3,271 research outputs found
Spectral method for matching exterior and interior elliptic problems
A spectral method is described for solving coupled elliptic problems on an
interior and an exterior domain. The method is formulated and tested on the
two-dimensional interior Poisson and exterior Laplace problems, whose solutions
and their normal derivatives are required to be continuous across the
interface. A complete basis of homogeneous solutions for the interior and
exterior regions, corresponding to all possible Dirichlet boundary values at
the interface, are calculated in a preprocessing step. This basis is used to
construct the influence matrix which serves to transform the coupled boundary
conditions into conditions on the interior problem. Chebyshev approximations
are used to represent both the interior solutions and the boundary values. A
standard Chebyshev spectral method is used to calculate the interior solutions.
The exterior harmonic solutions are calculated as the convolution of the
free-space Green's function with a surface density; this surface density is
itself the solution to an integral equation which has an analytic solution when
the boundary values are given as a Chebyshev expansion. Properties of Chebyshev
approximations insure that the basis of exterior harmonic functions represents
the external near-boundary solutions uniformly. The method is tested by
calculating the electrostatic potential resulting from charge distributions in
a rectangle. The resulting influence matrix is well-conditioned and solutions
converge exponentially as the resolution is increased. The generalization of
this approach to three-dimensional problems is discussed, in particular the
magnetohydrodynamic equations in a finite cylindrical domain surrounded by a
vacuum
Conformal Invariance and Shape-Dependent Conductance of Graphene Samples
For a sample of an arbitrary shape, the dependence of its conductance on the
longitudinal and Hall conductivity is identical to that of a rectangle. We use
analytic results for a conducting rectangle, combined with the semicircle model
for transport coefficients, to study properties of the monolayer and bilayer
graphene. A conductance plateau centered at the neutrality point, predicted for
square geometry, is in agreement with recent experiments. For rectangular
geometry, the conductance exhibits maxima at the densities of compressible
quantum Hall states for wide samples, and minima for narrow samples. The
positions and relative sizes of these features are different in the monolayer
and bilayer cases, indicating that the conductance can be used as a tool for
sample diagnostic.Comment: 9 pages, 6 figure
Strange metals and the AdS/CFT correspondence
I begin with a review of quantum impurity models in condensed matter physics,
in which a localized spin degree of freedom is coupled to an interacting
conformal field theory in d = 2 spatial dimensions. Their properties are
similar to those of supersymmetric generalizations which can be solved by the
AdS/CFT correspondence; the low energy limit of the latter models is described
by a AdS2 geometry. Then I turn to Kondo lattice models, which can be described
by a mean- field theory obtained by a mapping to a quantum impurity coupled to
a self-consistent environment. Such a theory yields a 'fractionalized Fermi
liquid' phase of conduction electrons coupled to a critical spin liquid state,
and is an attractive mean-field theory of strange metals. The recent
holographic description of strange metals with a AdS2 x R2 geometry is argued
to be related to such mean-field solutions of Kondo lattice models.Comment: 19 pages, 4 figures; Plenary talk at Statphys24, Cairns, Australia,
July 2010; (v2) added refs; (v3) more ref
An Overdetermined Problem in Potential Theory
We investigate a problem posed by L. Hauswirth, F. H\'elein, and F. Pacard,
namely, to characterize all the domains in the plane that admit a "roof
function", i.e., a positive harmonic function which solves simultaneously a
Dirichlet problem with null boundary data, and a Neumann problem with constant
boundary data. Under some a priori assumptions, we show that the only three
examples are the exterior of a disk, a halfplane, and a nontrivial example. We
show that in four dimensions the nontrivial simply connected example does not
have any axially symmetric analog containing its own axis of symmetry.Comment: updated version. 20 pages, 3 figure
Looking beyond the Thermal Horizon: Hidden Symmetries in Chiral Models
In thermal states of chiral theories, as recently investigated by H.-J.
Borchers and J. Yngvason, there exists a rich group of hidden symmetries. Here
we show that this leads to a radical converse of of the Hawking-Unruh
observation in the following sense. The algebraic commutant of the algebra
associated with a (heat bath) thermal chiral system can be used to reprocess
the thermal system into a ground state system on a larger algebra with a larger
localization space-time. This happens in such a way that the original system
appears as a kind of generalized Unruh restriction of the ground state sytem
and the thermal commutant as being transmutated into newly created ``virgin
space-time region'' behind a horizon. The related concepts of a ``chiral
conformal core'' and the possibility of a ``blow-up'' of the latter suggest
interesting ideas on localization of degrees of freedom with possible
repercussion on how to define quantum entropy of localized matter content in
Local Quantum Physics.Comment: 17 pages, tcilatex, still more typos removed and one reference
correcte
- …