465 research outputs found
The Optical Approach to Casimir Effects
We propose a new approach to the Casimir effect based on classical ray
optics. We define and compute the contribution of classical optical paths to
the Casimir force between rigid bodies. Our approach improves upon the
proximity force approximation. It can be generalized easily to arbitrary
geometries, different boundary conditions, to the computation of Casimir energy
densities and to many other situations. This is a brief introduction to the
method. Joint work with R.L.Jaffe.Comment: Talk given at the conference `Continuous Advances in QCD 2004',
University of Minnesot
The forward approximation as a mean field approximation for the Anderson and Many Body Localization transitions
In this paper we analyze the predictions of the forward approximation in some
models which exhibit an Anderson (single-) or many-body localized phase. This
approximation, which consists in summing over the amplitudes of only the
shortest paths in the locator expansion, is known to over-estimate the critical
value of the disorder which determines the onset of the localized phase.
Nevertheless, the results provided by the approximation become more and more
accurate as the local coordination (dimensionality) of the graph, defined by
the hopping matrix, is made larger. In this sense, the forward approximation
can be regarded as a mean field theory for the Anderson transition in infinite
dimensions. The sum can be efficiently computed using transfer matrix
techniques, and the results are compared with the most precise exact
diagonalization results available.
For the Anderson problem, we find a critical value of the disorder which is
off the most precise available numerical value already in 5 spatial
dimensions, while for the many-body localized phase of the Heisenberg model
with random fields the critical disorder is strikingly close
to the most recent results obtained by exact diagonalization. In both cases we
obtain a critical exponent . In the Anderson case, the latter does not
show dependence on the dimensionality, as it is common within mean field
approximations.
We discuss the relevance of the correlations between the shortest paths for
both the single- and many-body problems, and comment on the connections of our
results with the problem of directed polymers in random medium
Soap films with gravity and almost-minimal surfaces
Motivated by the study of the equilibrium equations for a soap film hanging
from a wire frame, we prove a compactness theorem for surfaces with
asymptotically vanishing mean curvature and fixed or converging boundaries. In
particular, we obtain sufficient geometric conditions for the minimal surfaces
spanned by a given boundary to represent all the possible limits of sequences
of almost-minimal surfaces. Finally, we provide some sharp quantitative
estimates on the distance of an almost-minimal surface from its limit minimal
surface.Comment: 34 pages, 6 figures. Version 2: more detailed description of the
proof of the estimates in Section 5 adde
Integrals of motion in the Many-Body localized phase
We construct a complete set of quasi-local integrals of motion for the
many-body localized phase of interacting fermions in a disordered potential.
The integrals of motion can be chosen to have binary spectrum , thus
constituting exact quasiparticle occupation number operators for the Fermi
insulator. We map the problem onto a non-Hermitian hopping problem on a lattice
in operator space. We show how the integrals of motion can be built, under
certain approximations, as a convergent series in the interaction strength. An
estimate of its radius of convergence is given, which also provides an estimate
for the many-body localization-delocalization transition. Finally, we discuss
how the properties of the operator expansion for the integrals of motion imply
the presence or absence of a finite temperature transition.Comment: 65 pages, 12 figures. Corrected typos, added reference
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