
Original Publications 


Prof. Claudius Gros 

Covid19 research 
Authors: Claudius Gros, Daniel Gros
Journalref.: to be published.
Delays in the availability of vaccines are costly as the pandemic continues.
However, in the presence of adjustment costs firms have an incentive to
increase production capacity only gradually. The existing contracts specify
only a fixed quantity to be supplied over a certain period and thus provide no
incentive for an accelerated buildup in capacity. A high price does not change
this. The optimal contract would specify a decreasing price schedule over time
which can replicate the social optimum.
Authors: C. Gros
Journalref.: Entropy 23, 157 (2021).
Human societies are characterized, besides others, by three
constituent features.
(A) Options, as for jobs and societal positions, differ with respect
to their associated monetary and nonmonetary payoffs.
(B) Competition leads to reduced payoffs when individuals
compete for the same option with others.
(C) People care how they are doing relatively to others.
The latter trait, the propensity to compare one's own success with
that of others, expresses itself as envy. It is shown that the
combination of (A)(C) leads to spontaneous class stratification.
Societies of agents split endogenously into two social
classes, an upper and a lower class, when envy becomes relevant.
A comprehensive analysis of the Nash equilibria characterizing
a basic reference game is presented. Class separation is due to
the condensation of the strategies of lowerclass agents, which
play an identical mixed strategy. Upper class agents do not
condense, following individualist pure strategies.
Model and results are sizeconsistent, holding for arbitrary
large numbers of agents and options. Analytic results are
confirmed by extensive numerical simulations. An analogy to
interacting confined classical particles is discussed.
Authors: C. Gros, R. Valenti, L. Schneider, B. Gutsche,
D. Markovic
Journalref.: PLOS One (in press).
The distinct ways the COVID19 pandemics has been unfolding in different
countries and regions suggest that local societal and governmental structures
play an essential role both for the baseline infection rate and the shortterm
and longterm reaction to the outbreak. Here we investigate how societies as a
whole, and governments, in particular, modulate the dynamics of a novel
epidemic using a generalisation of the SIR model, the controlled SIR model. We
posit that containment measures correspond to feedback between the status of
the outbreak (the daily or the cumulative number of cases and fatalities) and
the reproduction factor. We present the exact phase space solution of the
controlled SIR model and use it to quantify containment policies for a large
number of countries in terms of short and longterm control parameters.
Furthermore, we identified for numerous countries a relationship between the
number of fatalities within a fixed period before and after the peak in daily
fatalities. As the number of fatalities corresponds to the number of
hospitalised patients, the relationship can be used to predict the cumulative
medical load, once the effectiveness of outbreak suppression policies is
established with sufficient certainty.
Authors: F. Schubert, C. Gros
Journalref.: Frontiers In Computational Neuroscience
24, 587721 (2021).
Recurrent cortical network dynamics plays a crucial role for sequential
information processing in the brain. While the theoretical framework of
reservoir computing provides a conceptual basis for the understanding of
recurrent neural computation, it often requires manual adjustments of global
network parameters, in particular of the spectral radius of the recurrent
synaptic weight matrix. Being a mathematical and relatively complex quantity,
the spectral radius is not readily accessible to biological neural networks,
which generally adhere to the principle that information about the network
state should either be encoded in local intrinsic dynamical quantities (e.g.
membrane potentials), or transmitted via synaptic connectivity. We present two
synaptic scaling rules for echo state networks that solely rely on locally
accessible variables. Both rules work online, in the presence of a continuous
stream of input signals. The first rule, termed flow control, is based on a
local comparison between the mean squared recurrent membrane potential and the
mean squared activity of the neuron itself. It is derived from a global scaling
condition on the dynamic flow of neural activities and requires the
separability of external and recurrent input currents. We gained further
insight into the adaptation dynamics of flow control by using a mean field
approximation on the variances of neural activities that allowed us to describe
the interplay between network activity and adaptation as a twodimensional
dynamical system. The second rule that we considered, variance control,
directly regulates the variance of neural activities by locally scaling the
recurrent synaptic weights. The target set point of this homeostatic mechanism
is dynamically determined as a function of the variance of the locally measured
external input. This functional relation was derived from the same meanfield
approach that was used to describe the approximate dynamics of flow control.
The effectiveness of the presented mechanisms was tested numerically using
different external input protocols. The network performance after adaptation
was evaluated by training the network to perform a time delayed XOR operation
on binary sequences. As our main result, we found that flow control can
reliably regulate the spectral radius under different input statistics, but
precise tuning is negatively affected by interneural correlations. Furthermore,
flow control showed a consistent task performance over a wide range of input
strengths/variances. Variance control, on the other side, did not yield the
desired spectral radii with the same precision. Moreover, task performance was
less consistent across different input strengths.
Given the better performance and simpler mathematical form of flow control, we
concluded that a local control of the spectral radius via an implicit
adaptation scheme is a realistic alternative to approaches using classical “set
point” homeostatic feedback controls of neural firing.
