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Original Publications
Prof. Claudius Gros
publications of general interest
ORCID




Entrenched time delays versus accelerating opinion dynamics: are advanced democracies inherently unstable?

Authors: C. Gros
Journal-ref.: European Physical Journal B (in press)
Modern societies face the challenge that the time scale of opinion formation is continuously accelerating in contrast to the time scale of political decision making. With the latter remaining of the order of the election cycle we examine here the case that the political state of a society is determined by the continuously evolving values of the electorate. Given this assumption we show that the time lags inherent in the election cycle will inevitable lead to political instabilities for advanced democracies characterized both by an accelerating pace of opinion dynamics and by high sensibilities (political correctness) to deviations from mainstream values. Our result is based on the observation that dynamical systems become generically unstable whenever time delays become comparable to the time it takes to adapt to the steady state. The time needed to recover from external shocks grows in addition dramatically close to the transition. Our estimates for the order of magnitude of the involved time scales indicate that socio-political instabilities may develop once the aggregate time scale for the evolution of the political values of the electorate falls below 7-15 months.


Universal scaling relation for magnetic sails: momentum breaking in the limit of dilute interstellar media

Authors: C. Gros
Journal-ref.: to be published
The recent progress in laser propulsion research has advanced substantially the prospects to realize interstellar spaceflight within a few decades. Here we examine passive deceleration via momentum breaking from ionized interstellar media. The very large area to mass relations needed as a consequence of the low interstellar densities, of the order of 0.1 particles per cm3, or lower, are potentially realizable with magnetic sails generated by superconducting coils. Integrating the equations of motion for interstellar protons hitting a Biot Savart loop we evaluate the effective reflection area A(v) in terms of the velocity v of the craft. We find that the numerical data is fitted over two orders of magnitude by the scaling relation A(v) = 0.081ARlog3(I/(βIc)), where AR=πR2 is the bare sail area, I the current and β=v/c. The critical current Ic is 1.55 106 Ampere. The resulting universal deceleration profile can be evaluated analytically and mission parameters optimized for a minimal craft mass.

For the case of a sample high speed transit to Proxima Centauri we find that magnetic momentum breaking would involve daunting mass requirements of the order of 103 tons. A low speed mission to the Trappist-1 system could be realized on the other side already with a 1.5 ton spacecraft, which would be furthermore compatible with the specifications of currently envisioned directed energy launch systems. The extended cruising times of the order of 104 years imply however that a mission to the Trappist-1 system would be viable only for mission concepts for which time constrains are not relevant.


How to test for partially predictable chaos

Authors: H. Wernecke, B. Sandor, C. Gros
Journal-ref.: Scientific Reports 7, 1087 (2017).
For a chaotic system pairs of initially close-by trajectories become eventually fully uncorrelated on the attracting set. This process of decorrelation is split into an initial decrease, characterized by the maximal Lyapunov exponent, and a subsequent diffusive process on the chaotic attractor causing the final loss of predictability. The time scales of both processes can be either of the same or of very different orders of magnitude. In the latter case the two trajectories linger within a finite but small distance (with respect to the overall size of the attractor) for exceedingly long times and therefore remain partially predictable. Tests for distinguishing chaos from laminar flow widely use the time evolution of inter-orbital correlations as an indicator. Standard tests however yield mostly ambiguous results when it comes to distinguish partially predictable chaos and laminar flow, which are respectively characterized by attractors of fractally broadened braids and limit cycles respectively. For a resolution we introduce a novel 0-1 indicator for chaos based on the cross-distance scaling of pairs of initially close trajectories, showing that this test robustly discriminate chaos, including partially predictable chaos, from laminar flow. For a complete classification we use the finite time cross-correlation of pairs of initially close trajectories to also draw a distinction between chaos and partial predictability. We are thus able to identify all three types of dynamics in a 0-1 manner from the properties of pairs of trajectories.


Developing Ecospheres on Transiently Habitable Planets: The Genesis Project

Authors: C. Gros
Journal-ref.: Astrophysics and Space Science 361, 324 (2016).
Interviews:  How to Jumpstart Life Elsewhere in Our Galaxy
Q&A: Should we seed life on alien worlds?
Press:  brazilian, chinese, czech, dutch, english [video], french, german, italian, korean, polish, russian, spanish [video], turkish, vietnamese
Discussions:  Claudius Gros is the Johnny Appleseed of Extraterrestrial Life
Experts Debate Our Duty and Ability to Spread Life around the Cosmos
Genesis Project: Should We 'Gift' the Cosmos With Life?
Mit Mikro-Raumschiffen ins All
Semer la vie sur d'autres planétes en envoyant des microbes? Ça n'a rien de farfelu
See also: C. Gros: Terrestrial life for habitable oxygen worlds
It is often presumed, that life evolves relatively fast on planets with clement conditions, at least in its basic forms, and that extended periods of habitability are subsequently needed for the evolution of higher life forms. Many planets are however expected to be only transiently habitable. On a large set of otherwise suitable planets life will therefore just not have the time to develop on its own to a complexity level as it did arise on earth with the cambrian explosion. The equivalent of a cambrian explosion may however have the chance to unfold on transiently habitable planets if it would be possible to fast forward evolution by 3-4 billion years (with respect to terrestrial timescales). We argue here, that this is indeed possible when seeding the candidate planet with the microbial lifeforms, bacteria and unicellular eukaryotes alike, characterizing earth before the cambrian explosion. An interstellar mission of this kind, denoted the `Genesis project', could be carried out by a relatively low-cost robotic microcraft equipped with a on-board gene laboratory for the in situ synthesis of the microbes.

We review here our current understanding of the processes determining the timescales shaping the geo-evolution of an earth-like planet, the prospect of finding Genesis candidate planets and selected issues regarding the mission layout. Discussing the ethical aspects connected with a Genesis mission, which would be expressively not for human benefit, we will also touch the risk that a biosphere incompatibility may arise in the wake of an eventual manned exploration of a second earth.


Attractor metadynamics in a recurrent neural network: adiabatic vs. symmetry protected flow

Authors: H. Wernecke, B. Sandor, C. Gros
Journal-ref.: to be published
In dynamical systems with distinct time scales the time evolution in phase space may be influenced strongly by slow manifolds. Orbits then typically follow the slow manifold, which hence act as a transient attractor, performing in addition rapid transitions between distinct branches of the slow manifold on the time scales of the fast variables. These intermittent transitions correspond to state switching within transient state dynamics. A full characterization of slow manifolds is often difficult, e. g. in neural networks with a large number of dynamical variables, due to the generically complex shape. We therefore introduce here the concept of locally attracting points, the target points. The set of target points is, by definition, the subsets of the slow manifold guiding the time evolution of a given trajectory.

We consider here systems, in which the overall dynamics settles in the limit of long times either in a limit cycle switching between transient states, or in a chaotic attractor. The set of target points then decomposes into one-dimensional (or fractal) branches, which can be analyzed directly. Here we examine the role of target points as transiently stable attractors in an autonomously active recurrent neural network. We quantify their influence on the transient states by measuring the effective distance between trajectories and the corresponding target points in phase space. We also present an example of chaotic dynamics, discussing how the chaotic motion is related to the set of transient attractors.

The network considered contains, for certain parameters settings, symmetry protected solutions in the form of travelling waves. We find, that the slow manifold does not guide the flow in this regime, which we denote as non-adiabatic, even though there are up to four orders of magnitude difference between the slow and the fast time scales.


Drifting States and Synchronization Induced Chaos in Autonomous Networks of Excitable Neurons

Authors: R. Echeveste, C. Gros
Journal-ref.: Frontiers in Computational Neuroscience 10, 98 (2016).
The study of balanced networks of excitatory and inhibitory neurons has led to several open questions. On the one hand it is yet unclear whether the asynchronous state observed in the brain is autonomously generated, or if it results from the interplay between external drivings and internal dynamics. It is also not known, which kind of network variabilities will lead to irregular spiking and which to synchronous firing states. Here we show how isolated networks of purely excitatory neurons generically show asynchronous firing whenever a minimal level of structural variability is present together with a refractory period. Our autonomous networks are composed of excitable units, in the form of leaky integrators spiking only in response to driving currents, remaining otherwise quiet. For a non-uniform network, composed exclusively of excitatory neurons, we find a rich repertoire of self-induced dynamical states. We show in particular that asynchronous drifting states may be stabilized in purely excitatory networks whenever a refractory period is present. Other states found are either fully synchronized or mixed, containing both drifting and synchronized components. The individual neurons considered are excitable and hence do not dispose of intrinsic natural firing frequencies. An effective network-wide distribution of natural frequencies is however generated autonomously through self-consistent feedback loops. The asynchronous drifting state is, additionally, amenable to an analytic solution. We find two types of asynchronous activity, with the individual neurons spiking regularly in the pure drifting state, albeit with a continuous distribution of firing frequencies. The activity of the drifting component, however, becomes irregular in the mixed state, due to the periodic driving of the synchronized component. We propose a new tool for the study of chaos in spiking neural networks, which consists of an analysis of the time series of pairs of consecutive interspike intervals. In this space, we show that a strange attractor with a fractal dimension of about 1.8 is formed in the mentioned mixed state.


Closed-loop robots driven by short-term synaptic plasticity: Emergent explorative vs. limit-cycle locomotion

Authors: L. Martin, B. Sandor, C. Gros
Journal-ref.: Frontiers in Neurorobotics 10, 12 (2016).
See also: L. Martin, B. Sandor, C. Gros: The role of the sensorimotor loop for cognition, EUCognition 2016 proceedings Cognitive Robot Architectures
We examine the hypothesis, that short-term synaptic plasticity (STSP) may generate self-organized motor patterns. We simulated sphere-shaped autonomous robots, within the LPZRobots simulation package, containing three weights moving along orthogonal internal rods. The position of a weight is controlled by a single neuron receiving excitatory input from the sensor, measuring its actual position, and inhibitory inputs from the other two neurons. The inhibitory connections are transiently plastic, following physiologically inspired STSP-rules.

We find that a wide palette of motion patterns are generated through the interaction of STSP, robot, and environment (closed-loop configuration), including various forward meandering and circular motions, together with chaotic trajectories. The observed locomotion is robust with respect to additional interactions with obstacles. In the chaotic phase the robot is seemingly engaged in actively exploring its environment. We believe that our results constitute a concept of proof that transient synaptic plasticity, as described by STSP, may potentially be important for the generation of motor commands and for the emergence of complex locomotion patterns, adapting seamlessly also to unexpected environmental feedback.

Induced (by collisions) and spontaneous mode switching are observed. We find that locomotion may follow transiently unstable limit cycles. The degeneracy of the propagating state with respect to the direction of propagating is, in our analysis, one of the drivings for the chaotic wandering, which partly involves a smooth diffusion of the angle of propagation.


Emergent lattices with geometrical frustration in doped extended Hubbard models

Authors: R. Kaneko, L.F. Tocchio, R. Valenti, C. Gros
Journal-ref.: Physical Review B 94, 195111 (2016).
Spontaneous charge ordering occurring in correlated systems may be considered as a possible route to generate effective lattice structures with unconventional couplings. For this purpose we investigate the phase diagram of doped extended Hubbard models on two lattices: (i) the honeycomb lattice with on-site U and nearest neighbor V Coulomb interactions at 3/4 filling (n=3/2) and (ii) the triangular lattice with on-site U, nearest neighbor V, and next-nearest neighbor V′ Coulomb interactions at 3/8 filling (n=3/4). We consider various approaches including mean-field approximations, perturbation theory, and variational Monte Carlo (VMC). For the honeycomb case (i), charge order induces an effective triangular lattice at large values of U/t and V/t. The nearest-neighbor spin exchange interactions on this effective triangular lattice are antiferromagnetic in most of the phase diagram, while they become ferromagnetic when U is much larger than V. At U/t∼(V/t)3, ferromagnetic and antiferromagnetic exchange interactions nearly cancel out, leading to a system with four-spin ring-exchange interactions. On the other hand, for the triangular case (ii) at large U and finite V′, we find no charge order for small V, an effective kagome lattice for intermediate V, and one-dimensional charge order for large V. These results indicate that Coulomb interactions induce (case (i)) or enhance (case(ii)) emergent geometrical frustration of the spin degrees of freedom in the system, by forming charge order.


Spontaneous symmetry breaking in correlated wave functions

Authors: R. Kaneko, L.F. Tocchio, R. Valenti, F. Becca, C. Gros
Journal-ref.: Physical Review B 93, 125127 (2016).
We show that Jastrow-Slater wave functions, in which a density-density Jastrow factor is applied onto an uncorrelated fermionic state, may possess long-range order even when all symmetries are preserved in the wave function. This fact is mainly related to the presence of a sufficiently strong Jastrow term (also including the case of full Gutzwiller projection, suitable for describing spin models). Selected examples are reported, including the spawning of N\'eel order and dimerization in spin systems, and the stabilization of density and orbital order in itinerant electronic systems.


The sensorimotor loop as a dynamical system: How regular motion primitives may emerge from self-organized limit cycles

Authors: B. Sandor, T. Jahn, L. Martin, C. Gros
Journal-ref.: Frontiers in Robotics and AI 2, 31 (2015).
We investigate the sensorimotor loop of simple robots simulated within the LPZRobots environment from the point of view of dynamical systems theory. For a robot with a cylindrical shaped body and an actuator controlled by a single proprioceptual neuron we find various types of periodic motions in terms of stable limit cycles. These are self-organized in the sense, that the dynamics of the actuator kicks in only, for a certain range of parameters, when the barrel is already rolling, stopping otherwise. The stability of the resulting rolling motions terminates generally, as a function of the control parameters, at points where fold bifurcations of limit cycles occur. We find that several branches of motion types exist for the same parameters, in terms of the relative frequencies of the barrel and of the actuator, having each their respective basins of attractions in terms of initial conditions. For low drivings stable limit cycles describing periodic and drifting back-and-forth motions are found additionally. These modes allow to generate symmetry breaking explorative behavior purely by the timing of an otherwise neutral signal with respect to the cyclic back-and-forth motion of the robot.


