Lieb-Robinson Bounds and Quasi-locality for the Dynamics of Many-Body Quantum Systems

We review a recently proven Lieb-Robinson bound for general, many-body quantum systems with bounded interactions. Several basic examples are discussed as well as the connection between commutator estimates and quasi-locality.Comment: contribution to the proceedings of the conference QMath 11 (Hradec Kralove, September 2010

Lieb-Robinson Bounds in Quantum Many-Body Physics

We give an overview of recent results on Lieb-Robinson bounds and some of their applications in the study of quantum many-body models in condensed matter physics.Comment: Lecture Notes for the school "Entropy and the Quantum", 16-20 March 2009, Tucson, Arizona

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

Recent Progress in Quantum Spin Systems

Some recent developments in the theory of quantum spin systems are reviewed.Comment: To appear in Markov Processes and Related Fields (J.T. Lewis special edition

On the dynamics of lattice systems with unbounded on-site terms in the Hamiltonian

We supply the mathematical arguments required to complete the proofs of two previously published results: Lieb-Robinson bounds for the dynamics of quantum lattice systems with unbounded on-site terms in the Hamiltonian and the existence of the thermodynamic limit of the dynamics of such systems

Activity-dependent adenosine release may be linked to activation of Na+-K+ ATPase : an in vitro rat study

In the brain, extracellular adenosine increases as a result of neuronal activity. The mechanisms by which this occurs are only incompletely understood. Here we investigate the hypothesis that the Na+ influxes associated with neuronal signalling activate the Na+-K+ ATPase which, by consuming ATP, generates intracellular adenosine that is then released via transporters. By measuring adenosine release directly with microelectrode biosensors, we have demonstrated that AMPA-receptor evoked adenosine release in basal forebrain and cortex depends on extracellular Na+. We have simultaneously imaged intracellular Na+ and measured adenosine release. The accumulation of intracellular Na+ during AMPA receptor activation preceded adenosine release by some 90 s. By removing extracellular Ca2+, and thus preventing indiscriminate neuronal activation, we used ouabain to test the role of the Na+-K+ ATPase in the release of adenosine. Under conditions which caused a Na+ influx, brief applications of ouabain increased the accumulation of intracellular Na+ but conversely rapidly reduced extracellular adenosine levels. In addition, ouabain greatly reduced the amount of adenosine released during application of AMPA. Our data therefore suggest that activity of the Na+-K+ ATPase is directly linked to the efflux of adenosine and could provide a universal mechanism that couples adenosine release to neuronal activity. The Na+-K+ ATPase-dependent adenosine efflux is likely to provide adenosine-mediated activity-dependent negative feedback that will be important in many diverse functional contexts including the regulation of sleep

Critical Nodes In Directed Networks

Critical nodes or "middlemen" have an essential place in both social and economic networks when considering the flow of information and trade. This paper extends the concept of critical nodes to directed networks. We identify strong and weak middlemen. Node contestability is introduced as a form of competition in networks; a duality between uncontested intermediaries and middlemen is established. The brokerage power of middlemen is formally expressed and a general algorithm is constructed to measure the brokerage power of each node from the networks adjacency matrix. Augmentations of the brokerage power measure are discussed to encapsulate relevant centrality measures. We use these concepts to identify and measure middlemen in two empirical socio-economic networks, the elite marriage network of Renaissance Florence and Krackhardt's advice network.Comment: 28 pages, 6 figures, 2 table