97,733 research outputs found
Topological order from quantum loops and nets
I define models of quantum loops and nets which have ground states with
topological order. These make possible excited states comprised of deconfined
anyons with non-abelian braiding. With the appropriate inner product, these
quantum loop models are equivalent to net models whose topological weight
involves the chromatic polynomial. A useful consequence is that the models have
a quantum self-duality, making it possible to find a simple Hamiltonian
preserving the topological order. For the square lattice, this Hamiltonian has
only four-spin interactions
Wilson Loops, Bianchi Constraints and Duality in Abelian Lattice Models
We introduce new modified Abelian lattice models, with inhomogeneous local
interactions, in which a sum over topological sectors are included in the
defining partition function. The dual models, on lattices with arbitrary
topology, are constructed and they are found to contain sums over topological
sectors, with modified groups, as in the original model. The role of the sum
over sectors is illuminated by deriving the field-strength formulation of the
models in an explicitly gauge-invariant manner. The field-strengths are found
to satisfy, in addition to the usual local Bianchi constraints, global
constraints. We demonstrate that the sum over sectors removes these global
constraints and consequently softens the quantization condition on the global
charges in the system. Duality is also used to construct mappings between the
order and disorder variables in the theory and its dual. A consequence of the
duality transformation is that correlators which wrap around non-trivial cycles
of the lattice vanish identically. For particular dimensions this mapping
allows an explicit expression for arbitrary correlators to be obtained.Comment: LaTeX 30 pages, 6 figures and 2 tables. References updated and
connection with earlier work clarified, final version to appear in Nucl.
Phys.
Comments about quantum symmetries of SU(3) graphs
For the SU(3) system of graphs generalizing the ADE Dynkin digrams in the
classification of modular invariant partition functions in CFT, we present a
general collection of algebraic objects and relations that describe fusion
properties and quantum symmetries associated with the corresponding Ocneanu
quantum groupo\"{i}ds. We also summarize the properties of the individual
members of this system.Comment: 36 page
Transport Networks Revisited: Why Dual Graphs?
Deterministic equilibrium flows in transport networks can be investigated by
means of Markov's processes defined on the dual graph representations of the
network. Sustained movement patterns are generated by a subset of automorphisms
of the graph spanning the spatial network of a city naturally interpreted as
random walks. Random walks assign absolute scores to all nodes of a graph and
embed space syntax into Euclidean space.Comment: 12 page
A modular network treatment of Baars' Global Workspace consciousness model
Network theory provides an alternative to the renormalization and phase transition methods used in Wallace's (2005a) treatment of Baars' Global Workspace model. Like the earlier study, the new analysis produces the workplace itself, the tunable threshold of consciousness, and the essential role for embedding contexts, in an explicitly analytic 'necessary conditions' manner which suffers neither the mereological fallacy inherent to brain-only theories nor the sufficiency indeterminacy of neural network or agent-based simulations. This suggests that the new approach, and the earlier, represent different analytically solvable limits in a broad continuum of possible models, analogous to the differences between bond and site percolation or between the two and many-body limits of classical mechanics. The development significantly extends the theoretical foundations for an empirical general cognitive model (GCM) based on the Shannon-McMillan Theorem. Patterned after the general linear model which reflects the Central Limit Theorem, the proposed technique should be both useful for the reduction of expermiental data on consciousness and in the design of devices with capacities which may transcend those of conventional machines and provide new perspectives on the varieties of biological consciousness
Deciding the Borel complexity of regular tree languages
We show that it is decidable whether a given a regular tree language belongs
to the class of the Borel hierarchy, or equivalently whether
the Wadge degree of a regular tree language is countable.Comment: 15 pages, 2 figure
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