A versatile class of prototype dynamical systems for complex bifurcation cascades of limit cycles

Authors: B. Sandor, C. Gros
Journal-ref.: Scientific Reports 5, 12316 (2015).
We introduce a versatile class of prototype dynamical systems for the study of complex bifurcation cascades of limit cycles, including bifurcations breaking spontaneously a symmetry of the system, period doubling bifurcations and transitions to chaos induced by sequences of limit cycle bifurcations. The prototype system consist of a 2d-dimensional dynamical system with friction forces f(V(x)) functionally dependent exclusively on the mechanical potential V(x), which is typically characterized, here, by a finite number of local minima. We present examples for d=1,2 and simple polynomial friction forces f(V), where the zeros of f(V) regulate the relative importance of energy uptake and dissipation respectively, serving as bifurcation parameters. Starting from simple Hopf- and homoclinic bifurcations, complex sequences of limit cycle bifurcation are observed when energy uptake gains progressively in importance.


The Fisher Information as a Neural Guiding Principle for Independent Component Analysis

Authors: R. Echeveste, S. Eckmann, C. Gros
Journal-ref.: Entropy 17, 3838 (2015).
See also:  R. Echeveste, C. Gros: An objective function for self-limiting neural plasticity rules, ESANN 2015 proceedings.
The Fisher information constitutes a natural measure for the sensitivity of a probability distribution with respect to a set of parameters. An implementation of the stationarity principle for synaptic learning in terms of the Fisher information results in a Hebbian self-limiting learning rule for synaptic plasticity. In the present work, we study the dependence of the solutions to this rule in terms of the moments of the input probability distribution and find a preference for non-Gaussian directions, making it a suitable candidate for independent component analysis (ICA). We confirm in a numerical experiment that a neuron trained under these rules is able to find the independent components in the non-linear bars problem. The specific form of the plasticity rule depends on the transfer function used, becoming a simple cubic polynomial of the membrane potential for the case of the rescaled error function. The cubic learning rule is also an excellent approximation for other transfer functions, as the standard sigmoidal, and can be used to show analytically that the proposed plasticity rules are selective for directions in the space of presynaptic neural activities characterized by a negative excess kurtosis.


Two-trace model for spike-timing-dependent synaptic plasticity

Authors: R. Echeveste, C. Gros
Journal-ref.: Neural Computation March 27, 672 (2015).
We present an effective model for timing-dependent synaptic plasticity (STDP) in terms of two interacting traces, corresponding to the fraction of activated NMDA receptors and the Ca2+ concentration in the dendritic spine of the postsynaptic neuron. This model intends to bridge the worlds of existing simplistic phenomenological rules and highly detailed models, constituting thus a practical tool for the study of the interplay between neural activity and synaptic plasticity in extended spiking neural networks. For isolated pairs of pre- and postsynaptic spikes the standard pairwise STDP rule is reproduced, with appropriate parameters determining the respective weights and time scales for the causal and the anti-causal contributions. The model contains otherwise only three free parameters which can be adjusted to reproduce triplet nonlinearities in both hippocampal culture and cortical slices. We also investigate the transition from time-dependent to rate-dependent plasticity occurring for both correlated and uncorrelated spike patterns.


Generating functionals for computational intelligence: the Fisher information as an objective function for self-limiting Hebbian learning rules

Authors: R. Echeveste, C. Gros
Journal-ref.: Frontiers in Robotics and AI 1, 1 (2014).
See also: R. Echeveste, C. Gros: Corrigendum
Generating functionals may guide the evolution of a dynamical system and constitute a possible route for handling the complexity of neural networks as relevant for computational intelligence. We propose and explore a new objective function, which allows to obtain plasticity rules for the afferent synaptic weights. The adaption rules are Hebbian, self-limiting, and result from the minimization of the Fisher information with respect to the synaptic flux. We perform a series of simulations examining the behavior of the new learning rules in various circumstances. The vector of synaptic weights aligns with the principal direction of input activities, whenever one is present. A linear discrimination is performed when there are two or more principal directions; directions having bimodal firing-rate distributions, being characterized by a negative excess kurtosis, are preferred. We find robust performance and full homeostatic adaption of the synaptic weights results as a by-product of the synaptic flux minimization. This self-limiting behavior allows for stable online learning for arbitrary durations. The neuron acquires new information when the statistics of input activities is changed at a certain point of the simulation, showing however, a distinct resilience to unlearn previously acquired knowledge. Learning is fast when starting with randomly drawn synaptic weights and substantially slower when the synaptic weights are already fully adapted.


Attractor Metadynamics in Adapting Neural Networks

Authors: C. Gros, M. Linkerhand, V. Walther
Journal-ref.: Artificial Neural Networks and Machine Learning - ICANN 2014 , S. Wermter et al. (Eds), pp. 65-72. Springer (2014).
Slow adaption processes, like synaptic and intrinsic plasticity, abound in the brain and shape the landscape for the neural dynamics occurring on substantially faster timescales. At any given time the network is characterized by a set of internal parameters, which are adapting continuously, albeit slowly. This set of parameters defines the number and the location of the respective adiabatic attractors. The slow evolution of network parameters hence induces an evolving attractor landscape, a process which we term attractor metadynamics. We study the nature of the metadynamics of the attractor landscape for several continuous-time autonomous model networks. We find both first- and second-order changes in the location of adiabatic attractors and argue that the study of the continuously evolving attractor landscape constitutes a powerful tool for understanding the overall development of the neural dynamics.


One-dimensional spin liquid, collinear, and spiral phases from uncoupled chains to the triangular lattice

Authors: L.F. Tocchio, C. Gros, R. Valenti, F. Becca
Journal-ref.: Physical Review B 89, 235107 (2014).
We investigate the Hubbard model on the anisotropic triangular lattice with two hopping parameters t and t' in different spatial directions, interpolating between decoupled chains (t=0) and the isotropic triangular lattice (t=t'). Variational wave functions that include both Jastrow and backflow terms are used to compare spin-liquid and magnetic phases with different pitch vectors describing both collinear and coplanar (spiral) order. For relatively large values of the on-site interaction U/t'≥10 and substantial frustration, i.e., 0.3≤t/t'≤0.8, the spin-liquid state is clearly favored over magnetic states. Spiral magnetic order is only stable in the vicinity of the isotropic point, while collinear order is obtained in a wide range of inter-chain hoppings from small to intermediate frustration.


Phase diagram of the triangular extended Hubbard model

Authors: L.F. Tocchio, C. Gros, X.-F. Zhang, S. Eggert
Journal-ref.: Physical Review Letters 113, 246405 (2014).
We study the extended Hubbard model on the triangular lattice as a function of filling and interaction strength. The complex interplay of kinetic frustration and strong interactions on the triangular lattice leads to exotic phases where long-range charge order, antiferromagnetic order, and metallic conductivity can coexist. Variational Monte Carlo simulations show that three kinds of ordered metallic states are stable as a function of nearest neighbor interaction and filling. The coexistence of conductivity and order is explained by a separation into two functional classes of particles: part of them contributes to the stable order, while the other part forms a partially filled band on the remaining substructure. The relation to charge ordering in charge transfer salts is discussed.


Phase diagram of the square lattice bilayer Hubbard model: A variational Monte Carlo study

Authors: R. Rüger, L.F. Tocchio, R. Valenti, C. Gros
Journal-ref.: New Journal of Physics 16, 033010 (2014).
We investigate the phase diagram of the square lattice bilayer Hubbard model at half filling with the variational Monte Carlo method for both the magnetic and the paramagnetic case as a function of interlayer hopping t_perp and on-site Coulomb repulsion U. With this study we resolve some discrepancies in previous calculations based on the dynamical mean field theory, and we are able to determine the nature of the phase transitions between metal, Mott insulator and band insulator. In the magnetic case we find only two phases: An antiferromagnetic Mott insulator at small t_perp for any value of U and a band insulator at large t_perp. At large U values we approach the Heisenberg limit. The paramagnetic phase diagram shows at small t_perp a metal to Mott insulator transition at moderate U values and a Mott to band insulator transition at larger U values. We also observe a reentrant Mott insulator to metal transition and metal to band insulator transition for increasing t_perp in the range of 5.5t < U < 7.5t. Finally, we discuss the obtained phase diagrams in relation to previous studies based on different many-body approaches.


Exploration in Free Word Association Networks: Models and Experiment

Authors: G.A. Luduena, M.D. Behzad, C. Gros
Journal-ref.: Cognitive Processing 15, 195 (2014).
Free association is a task that requires a subject to express the first word to come to their mind when presented with a certain cue. It is a task which can be used to expose the basic mechanisms by which humans connect memories. In this work we have made use of a publicly available database of free associations to model the exploration of the averaged network of associations using a statistical and the \emph{ACT-R} model. We performed, in addition, an online experiment asking participants to navigate the averaged network using their individual preferences for word associations. We have investigated the statistics of word repetitions in this guided association task. We find that the considered models mimic some of the statistical properties, viz the probability of word repetitions, the distance between repetitions and the distribution of association chain lengths, of the experiment, with the \emph{ACT-R} model showing a particularly good fit to the experimental data for the more intricate properties as, for instance, the ratio of repetitions per length of association chains.


Powerlaws and Self-Organized Criticality in Theory and Nature

Authors: D. Markovic, C. Gros
Journal-ref.: Physics Reports 536, 41-74 (2014).
Powerlaws and distributions with heavy tails are common features of many experimentally studied complex systems, like the distribution of the sizes of earthquakes and solar flares, or the duration of neuronal avalanches in the brain. It had been tempting to surmise that a single general concept may act as a unifying underlying generative mechanism, with the theory of self organized criticality being a weighty contender.

On the theory side there has been, lively activity in developing new and extended models. Three classes of models have emerged. The first line of models is based on a separation between the time scales of drive and 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 approach 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 modeling proposes a non-critical state which is self-organizing, 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, like the theory of self-organized criticality, the interaction between theoretical development and experimental observations has been very fruitful, leading to a series of novel concepts and insights.


Generating functionals for guided self-organization

Authors: C. Gros
Journal-ref.: M. Prokopenko (ed.), Guided Self-Organization: Inception, 53-66, Springer (2014).
Time evolution equations for dynamical systems can often be derived from generating functionals. Examples are Newton's equations of motion in classical dynamics which can be generated within the Lagrange or the Hamiltonian formalism. We propose that generating functionals for self-organizing complex systems offer several advantages. Generating functionals allow to formulate complex dynamical systems systematically and the results obtained are typically valid for classes of complex systems, as defined by the type of their respective generating functionals. The generated dynamical systems tend, in addition, to be minimal, containing only few free and undetermined parameters. We point out that two or more generating functionals may be used to define a complex system and that multiple generating function may not, and should not, be combined into a single overall objective function. We provide and discuss examples in terms of adapting neural networks.


A large-scale study of the World Wide Web:
network correlation functions with scale-invariant boundaries

Authors: G.A. Luduena, H. Meixner, G. Kaczor, C. Gros
Journal-ref.: European Physical Journal B 86, 348 (2013).
We performed a large-scale crawl of the World Wide Web, covering 6.9 Million domains and 57 Million subdomains, including all high-traffic sites of the Internet. We present a study of the correlations found between quantities measuring the structural relevance of each node in the network (the in- and out-degree, the local clustering coefficient, the first-neighbor in-degree and the Alexa rank). We find that some of these properties show strong correlation effects and that the dependencies occurring out of these correlations follow power laws not only for the averages, but also for the boundaries of the respective density distributions. In addition, these scale-free limits do not follow the same exponents as the corresponding averages. In our study we retain the directionality of the hyperlinks and develop a statistical estimate for the clustering coefficient of directed graphs.

We include in our study the correlations between the in-degree and the Alexa traffic rank, a popular index for the traffic volume, finding non-trivial power-law correlations. We find that sites with more/less than about one Thousand links from different domains have remarkably different statistical properties, for all correlation functions studied, indicating towards an underlying hierarchical structure of the World Wide Web.


Generating functionals for autonomous latching dynamics in attractor relict networks

Authors: M. Linkerhand, C. Gros
Journal-ref.: Scientific Reports 3, 2042 (2013).
Coupling local, slowly adapting variables to an attractor network allows to destabilize all attractors, turning them into attractor ruins. The resulting attractor relict network may show ongoing autonomous latching dynamics. We propose to use two generating functionals for the construction of attractor relict networks, a Hopfield energy functional generating a neural attractor network and a functional based on information-theoretical principles, encoding the information content of the neural firing statistics, which induces latching transition from one transiently stable attractor ruin to the next.

We investigate the influence of stress, in terms of conflicting optimization targets, on the resulting dynamics. Objective function stress is absent when the target level for the mean of neural activities is identical for the two generating functionals and the resulting latching dynamics is then found to be regular. Objective function stress is present when the respective target activity levels differ, inducing intermittent bursting latching dynamics.


Observing scale-invariance in non-critical dynamical systems

Authors: C. Gros, D. Markovic
Journal-ref.: Physics, Computation and the Mind - Advances and Challenges at Interfaces, P.L. Garrido, J. Marro, J.J. Torres, J.M. Cortes (Eds). AIP (2013). .
Recent observation for scale invariant neural avalanches in the brain have been discussed in details in the scientific literature. We point out, that these results do not necessarily imply that the properties of the underlying neural dynamics are also scale invariant. The reason for this discrepancy lies in the fact that the sampling statistics of observations and experiments is generically biased by the size of the basins of attraction of the processes to be studied. One has hence to precisely define what one means with statements like `the brain is critical'.

