224 research outputs found
Non-abelian vortices on CP^1 and Grassmannians
Many properties of the moduli space of abelian vortices on a compact Riemann
surface are known. For non-abelian vortices the moduli space is less well
understood. Here we consider non-abelian vortices on the Riemann sphere CP^1,
and we study their moduli spaces near the Bradlow limit. We give an explicit
description of the moduli space as a Kahler quotient of a finite-dimensional
linear space. The dimensions of some of these moduli spaces are derived.
Strikingly, there exist non-abelian vortex configurations on CP^1, with
non-trivial vortex number, for which the moduli space is a point. This is in
stark contrast to the moduli space of abelian vortices. For a special class of
non-abelian vortices the moduli space is a Grassmannian, and the metric near
the Bradlow limit is a natural generalization of the Fubini--Study metric on
complex projective space. We use this metric to investigate the statistical
mechanics of non-abelian vortices. The partition function is found to be
analogous to the one for abelian vortices.Comment: minor corrections; some notation improve
Memory-efficient array redistribution through portable collective communication
Modern large-scale deep learning workloads highlight the need for parallel
execution across many devices in order to fit model data into hardware
accelerator memories. In these settings, array redistribution may be required
during a computation, but can also become a bottleneck if not done efficiently.
In this paper we address the problem of redistributing multi-dimensional array
data in SPMD computations, the most prevalent form of parallelism in deep
learning. We present a type-directed approach to synthesizing array
redistributions as sequences of MPI-style collective operations. We prove
formally that our synthesized redistributions are memory-efficient and perform
no excessive data transfers. Array redistribution for SPMD computations using
collective operations has also been implemented in the context of the XLA SPMD
partitioner, a production-grade tool for partitioning programs across
accelerator systems. We evaluate our approach against the XLA implementation
and find that our approach delivers a geometric mean speedup of ,
with maximum speedups as a high as , while offering provable memory
guarantees, making our system particularly appealing for large-scale models.Comment: minor errata fixe
Vortices on Hyperbolic Surfaces
It is shown that abelian Higgs vortices on a hyperbolic surface can be
constructed geometrically from holomorphic maps , where is also
a hyperbolic surface. The fields depend on and on the metrics of and
. The vortex centres are the ramification points, where the derivative of
vanishes. The magnitude of the Higgs field measures the extent to which
is locally an isometry.
Witten's construction of vortices on the hyperbolic plane is rederived, and
new examples of vortices on compact surfaces and on hyperbolic surfaces of
revolution are obtained. The interpretation of these solutions as
SO(3)-invariant, self-dual SU(2) Yang--Mills fields on is also given.Comment: Revised version: new section on four-dimensional interpretation of
hyperbolic vortices added
Thermal Conductivity Anisotropy in Superconducting
Recent thermal conductivity measurements on single crystals by
Lussier et al. indicate the existence of a strong b--c anisotropy in the
superconducting state. We calculate the thermal conductivity in various
unconventional candidate states appropriate for the ``B phase" and
compare with experiment, specifically the and states
predicted in some Ginzburg-Landau analyses of the phase diagram. For the
simplest realizations of these states over spherical or ellipsoidal Fermi
surfaces, the normalized conductivity is found, surprisingly, to be
completely isotropic. We discuss the effects of inelastic scattering and
realistic Fermi surface anisotropy, and deduce constraints on the symmetry
class of the ground state.Comment: 4 postscript pages, UFL102
Heat Transport and the Nature of the Order Parameter in Superconducting
Recent thermal conductivity data on the heavy fermion superconductor
have been interpreted as offering support for an model of the order
parameter as opposed to an model. In this paper, we analyze this issue
from a theoretical standpoint including the detailed effects of Fermi surface
and gap anisotropy. Our conclusion is that although current data put strong
constraints on the gap anisotropy, they cannot definitively distinguish between
these two models. Measurements on samples of varying quality could be decisive
in this regard, however.Comment: 8 pages, revtex, 15 uunencoded postscript figure
Transport Properties of d-Wave Superconductors in the Vortex State
We calculate the magnetic field dependence of quasiparticle transport
properties in the vortex state of a d-wave superconductor arising solely from
the quasiparticle's Doppler shift in the superflow field surrounding the
vortex. Qualitative features agree well with experiments on cuprate and heavy
fermion superconductors at low fields and temperatures. We derive scaling
relations in the variable valid at sufficiently low temperatures
and fields , but show that these relations depend on the scattering
phase shift, and are in general fulfilled only approximately even in the clean
limit, due to the energy dependence of the quasiparticle relaxation time.Comment: 5 pages, 2 Postscript figure
Universal Heat Conduction in YBa_2Cu_3O_6.9
The thermal conductivity of YBa_2Cu_3O_6.9 was measured at low temperatures
in untwinned single crystals with concentrations of Zn impurities from 0 to 3%
of Cu. A linear term kappa_0/T = 0.19 mW/K^2.cm is clearly resolved as T -> 0,
and found to be virtually independent of Zn concentration. The existence of
this residual normal fluid strongly validates the basic theory of transport in
unconventional superconductors. Moreover, the observed universal behavior is in
quantitative agreement with calculations for a gap function of d-wave symmetry.Comment: Latex file, 4 pages, 3 EPS figures, to appear in Physical Review
Letter
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