51,537 research outputs found
Topological quasiparticles and the holographic bulk-edge relation in 2+1D string-net models
String-net models allow us to systematically construct and classify 2+1D
topologically ordered states which can have gapped boundaries. We can use a
simple ideal string-net wavefunction, which is described by a set of F-matrices
[or more precisely, a unitary fusion category (UFC)], to study all the
universal properties of such a topological order. In this paper, we describe a
finite computational method -- Q-algebra approach, that allows us to compute
the non-Abelian statistics of the topological excitations [or more precisely,
the unitary modular tensor category (UMTC)], from the string-net wavefunction
(or the UFC). We discuss several examples, including the topological phases
described by twisted gauge theory (i.e., twisted quantum double ).
Our result can also be viewed from an angle of holographic bulk-boundary
relation. The 1+1D anomalous topological orders, that can appear as edges of
2+1D topological states, are classified by UFCs which describe the fusion of
quasiparticles in 1+1D. The 1+1D anomalous edge topological order uniquely
determines the 2+1D bulk topological order (which are classified by UMTC). Our
method allows us to compute this bulk topological order (i.e., the UMTC) from
the anomalous edge topological order (i.e., the UFC).Comment: 32 pages, 8 figures, reference updated, some refinement
Subfactors of index less than 5, part 2: triple points
We summarize the known obstructions to subfactors with principal graphs which
begin with a triple point. One is based on Jones's quadratic tangles
techniques, although we apply it in a novel way. The other two are based on
connections techniques; one due to Ocneanu, and the other previously
unpublished, although likely known to Haagerup.
We then apply these obstructions to the classification of subfactors with
index below 5. In particular, we eliminate three of the five families of
possible principal graphs called "weeds" in the classification from
arXiv:1007.1730.Comment: 28 pages, many figures. Completely revised from v1: many additional
or stronger result
On the packing dimension of box-like self-affine sets in the plane
We consider a class of planar self-affine sets which we call "box-like". A
box-like self-affine set is the attractor of an iterated function system (IFS)
of affine maps where the image of the unit square, [0,1]^2, under arbitrary
compositions of the maps is a rectangle with sides parallel to the axes. This
class contains the Bedford-McMullen carpets and the generalisations thereof
considered by Lalley-Gatzouras, Bara\'nski and Feng-Wang as well as many other
sets. In particular, we allow the mappings in the IFS to have non-trivial
rotational and reflectional components. Assuming a rectangular open set
condition, we compute the packing and box-counting dimensions by means of a
pressure type formula based on the singular values of the maps.Comment: 15 pages, 4 figure
Rigidity of frameworks on expanding spheres
A rigidity theory is developed for bar-joint frameworks in
whose vertices are constrained to lie on concentric -spheres with
independently variable radii. In particular, combinatorial characterisations
are established for the rigidity of generic frameworks for with an
arbitrary number of independently variable radii, and for with at most
two variable radii. This includes a characterisation of the rigidity or
flexibility of uniformly expanding spherical frameworks in .
Due to the equivalence of the generic rigidity between Euclidean space and
spherical space, these results interpolate between rigidity in 1D and 2D and to
some extent between rigidity in 2D and 3D. Symmetry-adapted counts for the
detection of symmetry-induced continuous flexibility in frameworks on spheres
with variable radii are also provided.Comment: 22 pages, 2 figures, updated reference
Dirac lattice
We study the emergence of Dirac fermionic field in the low energy description
of non-relativistic dynamical models on graphs admitting continuum limit. The
Dirac fermionic field appears as the effective field describing the excitations
above point-like Fermi surface. Together with the Dirac fermionic field an
effective space-time metric is also emerging. We analyze the conditions for
such Fermi points to appear in general, paying special attention to the cases
of two and three spacial dimensions.Comment: 26 pages, 4 figures; typo and grammatical corrections, new
reference(s) added, version accepted for publicatio
- …