We recapitulate the notion of criticality, as originally introduced in statistical physics for second order phase transitions, turning then to the discussion of critical dynamical systems. We elucidate in detail the difference between a 'critical system', viz a system on the verge of a phase transition, and a 'critical state', viz state with scale-invariant correlations, stressing the fact that the notion of universality is linked to critical states.

We then discuss rigorous results for two classes of critical dynamical systems, the Kauffman net and a vertex routing model, which both have non-critical states. However, an external observer that samples randomly the phase space of these two critical models, would find scale invariance. We denote this phenomenon as 'observational criticality' and discuss its relevance for the response properties of critical dynamical systems.


Mott correlated states in the underdoped two-dimensional Hubbard model: variational Monte Carlo versus a dynamical cluster approximation

Authors: L.F. Tocchio, H. Lee, H.O. Jeschke, R.Valentí, C. Gros
Journal-ref.: Physical Review B 87, 045111 (2013).
We investigate the properties of the frustrated underdoped Hubbard model on the square lattice using two complementary approaches, the dynamical cluster extension of dynamical mean field theory, and variational Monte Carlo simulations of Gutzwiller-Jastrow wavefunctions with backflow corrections. We find good agreement, apart from the location of the Mott-Hubbard transition which differs. At small dopings, we observe a rapid crossover from a weakly correlated metal at low interaction strength U to a non Fermi liquid correlated state with strong local spin correlations, which we identify as the pseudo-gap state of the high-Tc superconductors. Furthermore, we investigate the stability of the pseudo-gap state against phase separation. We observe phase separation only for large values of U or very large frustration. No phase separation is present for the parameter range relevant for the cuprates.


Spin-liquid versus spiral-order phases in the anisotropic triangular lattice

Authors: L.F. Tocchio, H. Feldner, F. Becca, R. Valentí, C. Gros
Journal-ref.: Physical Review B 87, 035143 (2013).
We study the competition between magnetic and spin-liquid phases in the Hubbard model on the anisotropic triangular lattice, which is described by two hopping parameters t and t' in different spatial directions and is relevant for layered organic charge-transfer salts. By using a variational approach that includes spiral magnetic order, we provide solid evidence that a spin-liquid phase is stabilized in the strongly-correlated regime and close to the isotropic limit t'/t=1. Otherwise, a magnetically ordered spiral state is found, connecting the (collinear) Néel and the (coplanar) 120° phases. The pitch vector of the spiral phase obtained from the unrestricted Hartree-Fock approximation is substantially renormalized in presence of electronic correlations, and the Néel phase is stabilized in a wide regime of the phase diagram, i.e., for t'/t < 0.75. We discuss these results in the context of organic charge-transfer salts.


Criticality in conserved dynamical systems:
Experimental observation vs. exact properties

Authors: D. Markovic, A. Schuelein, C. Gros
Journal-ref.: Chaos 23, 013106 (2013).
Conserved dynamical systems are generally considered to be critical. We study a class of critical routing models, equivalent to random maps, which can be solved rigorously in the thermodynamic limit. The information flow is conserved for these routing models and governed by cyclic attractors. We consider two classes of information flow, Markovian routing without memory and vertex routing involving a one-step routing memory. Investigating the respective cycle length distributions for complete graphs we find log corrections to power-law scaling for the mean cycle length, as a function of the number of vertices, and a sub-polynomial growth for the overall number of cycles.

When observing experimentally a real-world dynamical system one normally samples stochastically its phase space. The number and the length of the attractors are then weighted by the size of their respective basins of attraction. This situation is equivalent to `on the fly' generation of routing tables for which we find power law scaling for the weighted average length of attractors, for both conserved routing models. These results show that critical dynamical systems are generically not scale-invariant, but may show power-law scaling when sampled stochastically. It is hence important to distinguish between intrinsic properties of a critical dynamical system and its behavior that one would observe when randomly probing its phase space.


Self-organized stochastic tipping in slow-fast dynamical systems

Authors: M. Linkerhand, C. Gros
Journal-ref.: Mathematics and Mechanics of Complex Systems 1, 129 (2013).
Polyhomeostatic adaption occurs when evolving systems try to achieve a target distribution function for certain dynamical parameters, a generalization of the notion of homeostasis. Here we consider a single rate encoding leaky integrator neuron model driven by white noise, adapting slowly its internal parameters, the threshold and the gain, in order to achieve a given target distribution for its time-average firing rate. For the case of sparse encoding, when the target firing-rated distribution is bimodal, we observe the occurrence of spontaneous quasi-periodic adaptive oscillations resulting from fast transition between two quasi-stationary attractors. We interpret this behavior as self-organized stochastic tipping, with noise driving the escape from the quasi-stationary attractors.


A Self-Organized Neural Comparator

Authors: G.A. Luduena, C. Gros
Journal-ref.: Neural Computation 25, 1006 (2013).
Learning algorithms need generally the possibility to compare several streams of information. Neural learning architectures hence need a unit, a comparator, able to compare several inputs encoding either internal or external information, like for instance predictions and sensory readings. Without the possibility of comparing the values of prediction to actual sensory inputs, reward evaluation and supervised learning would not be possible. Comparators are usually not implemented explicitly, necessary comparisons are commonly performed by directly comparing one-to-one the respective activities. This implies that the characteristics of the two input streams (like size and encoding) must be provided at the time of designing the system.

It is however plausible that biological comparators emerge from self-organizing, genetically encoded principles, which allow the system to adapt to the changes in the input and in the organism. We propose an unsupervised neural circuitry, where the function of input comparison emerges via self-organization only from the interaction of the system with the respective inputs, without external influence or supervision.

The proposed neural comparator adapts, unsupervised, according to the correlations present in the input streams. The system consists of a multilayer feed-forward neural network which follows a local output minimization (anti-Hebbian) rule for adaptation of the synaptic weights. The local output minimization allows the circuit to autonomously acquire the capability of comparing the neural activities received from different neural populations, which may differ in the size of the population and in the neural encoding used. The comparator is able to compare objects never encountered before in the sensory input streams and to evaluate a measure of their similarity, even when differently encoded.


Emotional control - conditio sine qua non for advanced artificial intelligences?

Authors: C. Gros
Journal-ref.: Philosophy and Theory of Artificial Intelligence, 187-198; V.C. Müller (Ed.), Springer (2013).
Humans dispose of two intertwined information processing pathways, cognitive information processing via neural firing patterns and diffusive volume control via neuromodulation. The cognitive information processing in the brain is traditionally considered to be the prime neural correlate of human intelligence, clinical studies indicate that human emotions intrinsically correlate with the activation of the neuromodulatory system.

We examine here the question: Why do humans dispose of the diffusive emotional control system? Is this a coincidence, a caprice of nature, perhaps a leftover of our genetic heritage, or a necessary aspect of any advanced intelligence, being it biological or synthetic? We argue here that emotional control is necessary to solve the motivational problem, viz the selection of short-term utility functions, in the context of an environment where information, computing power and time constitute scarce resources.


Pushing the complexity barrier: diminishing returns in the sciences

Authors: C. Gros
Journal-ref.: Complex Systems 21, 183 (2012).
See also: Gros: Forschungsförderung quo vadis - Effizienz und Komplexitätsbarrieren in den Wissenschaften
Are the sciences not advancing at an ever increasing speed? We contrast this popular perspective with the view that science funding may actually see diminishing returns, at least regarding established fields. In order to stimulate a larger discussion, we investigate two exemplary cases, the linear increase in human life expectancy over the last 170 years and the advances in the reliability of numerical short and medium term weather predictions during the last 50 years. We argue that the outcome of science and technology funding in terms of measurable results is a highly sub-linear function of the amount of resources committed. Supporting a range of small to medium size research projects, instead of a few large ones, will be, as a corollary, a more efficient use of resources for science funding agencies.


Strong renormalization of the Fermi-surface topology close to the Mott transition

Authors: L.F. Tocchio, F. Becca, C. Gros
Journal-ref.: Physical Review B 86, 035102 (2012).
The underlying Fermi surface is a key concept for strongly-interacting electron models and has been introduced to generalize the usual notion of the Fermi surface to generic (superconducting or insulating) systems. By using improved correlated wave functions that contain backflow and Jastrow terms, we examine the two-dimensional t-t' Hubbard model and find a non-trivial renormalization of the topology of the underlying Fermi surface close to the Mott insulator. Moreover, we observe a sharp crossover region, which arises from the metal-insulator transition, from a weakly interacting metal at small coupling to a resonating valence-bond superconductor at intermediate coupling. A violation of the Luttinger theorem is detected at low hole dopings.


Intrinsic adaptation in autonomous recurrent neural networks

Authors: D. Markovic, C. Gros
Journal-ref.: Neural Computation 24, 523 (2012)
A massively recurrent neural network responds on one side to input stimuli and is autonomously active, on the other side, in the absence of sensory inputs. Stimuli and information processing depends crucially on the qualia of the autonomous-state dynamics of the ongoing neural activity. This default neural activity may be dynamically structured in time and space, showing regular, synchronized, bursting or chaotic activity patterns.
We study the influence of non-synaptic plasticity on the default dynamical state of recurrent neural networks. The non-synaptic adaption considered acts on intrinsic neural parameters, such as the threshold and the gain, and is driven by the optimization of the information entropy. We observe, in the presence of the intrinsic adaptation processes, three distinct and globally attracting dynamical regimes, a regular synchronized, an overall chaotic and an intermittent bursting regime. The intermittent bursting regime is characterized by intervals of regular flows, which are quite insensitive to external stimuli, interseeded by chaotic bursts which respond sensitively to input signals. We discuss these finding in the context of self-organized information processing and critical brain dynamics.


Neuropsychological constraints to human data production on a global scale

Authors: C. Gros, G. Kaczor, D. Markovic
Journal-ref.: European Physical Journal B 85, 28 (2012)
Synopsis: Europhysics News 43, 16 (2012)
Press: MIT Technology Review, Red Orbit, Pressetext, Inovação Tecnologica
Which are the factors underlying human information production on a global level? In order to gain an insight into this question we study a corpus of 252-633 Million publicly available data files on the Internet corresponding to an overall storage volume of 284-675 Terabytes. Analyzing the file size distribution for several distinct data types we find indications that the neuropsychological capacity of the human brain to process and record information may constitute the dominant limiting factor for the overall growth of globally stored information, with real-world economic constraints having only a negligible influence. This supposition draws support from the observation that the files size distributions follow a power law for data without a time component, like images, and a log-normal distribution for multimedia files, for which time is a defining qualia.


Backflow correlations in the Hubbard model:
an efficient tool for the metal-insulator transition and the large-U limit

Authors: L.F. Tocchio, F. Becca, C. Gros
Journal-ref.: Physical Review B 83, 195138 (2011).
We show that backflow correlations in the variational wave function for the Hubbard model greatly improve the previous results given by the Slater-Jastrow state, usually considered in this context. We provide evidence that, within this approach, it is possible to have a satisfactory connection with the strong-coupling regime. Moreover, we show that, for the Hubbard model on the lattice, backflow correlations are essentially short range, inducing an effective attraction between empty (holons) and doubly occupied sites (doublons). In presence of frustration, we report the evidence that the metal to Mott-insulator transition is marked by a discontinuity of the double occupancy, together with a similar discontinuity of the kinetic term that does not change the number of holons and doublons, while the other kinetic terms are continuous across the transition. Finally, we show the estimation of the charge gap, obtained by particle-hole excitations á la Feynman over the ground-state wave function.


Tunnelling matrix elements with antiferromagnetic Gutzwiller wave functions

Authors: A. Di Ciolo, L.F. Tocchio, C. Gros
Journal-ref.: Physical Review B 83, 165116 (2011).
We use a generalized Gutzwiller Approximation (GA) elaborated to evaluate matrix elements with partially projected wave functions and formerly applied to homogeneous systems. In the present paper we consider projected single-particle (hole) excitations for electronic systems with antiferromagnetic (AFM) order and obtain the corresponding tunnelling probabilities. The accuracy and the reliability of our analytical approximation is tested using the Variational Monte Carlo (VMC). Possible comparisons with experimental results are also discussed.


Self-organized chaos through polyhomeostatic optimization

Authors: D. Markovic, C. Gros
Journal-ref.: Physical Review Letters 105, 068702 (2010).
The goal of polyhomeostatic control is to achieve a certain target distribution of behaviors, in contrast to homeostatic regulation which aims at stabilizing a steady-state dynamical state. We consider polyhomeostasis for individual and networks of firing-rate neurons, adapting to achieve target distributions of firing rates maximizing information entropy. We show that any finite polyhomeostatic adaption rate destroys all attractors in Hopfield-like network setups, leading to intermittently bursting behavior and self-organized chaos. The importance of polyhomeostasis to adapting behavior in general is discussed.


Cognition and Emotion: Perspectives of a Closing Gap

Authors: C. Gros
Journal-ref.: Cognitive Computation 2, 78 (2010).
he primary tasks of a cognitive system is to survive and to maximize a life-long utility function, like the number of offsprings. A direct computational maximization of life-long utility is however not possible in complex environments, especially in the context, of real-world time constraints. The central role of emotions is to serve as an intermediate layer in the space of policies available to agents and animals, leading to a large dimensional reduction of complexity.

We review our current understanding of the functional role of emotions, stressing the role of the neuromodulators mediating emotions for the diffusive homeostatic control system of the brain. We discuss a recent proposal, that emotional diffusive control is characterized, in contrast to neutral diffusive control, by interaction effects, viz by interferences between emotional arousal and reward signaling. Several proposals for the realization of synthetic emotions are discussed in this context, together with key open issues regarding the interplay between emotional motivational drives and diffusive control.


