173 research outputs found
Coexistence of long-range order for two observables at finite temperatures
We give a criterion for the simultaneous existence or non existence of two
long-range orders for two observables, at finite temperatures, for quantum
lattice many body systems. Our analysis extends previous results of G.-S. Tian
limited to the ground state of similar models. The proof involves an inequality
of Dyson-Lieb-Simon which connects the Duhamel two-point function to the usual
correlation function. An application to the special case of the Holstein model
is discussed.Comment: 12 pages, accepted for publication in J. of Phys.
Proof of phase separation in the binary-alloy problem: the one-dimensional spinless Falicov-Kimball model
The ground states of the one-dimensional Falicov-Kimball model are
investigated in the small-coupling limit, using nearly degenerate perturbation
theory. For rational electron and ion densities, respectively equal to
, , with relatively prime to and
close enough to , we find that in the ground state
the ion configuration has period . The situation is analogous to the Peierls
instability where the usual arguments predict a period- state that produces
a gap at the Fermi level and is insulating. However for far
enough from , this phase becomes unstable against phase
separation. The ground state is a mixture of a period- ionic configuration
and an empty (or full) configuration, where both configurations have the same
electron density to leading order. Combining these new results with those
previously obtained for strong coupling, it follows that a phase transition
occurs in the ground state, as a function of the coupling, for ion densities
far enough from .Comment: 22 pages, typeset in ReVTeX and one encapsulated postscript file
embedded in the text with eps
Characterization of the Spectrum of the Landau Hamiltonian with Delta Impurities
We consider a random Schro\"dinger operator in an external magnetic field.
The random potential consists of delta functions of random strengths situated
on the sites of a regular two-dimensional lattice. We characterize the spectrum
in the lowest N Landau bands of this random Hamiltonian when the magnetic field
is sufficiently strong, depending on N. We show that the spectrum in these
bands is entirely pure point, that the energies coinciding with the Landau
levels are infinitely degenerate and that the eigenfunctions corresponding to
energies in the remainder of the spectrum are localized with a uniformly
bounded localization length. By relating the Hamiltonian to a lattice operator
we are able to use the Aizenman-Molchanov method to prove localization.Comment: To appear in Commun. Math. Phys. (1999
Ground States and Flux Configurations of the Two-dimensional Falicov-Kimball Model
The Falicov-Kimball model is a lattice model of itinerant spinless fermions
("electrons") interacting by an on-site potential with classical particles
("ions"). We continue the investigations of the crystalline ground states that
appear for various filling of electrons and ions, for large coupling. We
investigate the model for square as well as triangular lattices. New ground
states are found and the effects of a magnetic flux on the structure of the
phase diagram is studied. The flux phase problem where one has to find the
optimal flux configurations and the nuclei configurations is also solved in
some cases. Finaly we consider a model where the fermions are replaced by
hard-core bosons. This model also has crystalline ground states. Therefore
their existence does not require the Pauli principle, but only the on-site
hard-core constraint for the itinerant particles.Comment: 42 pages, uuencoded postscript file. Missing pages adde
Spectral flow and level spacing of edge states for quantum Hall hamiltonians
We consider a non relativistic particle on the surface of a semi-infinite
cylinder of circumference submitted to a perpendicular magnetic field of
strength and to the potential of impurities of maximal amplitude . This
model is of importance in the context of the integer quantum Hall effect. In
the regime of strong magnetic field or weak disorder it is known that
there are chiral edge states, which are localised within a few magnetic lengths
close to, and extended along the boundary of the cylinder, and whose energy
levels lie in the gaps of the bulk system. These energy levels have a spectral
flow, uniform in , as a function of a magnetic flux which threads the
cylinder along its axis. Through a detailed study of this spectral flow we
prove that the spacing between two consecutive levels of edge states is bounded
below by with , independent of , and of the
configuration of impurities. This implies that the level repulsion of the
chiral edge states is much stronger than that of extended states in the usual
Anderson model and their statistics cannot obey one of the Gaussian ensembles.
Our analysis uses the notion of relative index between two projections and
indicates that the level repulsion is connected to topological aspects of
quantum Hall systems.Comment: 22 pages, no figure
A (p,q)-deformed Landau problem in a spherical harmonic well: spectrum and noncommuting coordinates
A (p,q)-deformation of the Landau problem in a spherically symmetric harmonic
potential is considered. The quantum spectrum as well as space noncommutativity
are established, whether for the full Landau problem or its quantum Hall
projections. The well known noncommutative geometry in each Landau level is
recovered in the appropriate limit p,q=1. However, a novel noncommutative
algebra for space coordinates is obtained in the (p,q)-deformed case, which
could also be of interest to collective phenomena in condensed matter systems.Comment: 9 pages, no figures; updated reference
Griffiths Kelly Sherman correlation inequalities: a useful tool in the theory of error correcting codes
It is shown that a correlation inequality of statistical mechanics can be applied to linear low-density parity-check codes. Thanks to this tool we prove that, under a natural assumption, the exponential growth rate of regular low-density parity-check (LDPC) codes, can be computed exactly by iterative methods, at least on the interval where it is a concave function of the relative weight of code words. Then, considering communication over a binary input additive white Gaussian noise channel with a Poisson LDPC code we prove that, under a natural assumption, part of the GEXIT curve (associated to MAP decoding) can also be computed exactly by the belief propagation algorithm. The correlation inequality yields a sharp lower bound on the GEXIT curve. We also make an extension of the interpolation techniques that have recently led to rigorous results in spin glass theory and in the SAT problem
Geometric expansion of the log-partition function of the anisotropic Heisenberg model
We study the asymptotic expansion of the log-partition function of the
anisotropic Heisenberg model in a bounded domain as this domain is dilated to
infinity. Using the Ginibre's representation of the anisotropic Heisenberg
model as a gas of interacting trajectories of a compound Poisson process we
find all the non-decreasing terms of this expansion. They are given explicitly
in terms of functional integrals. As the main technical tool we use the cluster
expansion method.Comment: 38 page
The flux phase problem on the ring
We give a simple proof to derive the optimal flux which minimizes the ground
state energy in one dimensional Hubbard model, provided the number of particles
is even.Comment: 8 pages, to appear in J. Phys. A: Math. Ge
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