241 research outputs found
Hamiltonian Cycles in Polyhedral Maps
We present a necessary and sufficient condition for existence of a
contractible, non-separating and noncontractible separating Hamiltonian cycle
in the edge graph of polyhedral maps on surfaces. In particular, we show the
existence of contractible Hamiltonian cycle in equivelar triangulated maps. We
also present an algorithm to construct such cycles whenever it exists.Comment: 14 page
Contractible Hamiltonian Cycles in Polyhedral Maps
We present a necessary and sufficient condition for existence of a
contractible Hamiltonian Cycle in the edge graph of equivelar maps on surfaces.
We also present an algorithm to construct such cycles. This is further
generalized and shown to hold for more general maps.Comment: 9 pages, 1 figur
(3+1)-dimensional topological phases and self-dual quantum geometries encoded on Heegard surfaces
We apply the recently suggested strategy to lift state spaces and operators
for (2+1)-dimensional topological quantum field theories to state spaces and
operators for a (3+1)-dimensional TQFT with defects. We start from the
(2+1)-dimensional Turaev-Viro theory and obtain a state space, consistent with
the state space expected from the Crane-Yetter model with line defects. This
work has important applications for quantum gravity as well as the theory of
topological phases in (3+1) dimensions. It provides a self-dual quantum
geometry realization based on a vacuum state peaked on a homogeneously curved
geometry. The state spaces and operators we construct here provide also an
improved version of the Walker-Wang model, and simplify its analysis
considerably. We in particular show that the fusion bases of the
(2+1)-dimensional theory lead to a rich set of bases for the (3+1)-dimensional
theory. This includes a quantum deformed spin network basis, which in a loop
quantum gravity context diagonalizes spatial geometry operators. We also obtain
a dual curvature basis, that diagonalizes the Walker-Wang Hamiltonian.
Furthermore, the construction presented here can be generalized to provide
state spaces for the recently introduced dichromatic four-dimensional manifold
invariants.Comment: 27 pages, many figures, v2: minor correction
Some Triangulated Surfaces without Balanced Splitting
Let G be the graph of a triangulated surface of genus . A
cycle of G is splitting if it cuts into two components, neither of
which is homeomorphic to a disk. A splitting cycle has type k if the
corresponding components have genera k and g-k. It was conjectured that G
contains a splitting cycle (Barnette '1982). We confirm this conjecture for an
infinite family of triangulations by complete graphs but give counter-examples
to a stronger conjecture (Mohar and Thomassen '2001) claiming that G should
contain splitting cycles of every possible type.Comment: 15 pages, 7 figure
Universal symmetry-protected topological invariants for symmetry-protected topological states
Symmetry-protected topological (SPT) states are short-range entangled states
with a symmetry G. They belong to a new class of quantum states of matter which
are classified by the group cohomology in
d-dimensional space. In this paper, we propose a class of symmetry- protected
topological invariants that may allow us to fully characterize SPT states with
a symmetry group G (ie allow us to measure the cocycles in
that characterize the SPT states). We give
an explicit and detailed construction of symmetry-protected topological
invariants for 2+1D SPT states. Such a construction can be directly generalized
to other dimensions.Comment: 12 pages, 11 figures. Added reference
Structure of the Entanglement Entropy of (3+1)D Gapped Phases of Matter
We study the entanglement entropy of gapped phases of matter in three spatial
dimensions. We focus in particular on size-independent contributions to the
entropy across entanglement surfaces of arbitrary topologies. We show that for
low energy fixed-point theories, the constant part of the entanglement entropy
across any surface can be reduced to a linear combination of the entropies
across a sphere and a torus. We first derive our results using strong
sub-additivity inequalities along with assumptions about the entanglement
entropy of fixed-point models, and identify the topological contribution by
considering the renormalization group flow; in this way we give an explicit
definition of topological entanglement entropy in (3+1)D,
which sharpens previous results. We illustrate our results using several
concrete examples and independent calculations, and show adding "twist" terms
to the Lagrangian can change in (3+1)D. For the generalized
Walker-Wang models, we find that the ground state degeneracy on a 3-torus is
given by in terms of the topological
entanglement entropy across a 2-torus. We conjecture that a similar
relationship holds for Abelian theories in dimensional spacetime, with
the ground state degeneracy on the -torus given by
.Comment: 34 pages, 16 figure
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