Interaction induced Fermi-surface renormalization in the t1-t2 Hubbard model close to the Mott-Hubbard transition

Authors: L.F. Tocchio, F. Becca, C. Gros
Journal-ref.: Physical Review B 81, 205109 (2010).
We investigate the nature of the interaction-driven Mott-Hubbard transition of the half-filled t1-t2 Hubbard model in one dimension, using a full-fledged variational Monte Carlo approach including a distance-dependent Jastrow factor and backflow correlations. We present data for the evolution of the magnetic properties across the Mott-Hubbard transition and on the commensurate to incommensurate transition in the insulating state. Analyzing renormalized excitation spectra, we find that the Fermi surface renormalizes to perfect nesting right at the Mott-Hubbard transition in the insulating state, with a first-order reorganization when crossing into the conducting state.


Semantic learning in autonomously active recurrent neural networks

Authors: C. Gros, G. Kaczor
Journal-ref.: Logic Journal IGPL 81, 686 (2010).
The human brain is autonomously active, being characterized by a self-sustained neural activity which would be present even in the absence of external sensory stimuli. Here we study the interrelation between the self-sustained activity in autonomously active recurrent neural nets and external sensory stimuli.

There is no a priori semantical relation between the influx of external stimuli and the patterns generated internally by the autonomous and ongoing brain dynamics. The question then arises when and how are semantic correlations between internal and external dynamical processes learned and built up?

We study this problem within the paradigm of transient state dynamics for the neural activity in recurrent neural nets, i.e. for an autonomous neural activity characterized by an infinite time-series of transiently stable attractor states. We propose that external stimuli will be relevant during the sensitive periods, viz the transition period between one transient state and the subsequent semi-stable attractor. A diffusive learning signal is generated unsupervised whenever the stimulus influences the internal dynamics qualitatively.

For testing we have presented to the model system stimuli corresponding to the bars and stripes problem. We found that the system performs a non-linear independent component analysis on its own, being continuously and autonomously active. This emergent cognitive capability results here from a general principle for the neural dynamics, the competition between neural ensembles.


Spin-liquid and magnetic phases in the anisotropic triangular lattice:
the case of κ-(ET)2X

Authors: L.F. Tocchio, A. Parola, C. Gros, F. Becca
Journal-ref.: Physical Review B 80, 064419 (2009).
The two-dimensional Hubbard model on the anisotropic triangular lattice, with two different hopping amplitudes t and t', is relevant to describe the low-energy physics of κ-(ET)2X, a family of organic salts. The ground-state properties of this model are studied by using Monte Carlo techniques, on the basis of a recent definition of backflow correlations for strongly-correlated lattice systems. The results show that there is no magnetic order for reasonably large values of the electron-electron interaction U and frustrating ratio t'/t = 0.85, suitable to describe the non-magnetic compound with X=Cu2(CN)3. On the contrary, Néel order takes place for weaker frustrations, i.e., t'/t ~ 0.4,0.6, suitable for materials with X=Cu2(SCN)2, Cu[N(CN)2]Cl, or Cu[N(CN)2]Br.


Vertex Routing Models

Authors: D. Markovic, C. Gros
Journal-ref.: New Journal of Physics 11, 073002 (2009).
A class of models describing the flow of information within networks via routing processes is proposed and investigated, concentrating on the effects of memory traces on the global properties. The long-term flow of information is governed by cyclic attractors, allowing to define a measure for the information centrality of a vertex given by the number of attractors passing through this vertex. We find the number of vertices having a non-zero information centrality to be extensive/sub-extensive for models with/without a memory trace in the thermodynamic limit. We evaluate the distribution of the number of cycles, of the cycle length and of the maximal basins of attraction, finding a complete scaling collapse in the thermodynamic limit for the later. Possible implications of our results on the information flow in social networks are discussed.


Cognitive computation with autonomously active neural networks:
an emerging field

Authors: C. Gros
Journal-ref.: Cognitive Computation 1, 77 (2009).
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 modelling.

This review will start with a brief overview of the experimental efforts, together with a discussion of transient vs. 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 non-linear 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.


Emotions, diffusive emotional control and the motivational problem for autonomous cognitive systems

Authors: C. Gros
Journal-ref.: (Book Chapter) Handbook of Research on Synthetic Emotions and Sociable Robotics: New Applications in Affective Computing and Artificial Intelligence, J. Vallverdu, D. Casacuberta (Eds). IGI-Global (2009).
All self-active living beings need to solve the motivational problem: The question what to do at any moment of their live. For humans and non-human animals at least two distinct layers of motivational drives are known, the primary needs for survival and the emotional drives leading to a wide range of sophisticated strategies, such as explorative learning and socializing. Part of the emotional layer of drives has universal facets, being beneficial in an extended range of environmental settings. Emotions are triggered in the brain by the release of neuromodulators, which are, at the same time, the agents for meta-learning. This intrinsic relation between emotions, meta-learning and universal action strategies suggests a central importance for emotional control for the design of artificial intelligences and synthetic cognitive systems. An implementation of this concept is proposed in terms of a dense and homogeneous associative network (dHan).


Effect of external pressure on the Fe magnetic moment in undoped LaFeAsO from density functional theory: Proximity to a magnetic instability

Authors: I. Opahle, H. C. Kandpal, Y. Zhang, C. Gros, R. Valentí
Journal-ref.: Physical Review B 79, 024509 (2009).
We investigate the effect of external pressure on the Fe magnetic moment in undoped LaFeAsO within the framework of density-functional theory and show that this system is close to a magnetic instability. The Fe moment is found to drop by nearly a factor of 3 within a pressure range of ±5 GPa around the calculated equilibrium volume. While the Fe moments show an unusually strong sensitivity to the spin arrangement (type of antiferromagnetic structure), the low-temperature structural distortion is found to have only a minor influence on them. Analysis of the Fermi-surface topology and nesting features shows that these properties change very little up to pressures of at least 10 GPa. We discuss the magnetic instability in terms of the itinerancy of this system.


Proposed low energy model Hamiltonian for spin-gapped system CuTe2O5

Authors: H. Das, T. Saha-Dasgupta, C. Gros, R. Valentí
Journal-ref.: Physical Review B 77, 224437 (2008).
Using first-principles electronic structure calculations based on the Nth order muffin tin orbital (NMTO)-downfolding technique, we derived the low-energy spin model for CuTe2O5. Our study reveals that this compound is a 2D coupled spin-dimer system with the strongest Cu-Cu interaction mediated by two O-Te-O bridges. We checked the goodness of our model by computing the magnetic susceptibility with the Quantum Monte Carlo technique and by comparing it with available experimental data. We also present magnetization and specific heat results which may be compared with future experimental investigations. Our derived model is in disagreement with a recently proposed model for this compound [J. Deisenhofer et al, Physical Review B,74 (2006) 174421]. The situation needs to be settled in terms of further experimental investigations.


Evolving complex networks with conserved clique distributions

Authors: G. Kaczor, C. Gros
Journal-ref.: Physical Review E 78, 016107 (2008).
We propose and study a hierarchical algorithm to generate graphs having a predetermined distribution of cliques, the fully connected subgraphs. The construction mechanism may be either random or incorporate preferential attachment. We evaluate the statistical properties of the graphs generated, such as the degree distribution and network diameters, and compare them to some real-world graphs.


Complex and Adaptive Dynamical Systems, a Primer

Authors: C. Gros
Journal-ref.: Springer (2008/2010/2013/2015).
[100+]
We are living in an ever more complex world, an epoch where human actions can accordingly acquire far-reaching potentialities. Complex and adaptive dynamical systems are ubiquitous in the world surrounding us and require us to adapt to new realities and the way of dealing with them. This primer aims to convey a wide range of "commons-sense" knowledge in the field of quantitative complex system science at an introductory level, using modular and phenomenological approach. The chapters deal with
  • 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, Hypercyles and Game Theory;
  • Synchronization Phenomena;
  • Elements of Cognitive System Theory.
Prerequisites are a basic knowledge of ordinary and partial differential equations and of statistics. Exercises (with solutions) and suggestions for further reading are provided.


Gutzwiller-RVB Theory of High Temperature Superconductivity: Results from Renormalized Mean Field Theory and Variational Monte Carlo Calculations

Authors: B. Edegger, V.N. Muthukumar, C. Gros
Journal-ref.: Advances in Physics 56, 927 (2007)
[100+]
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 Anderson's RVB picture and its implementation by renormalised 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 tunneling microscopy (STM). We conclude by summarising 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

Authors: C. Gros
Journal-ref.: New Journal of Physics 9, 109 (2007).
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.


Spontaneous breaking of the Fermi surface symmetry in the t-J model: a numerical study

Authors: B. Edegger, C. Gros, V.N. Muthukumar
Journal-ref.: Physical Review B 74 165109 (2006).
We present a variational Monte Carlo (VMC) study of spontaneous Fermi surface symmetry breaking in the t-J model. We find that the variational energy of a Gutzwiller projected Fermi sea is lowered by allowing for a finite asymmetry between the x- and the y-directions. However, the best variational state remains a pure superconducting state with d-wave symmetry, as long as the underlying lattice is isotropic. Our VMC results are in good overall agreement with slave boson mean field theory (SBMFT) and renormalized mean field theory (RMFT), although apparent discrepancies do show up in the half-filled limit, revealing some limitations of mean field theories. VMC and complementary RMFT calculations also confirm the SBMFT predictions that many-body interactions can enhance any anisotropy in the underlying crystal lattice. Thus, our results may be of consequence for the description of strongly correlated superconductors with an anisotropic lattice structure.


Determining the underlying Fermi surface of strongly correlated superconductors

Authors: C. Gros, B. Edegger, V.N. Muthukumar, P.W. Anderson
Journal-ref.: PNAS 103, 14298 (2006).
The notion of a Fermi surface (FS) is one of the most ingenious concepts developed by solid state physicists during the past century. It plays a central role in our understanding of interacting electron systems. Extraordinary efforts have been undertaken, both by experiment and by theory, to reveal the FS of the high temperature superconductors (HTSC), the most prominent strongly correlated superconductors. Here, we discuss some of the prevalent methods used to determine the FS and show that they lead generally to erroneous results close to half filling and at low temperatures, due to the large superconducting gap (pseudogap) below (above) the superconducting transition temperature. Our findings provide a perspective on the interplay between strong correlations and superconductivity and highlight the importance of strong coupling theories for the characterization as well as the determination of the underlying FS in ARPES experiments.


Electronic structure of strongly correlated d-wave superconductors

Authors: B. Edegger, V.N. Muthukumar, C. Gros, P.W. Anderson
Journal-ref.: Physical Review Letters 96, 207002 (2006).
We study the electronic structure of a strongly correlated d-wave superconducting state. Combining a renormalized mean field theory with direct calculation of matrix elements, we obtain results for the nodal Fermi velocity, vF, the Fermi wave vector, kF, and momentum distribution, nk, as a function of hole doping in a Gutzwiller projected d-wave superconductor. We calculate the energy dispersion, Ek, and spectral weight, of the Gutzwiller-Bogoliubov quasiparticles, and find that the spectral weight associated with the quasiparticle excitation at the antinodal point shows a non monotonic behavior as a function of doping. Results are compared to angle resolved photoemission spectroscopy (ARPES) of the high temperature superconductors.


Self-Sustained Thought Processes in a Dense Associative Network

Authors: C. Gros
Journal-ref.: KI 2005, Springer Lecture Notes in Artificial Intelligence 3698, 366 (2005).
Several guiding principles for thought processes are proposed and a neural-network-type model implementing these principles is presented and studied. We suggest to consider thinking within an associative network built-up of overlapping memory states. We consider a homogeneous associative network as biological considerations rule out distinct conjunction units between the information (the memories) stored in the brain. We therefore propose that memory states have a dual functionality: They represent on one side the stored information and serve, on the other side, as the associative links in between the different dynamical states of the network which consists of transient attractors. We implement these principles within a generalized winners-take-all neural network with sparse coding and an additional coupling to local reservoirs. We show that this network is capable to generate autonomously a self-sustained time-series of memory states which we identify with a thought process. Each memory state is associatively connected with its predecessor.
This system shows several emerging features, it is able (a) to recognize external patterns in a noisy background, (b) to focus attention autonomously and (c) to represent hierarchical memory states with an internal structure.


Particle number renormalization in almost half filled Mott Hubbard superconductors

Authors: B. Edegger, N. Fukushima, C. Gros, V.N. Muthukumar
Journal-ref.: Physical Review B 72, 134504 (2005).
The effects of the Gutzwiller projection on a BCS wave function with varying particle number are considered. We show that a fugacity factor has to be introduced in these wave functions when they are Gutzwiller projected, and derive an expression for this factor within the Gutzwiller approximation. We examine the effects of the projection operator on BCS wave functions by calculating the average number of particles before and after projection. We also calculate particle number fluctuations in a projected BCS state. Finally, we point out the differences between projecting BCS wave functions in the micro and grand canonical schemes, and discuss the relevance of our results for variational Monte Carlo studies.


On the evaluation of matrix elements in partially projected wave functions

Authors: N. Fukushima, B. Edegger, V.N. Muthukumar, C. Gros
Journal-ref.: Physical Review B 72, 144505 (2005).
We generalize the Gutzwiller approximation scheme to the calculation of nontrivial matrix elements between the ground state and excited states. In our scheme, the normalization of the Gutzwiller wave function relative to a partially projected wave function with a single non projected site (the reservoir site) plays a key role. For the Gutzwiller projected Fermi sea, we evaluate the relative normalization both analytically and by variational Monte-Carlo (VMC). We also report VMC results for projected superconducting states that show novel oscillations in the hole density near the reservoir site.


