Claudius Gros

NaV2O5 as a Quarter-Filled Ladder Compound

A new X-ray diffraction study of the one-dimensional spin-Peierls compound \alpha-NaV_2O_5 reveals a centrosymmetric (Pmmn) crystal structure with one type of V site, contrary to the previously postulated non-centrosymmetric P2_1mn structure with two types of V sites (V^{+4} and V^{+5}). Density functional calculations indicate that NaV_2O_5 is a quarter-filled ladder compound with the spins carried by V-O-V molecular orbitals on the rungs of the ladder. Estimates of the charge-transfer gap and the exchange coupling agree well with experiment and explain the insulating behavior of NaV_2O_5 and its magnetic properties.

Power laws and self-organized criticality in theory and nature

Abstract Power laws and distributions with heavy tails are common features of many complex systems. Examples are the distribution of earthquake magnitudes, solar flare intensities and the sizes of neuronal avalanches. Previously, researchers surmised that a single general concept may act as an underlying generative mechanism, with the theory of self organized criticality being a weighty contender. The power-law scaling observed in the primary statistical analysis is an important, but by far not the only feature characterizing experimental data. The scaling function, the distribution of energy fluctuations, the distribution of inter-event waiting times, and other higher order spatial and temporal correlations, have seen increased consideration over the last years. Leading to realization that basic models, like the original sandpile model, are often insufficient to adequately describe the complexity of real-world systems with power-law distribution. Consequently, a substantial amount of effort has gone into developing new and extended models and, hitherto, three classes of models have emerged. The first line of models is based on a separation between the time scales of an external drive and an internal dissipation, and includes the original sandpile model and its extensions, like the dissipative earthquake model. Within this approach the steady state is close to criticality in terms of an absorbing phase transition. The second line of models is based on external drives and internal dynamics competing on similar time scales and includes the coherent noise model, which has a non-critical steady state characterized by heavy-tailed distributions. The third line of models proposes a non-critical self-organizing state, being guided by an optimization principle, such as the concept of highly optimized tolerance. We present a comparative overview regarding distinct modeling approaches together with a discussion of their potential relevance as underlying generative models for real-world phenomena. The complexity of physical and biological scaling phenomena has been found to transcend the explanatory power of individual paradigmal concepts. The interaction between theoretical development and experimental observations has been very fruitful, leading to a series of novel concepts and insights.

Physics of Projected Wavefunctions

Abstract We present and discuss a variational approach to the one band Hubbard model in the limit of a large on-site Coulomb repulsion. The trial wavefunctions are the projected wavefunctions, generalized Gutzwiller wavefunctions. We discuss in detail the definition of these wavefunctions, the numerical methods used to evaluate them, their properties, and their physical relevance. Depending on the kind of parametrization used, the projected wavefunctions can describe a nearly localized Fermi liquid, an antiferromagnetically ordered state, or a quantum spin liquid. The physics of these three types of wavefunctions is described in detail. We discuss their relation to a proposed phase diagram of the two-dimensional Hubbard model and to results obtained by other approaches to the Hubbard model. The results obtained by numerical evaluation of the projected wavefunction are reviewed. The method used for the numerical evaluation, the variational Monte Carlo method, is described in detail. Finally we discuss the relation between a quantum spin liquid and the resonating valence bond state, which has been proposed, by P.W. Anderson, as a reference state for the Cu-O superconductors. In particular, we examine the question whether a quantum spin liquid is intrisically superconducting or not.

Magnetic light scattering in low-dimensional quantum spin systems

Abstract An overview of one- and two-dimensional quantum spin systems based on transition-metal oxides and halides of current interest is given, such as spin-Peierls, spin-dimer, geometrically frustrated and ladder systems. The most significant and outstanding contributions of magnetic light scattering to the understanding of these materials are discussed and compared to results of other spectroscopies and thermodynamic measurements.

Complex and Adaptive Dynamical Systems

An thorough introduction is given at an introductory level to the field of quantitative complex system science, with special emphasis on emergence in dynamical systems based on network topologies. Subjects treated include graph theory and small-world networks, a generic introduction to the concepts of dynamical system theory, random Boolean networks, cellular automata and self-organized criticality, the statistical modeling of Darwinian evolution, synchronization phenomena and an introduction to the theory of cognitive systems. It inludes chapter on Graph Theory and Small-World Networks, Chaos, Bifurcations and Diffusion, Complexity and Information Theory, Random Boolean Networks, Cellular Automata and Self-Organized Criticality, Darwinian evolution, Hypercycles and Game Theory, Synchronization Phenomena and Elements of Cognitive System Theory.

