3,395 research outputs found
Multiorbital kinetic effects on charge ordering of frustrated electrons on the triangular lattice
The role of the multiorbital effects on the emergence of frustrated
electronic orders on the triangular lattice at half filling is investigated
through an extended spinless fermion Hubbard model. By using two complementary
approaches, unrestricted Hartree-Fock and exact diagonalizations, we unravel a
very rich phase diagram controlled by the strength of both local and off-site
Coulomb interactions and by the interorbital hopping anisotropy ratio .
Three robust unconventional electronic phases, a pinball liquid, an inverse
pinball liquid, and a large-unit-cell droplet
phase, are found to be generic in the triangular geometry, being controlled by
the band structure parameters. The latter are also stabilized in the isotropic
limit of our microscopic model, which recovers the standard SU(2) spinful
extended single-band Hubbard model.Comment: 10 pages, 6 figure
Metric for attractor overlap
We present the first general metric for attractor overlap (MAO) facilitating
an unsupervised comparison of flow data sets. The starting point is two or more
attractors, i.e., ensembles of states representing different operating
conditions. The proposed metric generalizes the standard Hilbert-space distance
between two snapshots to snapshot ensembles of two attractors. A reduced-order
analysis for big data and many attractors is enabled by coarse-graining the
snapshots into representative clusters with corresponding centroids and
population probabilities. For a large number of attractors, MAO is augmented by
proximity maps for the snapshots, the centroids, and the attractors, giving
scientifically interpretable visual access to the closeness of the states. The
coherent structures belonging to the overlap and disjoint states between these
attractors are distilled by few representative centroids. We employ MAO for two
quite different actuated flow configurations: (1) a two-dimensional wake of the
fluidic pinball with vortices in a narrow frequency range and (2)
three-dimensional wall turbulence with broadband frequency spectrum manipulated
by spanwise traveling transversal surface waves. MAO compares and classifies
these actuated flows in agreement with physical intuition. For instance, the
first feature coordinate of the attractor proximity map correlates with drag
for the fluidic pinball and for the turbulent boundary layer. MAO has a large
spectrum of potential applications ranging from a quantitative comparison
between numerical simulations and experimental particle-image velocimetry data
to the analysis of simulations representing a myriad of different operating
conditions.Comment: 33 pages, 20 figure
Poset pinball, GKM-compatible subspaces, and Hessenberg varieties
This paper has three main goals. First, we set up a general framework to
address the problem of constructing module bases for the equivariant cohomology
of certain subspaces of GKM spaces. To this end we introduce the notion of a
GKM-compatible subspace of an ambient GKM space. We also discuss
poset-upper-triangularity, a key combinatorial notion in both GKM theory and
more generally in localization theory in equivariant cohomology. With a view
toward other applications, we present parts of our setup in a general algebraic
and combinatorial framework. Second, motivated by our central problem of
building module bases, we introduce a combinatorial game which we dub poset
pinball and illustrate with several examples. Finally, as first applications,
we apply the perspective of GKM-compatible subspaces and poset pinball to
construct explicit and computationally convenient module bases for the
-equivariant cohomology of all Peterson varieties of classical Lie type,
and subregular Springer varieties of Lie type . In addition, in the Springer
case we use our module basis to lift the classical Springer representation on
the ordinary cohomology of subregular Springer varieties to -equivariant
cohomology in Lie type .Comment: 32 pages, 4 figure
Billey's formula in combinatorics, geometry, and topology
In this expository paper we describe a powerful combinatorial formula and its
implications in geometry, topology, and algebra. This formula first appeared in
the appendix of a book by Andersen, Jantzen, and Soergel. Sara Billey
discovered it independently five years later, and it played a prominent role in
her work to evaluate certain polynomials closely related to Schubert
polynomials.
Billey's formula relates many pieces of Schubert calculus: the geometry of
Schubert varieties, the action of the torus on the flag variety, combinatorial
data about permutations, the cohomology of the flag variety and of the Schubert
varieties, and the combinatorics of root systems (generalizing inversions of a
permutation). Combinatorially, Billey's formula describes an invariant of pairs
of elements of a Weyl group. On its face, this formula is a combination of
roots built from subwords of a fixed word. As we will see, it has deeper
geometric and topological meaning as well: (1) It tells us about the tangent
spaces at each permutation flag in each Schubert variety. (2) It tells us about
singular points in Schubert varieties. (3) It tells us about the values of
Kostant polynomials. Billey's formula also reflects an aspect of GKM theory,
which is a way of describing the torus-equivariant cohomology of a variety just
from information about the torus-fixed points in the variety.
This paper will also describe some applications of Billey's formula,
including concrete combinatorial descriptions of Billey's formula in special
cases, and ways to bootstrap Billey's formula to describe the equivariant
cohomology of subvarieties of the flag variety to which GKM theory does not
apply.Comment: 14 pages, presented at the International Summer School and Workshop
on Schubert Calculus in Osaka, Japan, 201
Benchmark problems for continuum radiative transfer. High optical depths, anisotropic scattering, and polarisation
Solving the continuum radiative transfer equation in high opacity media
requires sophisticated numerical tools. In order to test the reliability of
such tools, we present a benchmark of radiative transfer codes in a 2D disc
configuration. We test the accuracy of seven independently developed radiative
transfer codes by comparing the temperature structures, spectral energy
distributions, scattered light images, and linear polarisation maps that each
model predicts for a variety of disc opacities and viewing angles. The test
cases have been chosen to be numerically challenging, with midplane optical
depths up 10^6, a sharp density transition at the inner edge and complex
scattering matrices. We also review recent progress in the implementation of
the Monte Carlo method that allow an efficient solution to these kinds of
problems and discuss the advantages and limitations of Monte Carlo codes
compared to those of discrete ordinate codes. For each of the test cases, the
predicted results from the radiative transfer codes are within good agreement.
The results indicate that these codes can be confidently used to interpret
present and future observations of protoplanetary discs.Comment: 15 pages, 10 figures, accepted for publication in A&
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