Expanding advanced civilizations in the universe

Authors: C. Gros
Journal-ref.: JBIS 58, 108 (2005).
See also: peregrinus interstellar
The 1950 lunch-table remark by Enrico Fermi `Where is everybody' has started intensive scientific and philosophical discussions about what we call nowadays the `Fermi paradox': If there had been ever a single advanced civilization in the cosmological history of our galaxy, dedicated to expansion, it would have had plenty of time to colonize the entire galaxy via exponential growth. No evidence of present or past alien visits to earth are known to us, leading to the standard conclusion that no advanced expanding civilization has ever existed in the milky-way. This conclusion rest fundamentally on the ad-hoc assumption, that any alien civilizations dedicated to expansion at one time would remain dedicated to expansions forever. Considering our limited knowledge about alien civilizations we need however to relax this basic assumption. Here we show that a substantial and stable population of expanding advanced civilization might consequently exist in our galaxy.


Na2V3O7, a nanotubular system with spin-1/2 diamond rings

Authors: T. Saha-Dasgupta, R. Valentí, F. Capraro, C. Gros
Journal-ref.: Physical Review Letters 95, 107201 (2005).
Following the recent discussion on the nature of the interactions in the tubular system Na2V3O7, we present a detailed ab-initio microscopic analysis of its electronic and magnetic properties. We show by means of a downfolding study that, due to the special geometry of this material, the edge-sharing V-V hopping interactions are of the same order of magnitude as the corner-sharing paths within a ring and an order of magnitude bigger than the hopping interactions between rings in a tube. We propose an effective model in terms of weakly-coupled partially frustrated nine-site rings with the geometry of a spin-diamond necklace. We calculate the susceptibility by exact diagonalization and obtain good agreement with experimental observations.


Breakdown of the Luttinger sum-rule at the Mott-Hubbard transition in the one-dimensional t1-t2 Hubbard model

Authors: C. Gros, K. Hamacher, W. Wenzel
Journal-ref.: Europhys. Lett. 69, 616 (2005).
We investigate the momentum distribution function near the Mott-Hubbard transition in the one-dimensional t1-t2 Hubbard model (the zig-zag Hubbard chain), with the density-matrix renormalization-group technique. We show that for strong interactions the Mott-Hubbard transition occurs between the metallic-phase and an insulating dimerized phase with incommensurate spin excitations, suggesting a decoupling of magnetic and charge excitations not present in weak coupling. We illustrate the signatures for the Mott-Hubbard transition and the commensurate-incommensurate transition in the insulating spin-gapped state in their respective ground-state momentum distribution functions.


The Spin-SAF transition in NaV2O5 induced by spin-pseudospin coupling

Authors: C. Gros, G.Y. Chitov
Journal-ref.: Europhys. Lett. 69, 447 (2005).
We present microscopic estimates for the spin-spin and spin-speudospin interactions of the quarter-filled ladder compound NaV2O5, obtained by exactly diagonalizing appropriate clusters of the underlying generalized Hubbard Hamiltonian. We present evidence for a substantial interladder spin-pseudospin interaction term which would allow simultaneously for the superantiferroelectric (SAF) charge (pseudospin) ordering and spin dimerization. We discuss the values of the coupling constants appropriate for NaV2O5 and deduce the absence of a soft antiferroelectric mode.


Ordering in Two-Dimensional Ising Models with Competing Interactions

Authors: G.Y. Chitov, C. Gros
Journal-ref.: Low Temp. Phys. 31, 722 (2005)
We study the 2D Ising Model on a square lattice with additional non-equal diagonal next-nearest neighbor interactions. The cases of classical and quantum (transverse) models are considered. Possible phases and their locations in the space of three Ising couplings are analyzed. In particular, incommensurate phases, occurring only at non-equal diagonal couplings, are predicted. In a particular region of interactions, corresponding to the Ising model's super-antiferromagnetic (SAF) ground state, we also consider a spin-pseudospin model comprised of the quantum Ising model coupled to XY spin chains. The spin-pseudospin model's spin-SAF transition into the phase with co-existent the SAF Ising (pseudospin) long-range order and the spin gap, has the properties analogous to the reported earlier by us (cond-mat/0310494) for a simpler coupled model. Along with destruction of the quantum critical point of the transverse Ising model, the phase digram of the spin-pseudospin model can also demonstrate a re-entrance. A detailed study of the latter is presented. The mechanism of the re-entrance, due to interplay of interactions in the coupled model, and the conditions of its appearance are established.


On the Stacking Charge Order in NaV2O5

Authors: G.Y. Chitov, C. Gros
Journal-ref.: J. Phys.: Cond. Matt. 16, L415 (2004).
We propose a mechanism for the observed stacking charge order in the quarter-filled ladder compound NaV2O5. Via a standard mapping of the charge degrees of freedom onto Ising spins we explain the stacking order as a result of competition between couplings of the nearest and next-nearest planes with the 4-fold degenerate super-antiferroelectric in-plane order.


Quantum Monte Carlo simulation for the Coulomb drag of the spin conductance

Authors: K. Louis, C. Gros
Journal-ref.: New Journal of Physics 6, 187 (2004).
In the situation of two electrostatically coupled chains a current in one chain may induce a current in the other chain. We will study this phenomenon, called Coulomb drag, with the aid of a Monte Carlo (MC) approach to the conductance which we presented in a recent paper. We will consider the spin transport (spin drag) in different variants of the Hubbard chain (with/without impurity and additional inter- and intra-chain interactions) for different fillings.


TiOCl, an orbital-ordered system?

Authors: T. Saha-Dasgupta, R. Valentí, H. Rosner, C. Gros
Journal-ref.: Europhys. Lett. 67, 63 (2004).
We present first principles density functional calculations and downfolding studies of the electronic and magnetic properties of the layered quantum spin system TiOCl. We discuss explicitely the nature of the exchange pathes and attempt to clarify the concept of orbital ordering in this material. An analysis of the electronic structure of slightly distorted structures according to the phononic modes allowed in this material suggests that this system is subject to large orbital fluctuations driven by the electron-phonon coupling. Based on these results, we propose a microscopic explanation of the behavior of TiOCl near the phase transition to a spin-gapped system.


Simultaneous Charge Ordering and Spin Dimerization in Quasi-Two-Dimensional Quarter-Filled Ladders

Authors: G.Y. Chitov, C. Gros
Journal-ref.: Physical Review B 69, 104423 (2004).
We study the spin-pseudospin Hamiltonian of the Ising Model in Transverse Field (IMTF) for pseudospins, coupled to the XY-spins on a triangular lattice. This model appears from analyses of the quarter-filled ladder compound NaV2O5, and pseudospins represent its charge degrees of freedom. In the molecular-field approximation we find that the model possesses two phases: charge-disordered without spin gap; and a low-temperature phase containing both the anti-ferroelectric (zigzag) charge order and spin dimerization (spin gap). The phase transition is of the second kind, and the calculated physical quantities are as those one expects from the Landau theory. One of particular features of the phase diagram is that the inter-ladder spin-pseudospin coupling, responsible for the spin gap generation, also destroys the IMTF quantum critical point, resulting in the exponential behavior of Tc in the region of Ising's coupling where the IMTF is always disordered. We conclude that our mean-field results give a qualitatively correct description of the phase transition in NaV2O5, while a more sophisticated analysis is warranted in order to take into account the thermal fluctuations and, probably, the proximity of the IMTF quantum critical point.


Stochastic Cluster Series expansion for quantum spin systems

Authors: K. Louis, C. Gros
Journal-ref.: Physical Review B 70, R100410 (2004).
In this paper we develop a cluster-variant of the Stochastic Series expansion method (SCSE). For certain systems with longer-range interactions the SCSE is considerably more efficient than the standard implementation of the Stochastic Series Expansion (SSE), at low temperatures. As an application of this method we calculated the T=0-conductance for a linear chain with a (diagonal) next nearest neighbor interaction.


Quantum Monte Carlo simulation for the conductance of one-dimensional quantum spin systems

Authors: K. Louis, C. Gros
Journal-ref.: Physical Review B 68, 184424 (2003).
Recently, the stochastic series expansion (SSE) has been proposed as a powerful MC-method, which allows simulations at low T for quantum-spin systems. We show that the SSE allows to compute the magnetic conductance for various one-dimensional spin systems without further approximations. We consider various modifications of the anisotropic Heisenberg chain. We recover the Kane-Fisher scaling for one impurity in a Luttinger-liquid and study the influence of non-interacting leads for the conductance of an interacting system.


Magnetic Raman scattering of the ordered tetrahedral spin-1/2 clusters in Cu2Te2O5(Br1-x Clx)2 compounds

Authors: J. Jensen, P. Lemmens, C. Gros
Journal-ref.: Europhys. Lett. 64, 689 (2003).
Raman light-scattering experiments in the antiferromagnetic phase of the Cu2Te2O5(Br1-x Clx)2 compounds are analyzed in terms of a dimerized spin model for the tetrahedral Cu-clusters. It is shown that the longitudinal magnetic excitation in the pure Br system hybridizes with a localized singlet excitation due to the presence of a Dzyaloshinskii-Moriya anisotropy term. The drastic change of the magnetic scattering intensities observed when a proportion of Br is replaced by Cl ions, is proposed to be caused by a change of the magnetic order parameter. Instead of being parallel/antiparallel with each other, the spins in the two pairs of spin-1/2 order perpendicular to each other, when the composition x is larger than about 0.25.


Minimal charge gap in the ionic Hubbard model

Authors: K. Pozgajcic, C. Gros
Journal-ref.: Physical Review B 68, 085106 (2003).
We study the ionic Hubbard model at temperature T=0 within the mean-field approximation and show that the charge gap does not close completely at the ionic-band insulator to antiferromagnetic insulator transition, contrary to previous expectations. Furthermore, we find a new intermediate phase for on-site repulsions U>Uc for different lattices and calculate the phase diagram for the ionic Hubbard model with alternating U, corresponding to a Cu-O lattice.


Halogen-mediated exchange in the coupled-tetrahedra quantum spin systems Cu2Te2O5X2 (X=Br,Cl)

Authors: R. Valentí, T. Saha-Dasgupta, C. Gros, H. Rosner
Journal-ref.: Physical Review B 67, 245110 (2003).
Motivated by recent discussion on possible quantum critical behavior in the coupled Cu-tetrahedra system Cu2Te2O5Br2, we present a comparative ab initio study of the electronic properties of Cu2Te2O5Br2, and the isostructural Cu2Te2O5Cl2, A detailed investigation of the copper-copper interaction pathes reveals that the halogen-ions play an important role in the inter-tetrahedral couplings via X4-rings (X=Br, Cl). We find that, contrary to initial indications, both systems show a similar electronic behavior with long range exchange pathes mediated by the X_4-rings.


Longitudinal magnon in the tetrahedral spin system Cu2Te2O5Br2 near quantum criticality

Authors: C. Gros, P. Lemmens, M. Vojta, R. Valentí, K.-Y. Choi, H. Kageyama, Z. Hiroi, N.V. Mushnikov, T. Goto, M. Johnsson, P. Millet
Journal-ref.: Physical Review B 67, 174405 (2003).
We present a comprehensive study of the coupled tetrahedra-compound Cu2Te2O5Br2, by theory and experiments in external magnetic fields. We report the observation of a longitudinal magnon in Raman scattering in the ordered state close to quantum criticality. We show that the excited tetrahedral-singlet sets the energy scale for the magnetic ordering temperature TN. This energy is determined experimentally. The ordering temperature TN has an inverse-log dependence on the coupling parameters near quantum criticality.


Diverging magnetothermal response in the one-dimensional Heisenberg chain

Authors: K. Louis, C. Gros
Journal-ref.: Physical Review B 67, 224410 (2003).
A current of magnetic moments will flow in the spin-1/2 Heisenberg chain in the presence of an external magnetic field B and a temperature gradient Delta T along the chain. We show that this magnetothermal effect is strictly infinite for the integrable Heisenberg-model in one dimension. We set-up the response formalism and derive several new generalized Einstein relations for this magnetothermal effect which vanishes in the absence of an external magnetic field. We estimate the size of the magnetothermal response by exact diagonalization and Quantum Monte Carlo and make contact with recent transport measurements for the one-dimensional Heisenberg compound Sr2CuO.


Magnetic light scattering in low-dimensional quantum spin systems

Authors: P. Lemmens, G. Güntherodt, C. Gros
Journal-ref.: Physics Reports, 375 , 1-103 (2003)
[100+]
We review recent progress in magnetic light scattering in one- and two-dimensional quantum spin systems. We give an overview of low-dimensional transition-metal oxides of current interest, such as spin-Peierls, spin-dimer, geometrically frustrated and ladder systems. Light scattering experiments and other spectroscopic methods are discussed in context of the available inelastic neutron scattering data and thermodynamic measurments.


Quantum phase transition in the dioptase magnetic lattice

Authors: C. Gros, P. Lemmens, K.-Y. Choi, G. Güntherodt, M. Baenitz, H.H. Otto
Journal-ref.: Europhys. Lett. 60, 276 (2002)
The study of quantum phase transitions, which are zero-temperature phase transitions between distinct states of matter, is of current interest in research since it allows for a description of low-temperature properties based on universal relations. Here we show that the crystal green dioptase Cu6Si6O18 . 6H2O, known to the ancient Roman as the gem of Venus, has a magnetic crystal structure, formed by the Cu(II) ions, which allows for a quantum phase transition between an antiferromagnetically ordered state and a quantum spin liquid.


The spin-1/2 anisotropic Heisenberg-chain in longitudinal and transversal magnetic fields: a DMRG study

Authors: F. Capraro and C. Gros
Journal-ref.: Euro. Phys. J. B 29, 35 (2002)
Using the density matrix renormalization group technique, we evaluate the low-energy spectrum (ground state and first excited states) of the anisotropic antiferromagnetic spin-one-half chain under magnetic fields. We study both homogeneous longitudinal and transversal fields as well as the influence of a transversal staggered field on opening of a spin-gap. We find that only a staggered transversal field opens a substantial gap.


