity to boundary conditions and we find a strong
crossover from localized to delocalized states.Comment: 4 pages, 4 figure
Phase Transitions of Fermions Coupled to a Gauge Field: a Quantum Monte Carlo Approach
A grand canonical system of non-interacting fermions on a square lattice is
considered at zero temperature. Three different phases exist: an empty lattice,
a completely filled lattice and a liquid phase which interpolates between the
other two phases. The Fermi statistics can be changed into a Bose statistics by
coupling a statistical gauge field to the fermions. Using a quantum Monte Carlo
method we investigate the effect of the gauge field on the critical properties
of the lattice fermions. It turns out that there is no significant change of
the phase diagram or the density of particles due to the gauge field even at
the critical points. This result supports a recent conjecture by Huang and Wu
that certain properties of a three-dimensional flux line system (which is
equivalent to two-dimensional hard-core bosons) can be explained with
non-interacting fermion models.Comment: 12 pages, Plain-Tex, 5 figure
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