9,459 research outputs found
Locality Estimates for Quantum Spin Systems
We review some recent results that express or rely on the locality properties
of the dynamics of quantum spin systems. In particular, we present a slightly
sharper version of the recently obtained Lieb-Robinson bound on the group
velocity for such systems on a large class of metric graphs. Using this bound
we provide expressions of the quasi-locality of the dynamics in various forms,
present a proof of the Exponential Clustering Theorem, and discuss a
multi-dimensional Lieb-Schultz-Mattis Theorem.Comment: Contribution for the proceedings of ICMP XV, Rio de Janeiro, 200
Enrichment Procedures for Soft Clusters: A Statistical Test and its Applications
Clusters, typically mined by modeling locality of attribute spaces, are often evaluated for their ability to demonstrate āenrichmentā of categorical features. A cluster enrichment procedure evaluates the membership of a cluster for significant representation in pre-defined categories of interest. While classical enrichment procedures assume a hard clustering deļ¬nition, in this paper we introduce a new statistical test that computes enrichments for soft clusters. We demonstrate an application of this test in reļ¬ning and evaluating soft clusters for classification of remotely sensed images
Dynamics and the Emergence of Geometry in an Information Mesh
The idea of a graph theoretical approach to modeling the emergence of a
quantized geometry and consequently spacetime, has been proposed previously,
but not well studied. In most approaches the focus has been upon how to
generate a spacetime that possesses properties that would be desirable at the
continuum limit, and the question of how to model matter and its dynamics has
not been directly addressed. Recent advances in network science have yielded
new approaches to the mechanism by which spacetime can emerge as the ground
state of a simple Hamiltonian, based upon a multi-dimensional Ising model with
one dimensionless coupling constant. Extensions to this model have been
proposed that improve the ground state geometry, but they require additional
coupling constants. In this paper we conduct an extensive exploration of the
graph properties of the ground states of these models, and a simplification
requiring only one coupling constant. We demonstrate that the simplification is
effective at producing an acceptable ground state. Moreover we propose a scheme
for the inclusion of matter and dynamics as excitations above the ground state
of the simplified Hamiltonian. Intriguingly, enforcing locality has the
consequence of reproducing the free non-relativistic dynamics of a quantum
particle
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