Conductivity of quantum-spin chains: A Quantum Monte Carlo approach

Authors: J.V. Alvarez and Claudius Gros
Journal-ref.: Physical Review B 66, 094403 (2002)
We discuss zero-frequency transport properties of various spin-1/2 chains. We show, that a careful analysis of Quantum Monte-Carlo (QMC) data on the imaginary axis allows to distinguish between intrinsic ballistic and diffusive transport. We determine the Drude weight, current-relaxation life-time and the mean-free path for integrable and a non-integrable quantum-spin chain. We discuss, in addition, some phenomenological relations between various transport-coefficients and thermal response functions.


On the nature of the spin-singlet ground state in CaCuGe2O6

Authors: R. Valentí, T. Saha-Dasgupta and C. Gros
Journal-ref.: Physical Review B 66 , 054426 (2002)
We investigate by means of ab initio electronic structure analysis and Quantum Monte Carlo calculations the scenario where longer-ranged magnetic interactions dominate over short-ranged interactions in the physical description of compounds. This question is discussed, in particular, for the case of CaCuGe2O6, which shows a spin-singlet behavior induced by third nearest neighbor copper pairs.


Anomalous thermal conductivity of frustrated Heisenberg spin-chains and ladders

Authors: J.V. Alvarez and C. Gros
Journal-ref.: Physical Review Letters 89, 156603 (2002)
We study the thermal transport properties of quantum spin chains and ladders. We find a diverging thermal conductivity at finite temperatures, independent of microscopic details of the models. The temperature at which the non-diverging prefactor kappa(th)(T) peaks is in general substantially lower than the temperature at which the corresponding specific heat cV(T) is maximal. We show that this effect has far-reaching consequences for the magnetic mean-free path lambda extracted by analyzing recent experiments with the microscopic theory results.


Interaction induced collapse of a section of the Fermi sea in in the zig-zag Hubbard ladder

Authors: K. Hamacher, C. Gros and W. Wenzel
Journal-ref.: Physical Review Letters 88, 217203 (2002)
(selected for publication in the Virtual Journal of Nanoscale Science & Technology)
Using the next-nearest neighbor (zig-zag) Hubbard chain as an one dimemensional model, we investigate the influence of interactions on the position of the Fermi wavevectors with the density-matrix renormalization-group technique (DMRG). For suitable choices of the hopping parameters we observe that electron-electron correlations induce very different renormalizations for the two different Fermi wavevectors, which ultimately lead to a complete destruction of one section of the Fermi sea in a quantum critical point.


Low-temperature transport in Heisenberg chains

Authors: J.V. Alvarez and C. Gros
Journal-ref.: Physical Review Letters 88, 077203 (2002)
A technique to determine accurately transport properties of integrable and non-integrable 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 a Drude weight in the gapless regime of a non-integrable system with longer-ranged interactions is presented. We estimate, in addition, the effect of the non-integrability on the transport properties and compare with recent experiments on one-dimensional quantum-spin chains.


Evidence for an unconventional magnetic instability in the spin-tetrahedra system Cu2Te2O5Br2

Authors: P. Lemmens, K.-Y. Choi, E.E. Kaul, Ch. Geibel, K. Becker, W. Brenig, R. Valentí, C. Gros, M. Johnsson, P. Millet and F. Mila
Journal-ref.: Physical Review Letters 87, 227201 (2001)
Thermodynamic experiments as well as Raman scattering have been used to study the magnetic instabilities in the spin-tetrahedra systems Cu2Te2O5X2, X=Cl and Br. While the phase transition observed in the Cl system at To=18.2 K is consistent with 3D AF ordering, the phase transition at To=11.3 K in the Br system has several unusual features. We propose an explanation in terms of weakly coupled tetrahedra with a singlet-triplet gap and low lying singlets.


Fermi surface renormalization in Hubbard ladders

Authors: K. Louis, J.V. Alvarez and C. Gros
Journal-ref.: Physical Review B 64 , 113 106 (2001).
We derive the one-loop renormalization equations for the shift in the Fermi-wavevectors for one-dimensional interacting models with four Fermi-points (two left and two right movers) and two Fermi velocities v_1 and v_2. We find the shift to be proportional to (v_1-v_2)U^2, where U is the Hubbard-U. Our results apply to the Hubbard ladder and to the t_1-t_2 Hubbard model. The Fermi-sea with fewer particles tends to empty. The stability of a saddle point due to shifts of the Fermi-energy and the shift of the Fermi-wavevector at the Mott-Hubbard transition are discussed.


Modeling the electronic behavior of LiV2O5: a microscopic study

Authors: R. Valentí, T. Saha-Dasgupta, J.V. Alvarez, K. Pozgajcic and C. Gros
Journal-ref.: Physical Review Letters 86, 5381 (2001)
We determine the electronic structure of γ-LiV2O5, which has two inequivalent vanadium ions, V(1) and V(2), via density-functional calculations. We find a relative V(1)-V(2) charge ordering of roughly 70:30. We discuss the possible scenarios compatible with the experimentally observed magnetic behavior, which is that of a one-dimensional spin-1/2 Heisenberg antiferromagnet and give estimates of the basic hopping matrix elements. Comparison with the most studied α-NaV2O5 is presented.


Dzyaloshinskii-Moriya interaction in NaV2O5: a microscopic study

Authors: R. Valentí, C. Gros and W. Brenig
Journal-ref.: Physical Review B 62, 14 164 (2000)
We present a unified account of magnetic exchange and Raman scattering in the quasi-one-dimensional transition-metal oxide NaV2O5. Based on a cluster-model approach explicit expressions for the exchange integral and the Raman-operator are given. It is demonstrated that a combination of the electronic-structure and the Dzyaloshinskii-Moriya interaction, allowed by symmetry in this material, are responsible for the finite Raman cross-section giving rise to both, one- and two-magnon scattering amplitudes.


Test of a frustrated spin-cluster model for the low-temperature physics of NaV2O5

Authors: C. Gros, R. Valentí, J. V. Alvarez, K. Hamacher and W. Wenzel
Journal-ref.: Physical Review B 62, R14 617 (2000)
Recent experimental evidence suggest the existence of three distinct V-valence states (V+4, V+4.5 and V+5) in the low-temperature phase of NaV2O5 in apparent discrepancy with the observed spin-gap. We investigate a novel spin cluster model, consisting of weakly coupled, frustrated four-spin clusters aligned along the crystallographic b-axis that was recently proposed to reconcile these experimental observations. We have studied the phase diagram and the magnon dispersion relation of this model using DMRG, exact diagonalization and a novel cluster-operator theory. We find a spin-gap for all parameter values and two distinct phases, a cluster phase and a Haldane phase. We evaluate the size of the gap and the magnon dispersion and find no parameter regime which would reproduce the experimental results. We conclude that this model is inappropriate for the low-temperature regime of NaV2O5.


Effective Charge and Spin Hamiltonian for the Quarter-Filled Ladder Compound NaV2O5

Authors: Debanand Sa and C. Gros
Journal-ref.: Euro. Phys. J. B 18, 421 (2000)

An effective intra- and inter-ladder charge-spin hamiltonian for the quarter-filled ladder compound NaV2O5 has been derived by using the standard canonical transformation method. In the derivation, it is clear that a finite inter-site Coulomb repulsion is needed to get a meaningful result otherwise the perturbation becomes ill-defined. Various limiting cases depending on the values of the model parameters have been analyzed in detail and the effective exchange couplings are estimated. We find that the effective intra-ladder exchange may become ferromagnetic for the case of zig-zag charge ordering in a purely electronic model. We estimate the magnitude of the effective inter-rung Coulomb repulsion in a ladder and find it to be about one-order of magnitude too small in order to stabilize charge-ordering.


On the evaluation of the specific heat and general off-diagonal n-point correlation functions within the loop algorithm

Authors: J.V. Alvarez and C. Gros
Journal-ref.: Euro. Phys. J. B 15, 641 (2000)

We present an efficient way to compute diagonal and off-diagonal n-point correlation functions for quantum spin-systems within the loop algorithm. We show that the general rules for the evaluation of these correlation functions take an especially simple form within the framework of directed loops. These rules state that contributing loops have to close coherently. As an application we evaluate the specific heat for the case of spin chains and ladders.


A Generalized Ginzburg-Landau Approach to Second Harmonic Generation

Authors: Debanand Sa, R. Valentí and C. Gros
Journal-ref.: Euro. Phys. J. B 14, 301 (2000)
We develop a generalized Ginzburg-Landau theory for second harmonic generation (SHG) in magnets by expanding the free energy in terms of the order parameter in the magnetic phase and the susceptibility tensor in the corresponding high-temperature phase. The non-zero components of the SHG susceptibility in the ordered phase are derived from the symmetries of the susceptibility tensor in the high-temperature phase and the symmetry of the order parameter. In this derivation, the dependence of the SHG susceptibility on the order parameter follows naturally, and therefore its nonreciprocal optical properties.
We examine this phenomenology for the magnetoelectric compound Cr2O3 as well as for the ferroelectromagnet YMnO3.


The microscopic spin-phonon coupling constants in CuGeO3

Authors: R. Werner, C. Gros and M. Braden
Journal-ref.: Physical Review B 59,14 356 (1999)

Using RPA results, mean field theory, and refined data for the polarization vectors we determine the coupling constants of the four Peierls-active phonon modes to the spin chains of CuGeO3. We then derive the values of the coupling of the spin system to the linear ionic displacements, the bond lengths and the angles between bonds. Our values are consistent with microscopic theories and various experimental results. We discuss the applicability of static approaches to the spin-phonon coupling. The c-axis anomaly of the thermal expansion is explained. We give the values of the coupling constants in an effective one-dimensional Hamiltonian.


Magnon splitting induced by charge ordering in NaV2O5

Authors: C. Gros and R. Valentí
Journal-ref.: Physical Review Letters 82, 976 (1999)

We consider the effects of charge ordering in NaV2O5 (below TSP) on the exchange constants and on the magnon dispersion. We show that the experimentally observed splitting of the magnon branches along the a direction is induced by charge ordering. We find that one can distinguish between the proposed 'zig-zag' and 'in-line' patterns of charge ordering. Only the zig-zag ordering is consistent with the experimental results regarding (i) the unusual intensity modulation observed in magnetic neutron scattering, (ii) the reduction in the intra-ladder exchange constant below TSP, and (iii) the magnon dispersion along a. We estimate the inter-ladder exchange constant to be 1.01meV=11.7K for T>TSP.


Novel Nonreciprocal Acoustic Effects in Antiferromagnets

Authors: R. Valentí, C. Gros and V. N. Muthukumar
Journal-ref.: Euro. Phys. Lett. 45, 242 (1999)

The possible occurrence of nonreciprocal acoustic effects in antiferromagnets in the absence of an external magnetic field is investigated using both (i) a microscopic formulation of the magnetoelastic interaction between spins and phonons and (ii) symmetry arguments. We predict for certain antiferromagnets the existence of two new nonreciprocal (non-time invariant) effects:
A boundary-condition induced nonreciprocal effect and the occurrence of transversal phonon modes propagating in opposite directions having different velocities. Estimates are given and possible materials for these effects to be observed are suggested.


Structure the Hilbert-space of the infinite-dimensional Hubbard model

Authors: C. Gros and W. Wenzel
Journal-ref.: Euro. Phys. J. B 8, 569 (1999)

An iterative procedure for the explicit construction of the nontrivial subspace of all symmetry-adapted configurations with non-zero weight in the ground-state of the infinite-dimensional Hubbard model is developed on the basis of a symmetrized representation of the transition operators on a sequence of Bethe-Lattices of finite depth. The relation ship between these operators and the well known mapping of the infinite-dimensional Hubbard model onto an effective impurity problem coupled to a (self-consistent) bath on non-interacting electrons is given. As an application we calculate the properties of various Hubbard stars and give estimates for the half-filled Hubbard model with up to 0.1% accuracy.


Dynamics of the Peierls-active phonon modes in CuGeO3

Authors: R. Werner and C. Gros
Journal-ref.: Physical Review B 58, R14 677 (1998)

We reconsider the Cross and Fischer approach to spin-Peierls transitions. We show that a soft phonon occurs only if Omega_0<2.2 TSP. For CuGeO3 this condition is not fulfilled and the calculated temperature dependence of the Peierls-active phonon modes is in excellent agreement with experiment. A central peak of a width ~0.2 meV is predicted at TSP. Good agreement is found between theory and experiment for the pretransitional Peierls-fluctuations. Finally, we consider the problem of quantum criticality in CuGeO3.


NaV2O5 as a quarter-filled ladder compound

Authors: H. Smolinski, C. Gros, W. Weber, U. Peuchert, G. Roth, M. Weiden, C. Geibel
Journal-ref.: Physical Review Letters 80, 5164 (1998)
[100+]
A new X-ray diffraction study of the one-dimensional spin-Peierls compound α-NaV2O5 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 NaV2O5 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 NaV2O5 and its magnetic properties.


Molecular-field approach to the spin-Peierls transition in CuGeO3

Authors: R. Werner and C. Gros
Journal-ref.: Physical Review B 57, 2897 (1998)
We present a theory for the spin-Peierls transition in CuGeO3. We map the elementary excitations of the dimerized chain (solitons) on an effective Ising model. Inter-chain coupling (or phonons) then introduce a linear binding potential between a pair of soliton and anti-soliton, leading to a finite transition temperature. We evaluate, as a function of temperature, the order parameter, the singlet-triplet gap, the specific heat, and the susceptibility and compare with experimental data on CuGeO3. We find that CuGeO3 is close to a first-order phase transition. We point out, that the famous scaling law ~δ(2/3) of the triplet gap is a simple consequence of the linear binding potential between pairs of solitons and anti-solitons in dimerized spin chains.


