12,014 research outputs found
Mass Partitions via Equivariant Sections of Stiefel Bundles
We consider a geometric combinatorial problem naturally associated to the
geometric topology of certain spherical space forms. Given a collection of
mass distributions on , the existence of affinely independent
regular -fans, each of which equipartitions each of the measures, can in
many cases be deduced from the existence of a -equivariant
section of the Stiefel bundle over , where
is the Stiefel manifold of all orthonormal -frames in
or , and
is the corresponding unit sphere. For example, the
parallelizability of when , or implies that any
two masses on can be simultaneously bisected by each of
pairwise-orthogonal hyperplanes, while when or 4, the triviality of the
circle bundle over the standard Lens Spaces
yields that for any mass on , there exist a pair of
complex orthogonal regular -fans, each of which equipartitions the mass.Comment: 11 pages, final versio
Effect of Landau Level Mixing on Braiding Statistics
We examine the effect of Landau level mixing on the braiding statistics of
quasiparticles of abelian and nonabelian quantum Hall states. While path
dependent geometric phases can perturb the abelian part of the statistics, we
find that the nonabelian properties remain unchanged to an accuracy that is
exponentially small in the distance between quasiparticles.Comment: 4 page
Exact Solutions of Fractional Chern Insulators: Interacting Particles in the Hofstadter Model at Finite Size
We show that all the bands of the Hofstadter model on the torus have an
exactly flat dispersion and Berry curvature when a special system size is
chosen. This result holds for any hopping and Chern number. Our analysis
therefore provides a simple rule for choosing a particularly advantageous
system size when designing a Hofstadter system whose size is controllable, like
a qubit lattice or an optical cavity array. The density operators projected
onto the flat bands obey exactly the Girvin-MacDonald-Platzman algebra, like
for Landau levels in the continuum in the case of , or obey its
straightforward generalization in the case of . This allows a mapping
between density-density interaction Hamiltonians for particles in the
Hofstatder model and in a continuum Landau level. By using the well-known
pseudopotential construction in the latter case, we obtain fractional Chern
insulator phases, the lattice counterpart of fractional quantum Hall phases,
that are exact zero-energy ground states of the Hofstadter model with certain
interactions. Finally, the addition of a harmonic trapping potential is shown
to lead to an appealingly symmetric description in which a new Hofstadter model
appears in momentum space.Comment: 15 pages, 8 figures; Published versio
Exactly Solvable Lattice Models with Crossing Symmetry
We show how to compute the exact partition function for lattice
statistical-mechanical models whose Boltzmann weights obey a special "crossing"
symmetry. The crossing symmetry equates partition functions on different
trivalent graphs, allowing a transformation to a graph where the partition
function is easily computed. The simplest example is counting the number of
nets without ends on the honeycomb lattice, including a weight per branching.
Other examples include an Ising model on the Kagome' lattice with three-spin
interactions, dimers on any graph of corner-sharing triangles, and non-crossing
loops on the honeycomb lattice, where multiple loops on each edge are allowed.
We give several methods for obtaining models with this crossing symmetry, one
utilizing discrete groups and another anyon fusion rules. We also present
results indicating that for models which deviate slightly from having crossing
symmetry, a real-space decimation (renormalization-group-like) procedure
restores the crossing symmetry
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