2,692 research outputs found
Fracton topological order via coupled layers
In this work, we develop a coupled layer construction of fracton topological
orders in spatial dimensions. These topological phases have sub-extensive
topological ground-state degeneracy and possess excitations whose movement is
restricted in interesting ways. Our coupled layer approach is used to construct
several different fracton topological phases, both from stacked layers of
simple topological phases and from stacks of fracton topological
phases. This perspective allows us to shed light on the physics of the X-cube
model recently introduced by Vijay, Haah, and Fu, which we demonstrate can be
obtained as the strong-coupling limit of a coupled three-dimensional stack of
toric codes. We also construct two new models of fracton topological order: a
semionic generalization of the X-cube model, and a model obtained by coupling
together four interpenetrating X-cube models, which we dub the "Four Color Cube
model." The couplings considered lead to fracton topological orders via
mechanisms we dub "p-string condensation" and "p-membrane condensation," in
which strings or membranes built from particle excitations are driven to
condense. This allows the fusion properties, braiding statistics, and
ground-state degeneracy of the phases we construct to be easily studied in
terms of more familiar degrees of freedom. Our work raises the possibility of
studying fracton topological phases from within the framework of topological
quantum field theory, which may be useful for obtaining a more complete
understanding of such phases.Comment: 20 pages, 18 figures, published versio
Hamiltonian submanifolds of regular polytopes
We investigate polyhedral -manifolds as subcomplexes of the boundary
complex of a regular polytope. We call such a subcomplex {\it -Hamiltonian}
if it contains the full -skeleton of the polytope. Since the case of the
cube is well known and since the case of a simplex was also previously studied
(these are so-called {\it super-neighborly triangulations}) we focus on the
case of the cross polytope and the sporadic regular 4-polytopes. By our results
the existence of 1-Hamiltonian surfaces is now decided for all regular
polytopes.
Furthermore we investigate 2-Hamiltonian 4-manifolds in the -dimensional
cross polytope. These are the "regular cases" satisfying equality in Sparla's
inequality. In particular, we present a new example with 16 vertices which is
highly symmetric with an automorphism group of order 128. Topologically it is
homeomorphic to a connected sum of 7 copies of . By this
example all regular cases of vertices with or, equivalently, all
cases of regular -polytopes with are now decided.Comment: 26 pages, 4 figure
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Polyhedra of small order and their Hamiltonian properties
We describe the results of an enumeration of several classes of polyhedra. The enumerated classes include polyhedra with up to 12 vertices and up to 26 edges, simplical polyhedra with up to 16 vertices, 4-connected polyhedra with up to 15 vertices, and bipartite polyhedra with up to 22 vertices.The results of the enumeration were used to systematically search for certain minimal non-Hamiltonian polyhedra. In particular, the smallest polyhedra satisfying certain toughness-like properties are presented here, as are the smallest non-Hamiltonian, 3-connected, Delaunay tessellations and triangulations. Improved upper and lower bounds on the size of the smallest non-Hamiltonian, inscribable polyhedra are also given
An extensive English language bibliography on graph theory and its applications, supplement 1
Graph theory and its applications - bibliography, supplement
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