Effects of in-chain and off-chain substitutions on spin fluctuations in the spin-Peierls compound CuGeO3

Authors: P. Lemmens, M. Fischer, G. Güntherodt, C. Gros, P. G. J. van Dongen, M. Weiden, W. Richter, C. Geibel, F. Steglich
Journal-ref.: Physical Review B 55, 15076 (1997)
The effect of in-chain and off-chain substitutions on 1D spin fluctuations in the spin-Peierls compound CuGeO3 has been studied using Raman scattering in order to understand the interplay between defect induced states, enhanced spin-spin correlations and the ground state of low dimensional systems. In-chain and off-chain substitutions quench the spin-Peierls state and induce 3D antiferromagnetic order at T≤ 5 K. Consequently a suppression of a 1D gap-induced mode as well as a constant intensity of a spinon continuum are observed at low temperatures. A 3D two-magnon density of states now gradually extends to higher temperatures T≤ 60K compared with pure CuGeO3. This effect is more pronounced in the case of off-chain substitutions (Si) for which a Néel state occurs over a larger substitution range, starting at very low concentrations.
Besides, additional low energy excitations are induced. These effects, i.e. the shift of a dimensional crossover to higher temperatures are due to an enhancement of the spin-spin correlations induced by a small amount of substitutions. The results are compared with recent Monte Carlo studies on substituted spin ladders, pointing to a similar instability of coupled, dimerized spin chains and spin ladders upon substitution.


Magnon-magnon interactions in the Spin-Peierls compound CuGeO3

Authors: C. Gros, W. Wenzel, A. Fledderjohann, P. Lemmens, M. Fischer, G. Güntherodt, M. Weiden, C. Geibel, F. Steglich
Journal-ref.: Physical Review B 55, 15048 (1997)
In a magnetic substance the gap in the Raman spectrum, DeltaR, is approximatively twice the value of the neutron scattering gap, DeltaS, if the the magnetic excitations (magnons) are only weakly interacting. But for CuGeO3 the experimentally observed ratio DeltaR/DeltaS is approximatively 1.49-1.78, indicating attractive magnon-magnon interactions in the quasi-1D Spin-Peierls compound CuGe3.

We present numerical estimates for DeltaR/DeltaS from exact diagonalization studies for finite chains and find agreement with experiment for intermediate values of the frustration parameter alpha. An analysis of the numerical Raman intensity leads us to postulate a continuum of two-magnon bound states in the Spin-Peierls phase. We discuss in detail the numerical method used, the dependence of the results on the model parameters and a novel matrix-element effect due to the dimerization of the Raman-operator in the Spin-Peierls phase.


The J1-J2 model revisited: Phenomenology of CuGeO3

Authors: V.N. Muthukumar, C. Gros, R. Valentí, M. Weiden, C. Geibel, F. Steglich, P. Lemmens, M. Fischer, G. Güntherodt
Journal-ref.: Physical Review B, 55, 5944 (1997)
We present a mean field solution of the antiferromagnetic Heisenberg chain with nearest (J1) and next to nearest neighbor (J2) interactions. This solution provides a way to estimate the effects of frustration. We calculate the temperature-dependent spin-wave velocity, vs(T) and discuss the possibility to determine the magnitude of frustration J2/J1 present in quasi-1D compounds from measurements of vs(T). We compute the thermodynamic susceptibility at finite temperatures and compare it with the observed susceptibility of the spin-Peierls compound CuGeO3. We also use the method to study the two-magnon Raman continuum observed in CuGeO3 above the spin-Peierls transition.


Spin dynamics of dimerized Heisenberg chains

Authors: A. Fledderjohann, C. Gros
Journal-ref.: Europhys. Lett. 37, 189 (1997)
We study numerically the dimerized Heisenberg model with frustration appropriate for the quasi-1D spin-Peierls compound CuGeO3. We present evidence for a bound state in the dynamical structure factor for any finite dimerization delta and estimate the respective spectral weight. For the homogeneous case (alpha=0) we show that the spin-wave velocity vs is renormalized by the n.n.n. frustration term α as vs=pi/2 J(1-b alpha), with b~1.12


Frustration induced Raman scattering in CuGeO3

Authors: V.N. Muthukumar, C. Gros, W. Wenzel, R. Valentí, P. Lemmens, B. Eisener, G. Güntherodt, M. Weiden, C. Geibel, F. Steglich
Journal-ref.: Physical Review B, 54, R9635 (1996)
We present experimental data for the Raman intensity in the spin-Peierls compound CuGeO3 and theoretical calculations from a one-dimensional frustrated spin model. The theory is based on (a) exact diagonalization and (b) a recently developed solitonic mean field theory. We find good agreement between the 1D-theory in the homogeneous phase and evidence for a novel dimerization of the Raman operator in the spin-Peierls state. Finally we present evidence for a coupling between the interchain exchange, the spin-Peierls order parameter and the magnetic excitations along the chains.


The spin 1/2 Heisenberg star with frustration II: The influence of the embedding medium

Authors: J. Richter, A. Voigt, S.E. Krüger, C. Gros
Journal-ref.: J. Phys. A: Math. Gen. 29, 825 (1996)
We investigate the spin 1/2 Heisenberg star introduced in J. Richter and A. Voigt, J. Phys. A: Math. Gen. 27, 1139 (1994). The model is defined by H=J1i=1N S0Si + J2 HR{Si}, with J1, J2 ≥ 0 and i=1,...,N. In extension to a previous publication we consider a more general HR{Si} describing the properties of the spins surrounding the central spin S0. The Heisenberg star may be considered as an essential structure element of a lattice with frustration (namely a spin embedded in a magnetic matrix HR) or, alternatively, as a magnetic system HR with a perturbation by an extra spin. We present some general features of the eigenvalues, the eigenfunctions as well as the spin correlation ⟨S0Si⟩ of the model. For HR being a linear chain, a square lattice or a Lieb-Mattis type system we present the ground state properties of the model in dependence on the frustration parameter α=J2/J1. Furthermore the thermodynamic properties are calculated for HR being a Lieb-Mattis antiferromagnet.


Control of the finite size corrections in exact diagonalization studies

Authors: C. Gros
Journal-ref.: Physical Review B, 53 6865(BR) (1996)
We study the possibility of controlling the finite size corrections in exact diagonalization studies quantitatively. We consider the one- and two dimensional Hubbard model. We show that the finite-size corrections can be be reduced systematically by a grand-canonical integration over boundary conditions. We find, in general, an improvement of one order of magnitude with respect to studies with periodic boundary conditions only. We present results for ground-state properties of the 2D Hubbard model and an evaluation of the specific heat for the 1D and 2D Hubbard model.


Theory of Non-Reciprocal Optical Effects in Antiferromagnets: The Case Cr2O3

Authors: V.N. Muthukumar, R. Valentí, C. Gros
Journal-ref.: Physical Review B, 54, 433 (1996)
A microscopic model of non-reciprocal optical effects in antiferromagnets is developed by considering the case of Cr2O3 where such effects have been observed. These effects are due to a direct coupling between light and the antiferromagnetic order parameter. This coupling is mediated by the spin-orbit interaction and involves an interplay between the breaking of inversion symmetry due to the antiferromagnetic order parameter and the trigonal field contribution to the ligand field at the magnetic ion. We evaluate the matrix elements relevant for the non-reciprocal second harmonic generation and gyrotropic birefringence.


A Microscopic Model of Non-Reciprocal Optical Effects in Cr2O3

Authors: V.N. Muthukumar, R. Valentí, C. Gros
Journal-ref.: Physical Review Letters 75, 2766 (1995)
This manuscript deals with the question "How does light couple to an antiferromagnetic order parameter"? For that we develop a microscopic model that explains the non-reciprocal optical effects in centrosymmetric Cr2O3. It is shown that light can couple directly to the antiferromagnetic order parameter. This coupling is mediated by the spin-orbit interaction and involves an interplay between the breaking of inversion symmetry due to the antiferromagnetic order parameter and the trigonal field contribution to the ligand field at the Cr3+ ion.


The transition from an ordered antiferromagnet to a quantum disordered spin liquid in a solvable bilayer model

Authors: C. Gros, W. Wenzel, J. Richter
Journal-ref.: Europhys. Lett. 32, 747 (1995)
We present a spin-1/2 bilayer model for the quantum order-disorder transition which (i) can be solved by mean-field theory for bulk quantities, (ii) becomes critical at the transition, and (iii) allows to include intralayer frustration. We present numerical data (for systems with up to 240 sites) and analytical results for the critical coupling strength, ground-state energy, order parameter and for the gap. We show that the critical coupling decreases linearly with frustration.


Spin-charge separation at small lengthscales in the 2D t-J mode

Authors: C. Gros, R. Valentí
Journal-ref.: Journal of Low Temperature Physics 99, 509 (1995)
We consider projected wave functions for the two-dimensional t-J model. For various wave functions, including correlated Fermi-liquid and Luttinger-type wave functions, we present the static charge-charge and spin-spin structure factors. Comparisons with recent results from a high-temperature expansion by Putikka et al. indicates spin-charge separation at small length scales.


Equation-of-motion approach to the Hubbard model in infinite dimensions

Authors: C. Gros
Journal-ref.: Physical Review B 50, 7295 (1994)
We consider the Hubbard model on the infinite-dimensional Bethe lattice and construct a systematic series of self-consistent approximations to the one-particle Green's function, G(n)(omega), n=2,3,... . The first n-1 equations of motion are exactly fullfilled by G(n)(omega) and the nth equation of motion is decoupled following a simple set of decoupling rules. G(2)(omega) corresponds to the Hubbard-III approximation. We present analytic and numerical results for the Mott-Hubbard transition at half filling for n=2,3,4.


Integrable models of interacting quantum spins with competing interactions

Authors: J. Richter, S.E. Krüger, A. Voigt, C. Gros
Journal-ref.: Europhys. Lett. 28 (1994) 363
We present a class of exactly solvable quantum spin models which consist of two Heisenberg-subsystems coupled via a long-range Lieb-Mattis interaction. The total system is exactly solvable whenever the individual subsystems are solvable and allows to study the effects of frustration. We consider (i) the antiferromagnetic linear chain and (ii) the Lieb-Mattis antiferromagnet for the subsystem-Hamiltonians and present (i) the complete ground-state phase diagram and (ii) the full thermodynamic phase diagram. We find a novel phase which exhibits order from disorder phenomena.


The Mott-Hubbard Transition on the D=∞ Bethe Lattice

Authors: C. Gros, W. Wenzel, R. Valentí, G. Hülsenbeck, J. Stolze
Journal-ref.: Europhys. Lett. 27, 299 (1994)
In view of a recent controversy we investigate the Mott-Hubbard transition in D=∞ with a novel cluster approach. i) We show that any truncated Bethe lattice of order n can be mapped exactly to a finite Hubbard-like cluster. ii) We evaluate the self-energy numerically for n=0,1,2 and compare with a series of self-consistent equation-of-motion solutions. iii) We find the gap to open continously at the critical Uc~2.5t*. iv) A low-energy theory for the Mott-Hubbard transition is developed and relations between critical exponents are presented.


A self-consistent cluster study of the Emery model

Authors: C. Gros, R. Valentí
Journal-ref.: Ann. Phys. 3, 460 (1994)
We calculate Fermi-surface properties of the Cuprate superconductors within the three-band Hubbard model, using a cluster expansion for the proper self-energy. The Fermi-surface topology is in agreement with angular-resolved photoemission data for dopings ~20%. We discuss possible violations of the Luttinger sum-rule for smaller dopings and the role of van-Hove singularities in the density of states of the Zhang-Rice singlets. We calculate the shift in the chemical potential upon doping and find quantitative agreement with recent experiments.


Cluster expansion for the self-energy: A simple many-body method for interpreting the photoemission spectra of correlated Fermi systems

Authors: C. Gros, R. Valentí
Journal-ref.: Physical Review B 48, 418 (1993)
The self-energy of a translational invariant system of interacting fermions may be expanded in diagrams contributing to the self-energy of finite clusters with open boundary conditions. The exact solution of small clusters might therefore be used to construct a systematic approximation to the self-energy of the infinite system. This approximation incorporates both the local and the itinerant degrees of freedom on an equal footing. We develop this method for the one-band Hubbard Hamiltonian and apply it to the three-band Hamiltonian of the CuO superconductors. Already the lowest nontrivial approximation yields interesting results for the spectral density useful for the interpretation of photoemission experiments. We find (i) transfer of spectral weight from the upper to the lower Hubbard band upon doping, (ii) the formation of an isolated band of Zhang-Rice singlets separated from the band of triplet states by a many-body gap, and (iii) creation of density of states above the top of the oxygen band upon doping.


Luttinger-Liquid Behaviour in 2D: The variational Approach

Authors: C. Gros, R. Valentí
Journal-ref.: Mod. Phys. Lett. 7, 3 (1993)
We study a variational formulation of the Luttinger-liquid concept in two dimensions. We show that a Luttinger-liquid wavefunction with an algebraic singularity at the Fermi-edge is given by a Jastrow-Gutzwiller type wavefunction, which we evaluate by variational Monte Carlo for lattices with up to 38x38=1444 sites. We therefore find that, from a variational point of view, the concept of a Luttinger liquid is well defined even in 2D. We also find that the Luttinger-liquid state is energetically favoured by the proejected kinetic energy in the context of the 2D t-J model. We study and find coexistence of d-wave superconductivity and Luttinger-liquid behaviour in two-dimensional projected wavefunctions. We then argue that generally, any two-dimensional d-wave superconductor should be unstable against Luttinger-liquid type correlations along the (quasi-1D) nodes of the d-wave order parameter, at temperatures small compared to the gap.