Superconducting Instability in the Large-U Limit of the Two-Dimensional Hubbard Model

We have investigated numerically the pairing instabilities of Gutzwiller wavefunctions. These are equivalent to a certain form of the resonant valence bond wavefunction. The case considered is a nearly half-filled two dimensional band with interactions given by a Hubbard model with large on-site Coulomb interactions. We find that the paramagnetic normal state is unstable tod-wave pairing but stable againsts-wave pairing. The antiferromagnetic state is marginally stable against both types of pairing. These results can be explained as an interference effect resulting in enhanced antiferromagnetic spin correlation in the paired state.

Cognitive Computation with Autonomously Active Neural Networks: An Emerging Field

The human brain is autonomously active. To understand the functional role of this self-sustained neural activity, and its interplay with the sensory data input stream, is an important question in cognitive system research and we review here the present state of theoretical modeling. This review will start with a brief overview of the experimental efforts, together with a discussion of transient versus self-sustained neural activity in the framework of reservoir computing. The main emphasis will be then on two paradigmal neural network architectures showing continuously ongoing transient-state dynamics: saddle point networks and networks of attractor relics. Self-active neural networks are confronted with two seemingly contrasting demands: a stable internal dynamical state and sensitivity to incoming stimuli. We show, that this dilemma can be solved by networks of attractor relics based on competitive neural dynamics, where the attractor relics compete on one side with each other for transient dominance, and on the other side with the dynamical influence of the input signals. Unsupervised and local Hebbian-style online learning then allows the system to build up correlations between the internal dynamical transient states and the sensory input stream. An emergent cognitive capability results from this set-up. The system performs online, and on its own, a nonlinear independent component analysis of the sensory data stream, all the time being continuously and autonomously active. This process maps the independent components of the sensory input onto the attractor relics, which acquire in this way a semantic meaning.

Gutzwiller–RVB theory of high-temperature superconductivity: Results from renormalized mean-field theory and variational Monte Carlo calculations

We review the resonating valence bond (RVB) theory of high-temperature superconductivity using Gutzwiller projected wave functions that incorporate strong correlations. After a general overview of the phenomenon of high-temperature superconductivity, we discuss Andersons RVB picture and its implementation by renormalized mean-field theory (RMFT) and variational Monte Carlo (VMC) techniques. We review RMFT and VMC results with an emphasis on recent developments in extending VMC and RMFT techniques to excited states. We compare results obtained from these methods with angle-resolved photoemission spectroscopy (ARPES) and scanning tunnelling microscopy (STM). We conclude by summarizing recent successes of this approach and discuss open problems that need to be solved for a consistent and complete description of high-temperature superconductivity using Gutzwiller projected wave functions.

Neural networks with transient state dynamics

We investigate dynamical systems characterized by a time series of distinct semi-stable activity patterns, as they are observed in cortical neural activity patterns. We propose and discuss a general mechanism allowing for an adiabatic continuation between attractor networks and a specific adjoined transient-state network, which is strictly dissipative. Dynamical systems with transient states retain functionality when their working point is autoregulated?avoiding prolonged periods of stasis or drifting into a regime of rapid fluctuations. We show, within a continuous-time neural network model, that a single local updating rule for online learning allows simultaneously (i) for information storage via unsupervised Hebbian-type learning, (ii) for adaptive regulation of the working point and (iii) for the suppression of runaway synaptic growth. Simulation results are presented; the spontaneous breaking of time-reversal symmetry and link symmetry are discussed.

Low-temperature transport in Heisenberg chains.

A technique to determine accurately transport properties of integrable and nonintegrable quantum-spin chains at finite temperatures by quantum Monte Carlo is presented. The reduction of the Drude weight by interactions in the integrable gapless regime is evaluated. Evidence for the absence of Drude weight in the gapless regime of a nonintegrable system with longer-ranged interactions is presented. We estimate the effect of the nonintegrability on the transport properties and compare with recent experiments on one-dimensional quantum-spin chains.