Perovskites in high dimenensions: Heavy-fermion vs. t-J fixed point

Authors: R. Valentí, C. Gros
Journal-ref.: Z. Phys. B 90, 161 (1993)
We study the (D+1)-band Hubbard model on generalized D-dimensional perovskite structures. We show that in the limit of high dimensions the possible scaling behaviour is uniquely determined via the band-structure and that the model without direct oxygen-oxygen hopping necessarily scales to the cluster limit. A 1/dimension expansion the leads to a t-J like Hamiltonian and the Zhang-Rice analysis becomes rigorous.
The large dimension fixed point, in general, still remains the cluster model even when a hopping term between the n.n. oxygen-sites is included. Only for a unique ratio of the oxygen onsite energies to the oxygen-oxygen hopping amplitude is a new fixed point possible, corresponding to a heavy-Fermion Hamiltonian.


Luttinger liquid instability of the 2D t-J model: A variational study

Authors: R. Valentí, C. Gros
Journal-ref.: Physical Review Letters 68, 2402 (1992)
See also: Erratum
We study variationally the possible occurrence of a Luttinger liquid in the normal state of the 2D t-J model. For this, we generalize to 2D a Luttinger-Jastrow-Gutzwiller-type wave function introduced by Hellberg and Mele for the 1D t-J model. We show that this wave function does show also in 2D the characteristic correlations of a Luttinger liquid and the gains in kinetic energy stabilize the Luttinger liquid state with respect to Fermi liquid states with short-range correlations only. In addition, we provide rigorous lower bounds to the transition to the fully phase separated state at larger ratios J/t.


The boudary condition integration technique: Results for the Hubbard model in 1D and 2D

Authors: C. Gros
Journal-ref.: Z. Phys. B 86, 359 (1992)
We study models of strongly correlated electrons in one- and two- dimensions. We exactly diagonalize small clusters with general boundary conditions (BC) and integrate over all possible BC. This techinque recovers the kinetic energy part of the (extended lattice) Hamiltonian exactly in a grand-canonical formulation. A continous range of particle densities may be described with this techinque and the momentum space can be probed for arbitrary momenta.
For the Hubbard Hamiltonian we recover details of the Mott-isulating behaviour for the momentum distribution function at half filling, both in 1D and 2D.
Off half-filling the shape of the canonical Fermi surface is strongly distored in 2D with respect to the grand-canonical Fermi-surface. The shape of the grand-canonical Fermi-surface obtained by this finite-size techinque reduces in the weak-coupling limit exactly to that of the infinite-lattice Fermi-sea.


Rigorous bounds for ground-state properties of correlated Fermi systems

Authors: R. Valentí, C. Gros, P.J. Hirschfeld, W. Stephan
Journal-ref.: Physical Review B 44, 13203 (1991)
We show that upper and lower bounds on the ground-state energy of models describing correlated Fermi systems may be combined to produce bounds on the ground-state magnetization and chemical potential. Such bounds are obtainable through standard variational techniques and through recently developed methods involving exact diagonalization of finite-size clusters. For the Hubbard model on the square lattice, we give rigorous bounds for the magnetization at nonzero magnetic field B and for the chemical potential at nonzero hole density 1-n. The quality of these bounds degrades as B-->0 and n-->1, precluding rigorous statements about the stability of the ferromagnetic state or the existence of a Mott-Hubbard gap. Nevertheless, the tendency towards large-U ferromagnetism and localization is evident. We discuss ways of improving these bounds, including the use of kinetic frustration, nonuniform clusters, and averaging over boundary conditions.


Chiral ordering in a frustrated quantum spin system

Authors: J. Richer, C. Gros, W. Weber
Journal-ref.: Physical Review B 44, 906 (1991)
We examine the J1-J2, spin-1/2 Heisenberg model on a square lattice with 16 and 20 sites. We evaluate the ground-state correlations of an operator, measuring the handiness of plaquettes. We find a strong, physically relevant enhancement in these chiral correlations for intermediate values of J2/J1. We compare with the known results for correlations between column-wise-ordered singlets. We find both types of correlations to be viable candidates for the ground-state correlations in the thermodynamic limit.


Geometry-controlled conserving approximations for the t-J model

Authors: M.D. Johnson, C. Gros
Journal-ref.: Physical Review B 43, 11207 (1991)
We present in detail a Green's-function approach for studying charged-spin systems which preserves the local constraints prohibiting double occupancy. This approach satisfies Wick's theorem, uses a fermionic expansion around a singly occupied Néel state, and treats charge and spin degrees of freedom on an equal footing. For the antiferromagnetic Heisenberg model we recover gapless spin excitations (renormalized spin waves) in a straightforward real-space random-phase-approximation approach. This expansion is strictly controlled by a geometrical factor, 1/z, where z is the coordination number. We describe the incoherent motion of charges (holes) in the t-J model by a self-retracing-path approximation and consider two competing contributions to the coherent hole propagation. These approximations are made conserving in a constructive fashion by mapping Feynman diagrams to an equivalent tight-binding model. To study the accuracy of this procedure, we have made a detailed numerical check against the results obtained by exact diagonalization of a 4 x 4 system with one hole, finding excellent agreement both in and near the Ising limit.


Variational theorem for vector-mean-field theories of statistical transmutation

Authors: C. Gros, S.M. Girvin, G.S. Canright, M. D. Johnson
Journal-ref.: Physical Review B 43, 5883 (1991)
e examine the validity of vector-mean-field theory (VMFT) for statistical transmutation on large lattices with a high density of particles per site (1/2 and 1/4). We take as a difficult test case the representation of hard-core bosons as fermions plus attached flux tubes. We use a variational Monte Carlo method to test the variational properties of the mean-field ground-state wave function against the predictions of the VMFT. We find a discrepancy of order 1 in the thermodynamic limit. This leads us to postulate that a better formulation on a lattice may be that of a renormalized vector-mean-field theory. We show that the renormalization coefficients can be understood by an analysis of the phase fluctuations (whose magnitude we estimate) of the long-range gauge interaction. These phase fluctuations are of order pi on the lattice (thus leading to a breakdown of VMFT on the lattice) while they vanish in a continuum formulation. We give a detailed discussion of the qualitative differences of VMFT on the lattice versus the continuum. In particular, we examine the effect of having lines of zeros (lattice) versus points of zeros (continuum) for the nodes of the many-body wave function. In addition, a remarkable variational theorem is discovered for the ground-state wave function of the VMFT.


Phases of the t-J model from variational Monte Carlo studies: Occurrence of time-reversal symmetry breaking

Authors: G.J. Chen, R.J. Joynt, F.C. Zhang, C. Gros
Journal-ref.: Physical Review B 42, 2662 (1990)
We have numerically evaluated the energy of several kinds of wave functions considered to be candidate ground states of the two-dimensional t-J model at various hole densities. We searched a parameter space which includes d-wave and s-wave superconductivity and spin-density-wave ordering as well as the projected Fermi-liquid state. Coexistence of different orderings, such as the s+id state and d-wave spin-density-wave state, were found to be stable states. We find a phase diagram in the density-t/J plane which has coexistence of antiferromagnetism and superconductivity at very low hole concentrations and superconductivity up to rather high values of density-about 40%. At intermediate concentrations, the time-reversal symmetry-breaking s+id state is found.


Criterion for a good variational wave function

Authors: C. Gros
Journal-ref.: Physical Review B 42, 6835 (1990)
The variance of the Hamiltonian in a given variational wave function measures how good an eigenstate this wave function is. In some instances, as for the two-dimensional antiferromagnetic Heisenberg Hamiltonian (2D AFH), the energy expectation value is not enough to distinguish between different trial Ansätze. Here we propose the variance as a simple criterion, which allows for further differentiation between degenerate trial wave functions. We show that this criterion establishes the projected wave functions as candidates for the ground state of the 2D AFH. A strong interference effect is discovered in computer experiment.


An Exact Mapping of the t-J Model to the Unrestricted Hilbert space

Authors: C. Gros, M.D. Johnson
Journal-ref.: Physica B 165 & 166, 985 (1990)
We present an exact mapping of the thermodynamical properties of the t-J model to the unrestricted fermionic Hilbert space. At half filling this is accomplished by the introduction of a complex chemical potential. At finite hole concentration we generalize the t-J model to a particle-hole symmetric form. Identifying a symmetrized combination of a hole an a doubly-occupied site with the charge carrier, we prove that the thermodynamical properties of original and the generalzed t-J model are identical.


Conjecture concerning the fractional Hall hierarchy

Authors: C. Gros, A.H. MacDonald
Journal-ref.: Physical Review B 42, 9514 (1990)
We present numerical evidence in support of a conjecture concerning the hierarchy of incompressible states that are responsible for the fractional quantum Hall effect (FQHE). We propose that for filling factors in the range 1/3 %3C= nu %3C= 2) / 3 , the FQHE occurs only when nu = nu n=n/(2n+1) (or when nu =1- nu n) and at no other fractional filling factors with odd denominators. If correct, this conjecture would imply that important qualitative features of the hierarchy physics of the FQHE are not understood


Wick's theorem for charged spin systems

Authors: C. Gros, M.D. Johnson
Journal-ref.: Physical Review B 40, 9423 (1989)
We present a new Green's-function approach to charged spin systems which preserves the local constraints prohibiting double occupancy. It is a systematic fermionic expansion and yields 1/(2z) as a control parameter for the Heisenberg model. For the t-J model the spin and hole Green's functions are treated on an equal footing. In the Ising limit, the Brinkman-Rice approximation and a bandwidth ~Jz are recovered for, respectively, the incoherent and coherent hole motion. A new picture for the coherent hole propagation is obtained in the Heisenberg limit.


Physics of Projected Wavefunctions

Authors: C. Gros
Journal-ref.: Annals of Physics (NY) 189, 53 (1989)
[100+]
We present and discuss a variational approach to the one band Hubbard model in the limit of 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 method used to evaluate them, their properties, and their physical relevance.
Depending on the kind of parameterization 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 an to results obtained by other approaches to the Hubbard model.
The results obtained by numerical evaluation of the projected wavefunctions 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 wether a quantum spin liquid is intrisically superconducting or not.


Superconductivity in correlated wave functions

Authors: C. Gros
Journal-ref.: Physical Review B 38, 931 (1988)
[100+]
We describe a new method to numerically evaluate the properties of correlated superconducting wave functions. We have applied it to the resonating-valence-bond (RVB) wave function for the Hubbard model on the square lattice. For the half-filled case we find that the d-wave RVB state and the antiferromagnetic ordered state have the same energy within numerical accuracy. At 10% doping we find d-wave superconductivity, consistent with previous studies. We show that the superconducting order parameter is proportional to the number of holes, for small hole concentrations.


A renormalized Hamiltonian approach to a resonant valence bond wavefunction

Authors: F.C. Zhang, C. Gros, T.M. Rice, H. Shiba
Journal-ref.: Supercond. Sci. Technol. 1, 36 (1988)
[100+]
The effective Hamiltonian of strongly correlated electrons on a square lattice is replaced by a renormalized Hamiltonian and the factors that renormalise the kinetic energy of holes and the Heisenberg spin-spin coupling are calculated using a Gutzwiller approximation scheme. The accuracy of this renormalisation procedure is tested nuermically and found to be qualitatively excellent. Within the scheme a resonant valence bond (RVB) wavefunction is found at half-filling to be lower in energy that the antiferromagnetic state.
If the wavefunction is expressed in fermion operators, local SU(2) and U(1) invariance leads to a redundancy in the representation. The introduction of holes removes these local invariances and we find that a d-wave RVB state is lowest in energy. This state has a superconducting order-parameter whose amplitude is linear in the density of holes.


Superconducting instability in the large-U limit of the two-dimensional Hubbard model

Authors: C. Gros, R. Joynt, T.M. Rice
Journal-ref.: Z. Phys. 68, 425 (1987)
[100+]
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 to d-wave pairing but stable against s-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 correlations in the paired state.


Antiferromagnetic correlations in almost-localized Fermi liquids

Authors: C. Gros, R. Joynt, T.M. Rice
Journal-ref.: Physical Review B 36, 381 (1987)
[100+]
A Monte Carlo method is used to calculate various properties of one-band Gutzwiller wave functions which are formed by restricting the charge fluctuations in noninteracting wave functions. Gutzwiller's approximate formula for the kinetic energy is tested both for the ground state and excited states. The ground state is found to have strong antiferromagnetic short-range spin-spin correlations for nearly-half-filled bands, thus extending previous work on the half-filled case. These correlations are very sensitive to the choice of occupied Bloch states and when the occupation is distributed uniformly over the band they disappear. From this fact we conclude that correlations are present only at temperatures low compared to the coherence temperature. In the almost-localized limit it is advantageous to describe the system by an effective Hamiltonian which separates into a term due to the kinetic energy of the charge carriers and one due to the Heisenberg spin-spin coupling. We show that the almost-localized Fermi liquid can gain energy from both terms in the effective Hamiltonian. In other words the restrictions on charge fluctuations can cause spin correlations which in turn can stabilize the Fermi-liquid ground state.


Landau parameters of almost-localized Fermi liquids

Authors: D. Baeriswyl, C. Gros, T.M. Rice
Journal-ref.: Physical Review B 35, 8391 (1987)
We derive partial sum rules for the intraband contributions to the charge and spin conductivities for almost-localized Fermi liquids in a lattice. From this we conclude that the l=1 Landau parameters have small values.


Crossover from Fermi Liquid to Classical Behavior of Normal 3He in the Model of Almost Localized Fermions

Authors: K. Seiler, C. Gros, T.M. Rice, K. Ueda, D. Vollhardt
Journal-ref.: J. Low. Temp. Phys. 64, 195 (1986).
A phenomenological extension of the model of almost localised fermions to finite temperatures is presented. It is used to calculate thermodyanamic properties of the normal state of 3He. No new adjustable parameters are introduced and the effective interaction strength is the same a used by Vollhardt. A good qualitative description of the crossover from Fermi liquid to classical behavior in the specific heat, spin susceptibility, and temperature-dependent pressure (or equivalently thermal expansion) is obtained. In particular, key results, such as the change in specific heat when the spin entropy saturates and the change from thermal expansion to thermal contraction at low temperatures are reproduced.




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