Authors: C. Gros, R. Valenti, L. Schneider, K. Valenti, D. Gros
Journalref.: Scientific Reports (in press).
The rapid spread of the Coronavirus (COVID19) confronts policy makers with
the problem of measuring the effectiveness of containment strategies and the
need to balance public health considerations with the economic costs of a
persistent lockdown. We introduce a modified epidemic model, the controlledSIR
model, in which the disease reproduction rate evolves dynamically in response
to political and societal reactions. An analytic solution is presented. The
model reproduces official COVID19 cases counts of a large number of regions
and countries that surpassed the peak of the outbreak. A single unbiased
feedback parameter is extracted from field data and used to formulate an index
that measures the efficiency of containment policies (the CEI index). CEI
values for a range of countries are given. For two variants of the
controlledSIR model, detailed estimates of the total medical and
socioeconomic costs are evaluated over the entire course of the epidemic.
Costs comprise medical care cost, the economic cost of social distancing, as
well as the economic value of lives saved. Under plausible parameters, strict
measures fare better than a handsoff policy. Strategies based on actual case
numbers lead to substantially higher total costs than strategies based on the
overall history of the epidemic.
Authors: C. Gros, D. Gros
Journalref.: Covid Economics 62, 74 (2020).
We analyze 'stop and go' containment policies which produces infection cycles
as periods of tight lockdowns are followed by periods of falling infection
rates, which then lead to a relaxation of containment measures, allowing cases
to increase again until another lockdown is imposed. The policies followed by
several European countries seem to fit this pattern. We show that 'stop and go'
should lead to lower medical costs than keeping infections at the midpoint
between the highs and lows produced by 'stop and go'. Increasing the upper and
reducing the lower limits of a stop and go policy by the same amount would
lower the average medical load. But increasing the upper and lowering the lower
limit while keeping the geometric average constant would have the opposite
impact. We also show that with economic costs proportional to containment, any
path that brings infections back to the original level (technically a closed
cycle) has the same overall economic cost.
Authors: O. Neumann, C. Gros
Journalref.: Learning Meets Combinatorial Algorithms
at NeurIPS2020 (2020).
We seek to determine the metabolic cost of a larger brain
compared to the advantage it gives in solving these problems,
where advantage is defined as performance in comparison to
competitors. For this purpose a twoplayer game based on the
knapsack problem is used. In effect, two players compete
over shared resources, with the goal to collect more
resources than the opponent. Neural nets with respectively
N_{A} and N_{B} hidden neurons are trained
using a variant of the AlphaGo Zero algorithm. A surprisingly
simple relation, N_{A}/(N_{A}+N_{B}),
is found for the relative win rate.
Success increases linearly with investments in additional
resources when the networks sizes are very different, i.e.
when N_{A} ≪ N_{B}, with diminishing
returns when both networks become comparable in size.
Authors: C. Gros
Journalref.: Royal Society Open Science 7, 200411 (2020).
Envy, the inclination to compare rewards, can be expected to unfold when
inequalities in terms of payoff differences are generated in competitive
societies. It is shown that increasing levels of envy lead inevitably to a
selfinduced separation into a lower and an upper class. Class stratification
is Nash stable and strict, with members of the same class receiving identical
rewards. Upper class agents play exclusively pure strategies, all lower class
agents the same mixed strategy. The fraction of upper class agents decreases
progressively with larger levels of envy, until a single upper class agent is
left. Numerical simulations and a complete analytic treatment of a basic
reference model, the shopping trouble model, are presented. The properties of
the classstratified society are universal and only indirectly controllable
through the underlying utility function, which implies that class stratified
societies are intrinsically resistant to political control. Implications for
human societies are discussed. It is pointed out that the repercussions of envy
are amplified when societies become increasingly competitive.
Authors: M. Göbel, C. Gros
Journalref.: Journal of Physics A 53, 035003 (2020).
We introduce and study a nonconserving sandpile model, the autonomously
adapting sandpile (AAS) model, for which a site topples whenever it has two or
more grains, distributing three or two grains randomly on its neighboring
sites, respectively with probability p and (1−p). The toppling process is
independent of the actual number of grains z_{i} of the toppling site, as long as
z_{i}≥2. For a periodic lattice the model evolves into an inactive state for small
p, with the number of active sites becoming stationary for larger values of p.
In one and two dimensions we find that the absorbing phase transition occurs
for p_{c}≈0.717 and p_{c}≈0.275.
The symmetry of bipartite lattices allows states in which all active sites
are located alternatingly on one of the two sublattices, A and B, respectively
for even and odd times. We show that the ABsublattice symmetry is
spontaneously broken for the AAS model, an observation that holds also for the
Manna model. One finds that a metastable ABsymmetry conserving state is
transiently observable and that it has the potential to influence the width of
the scaling regime, in particular in two dimensions.
The AAS model mimics the behavior of integrateandfire neurons which
propagate activity independently of the input received, as long as the
threshold is crossed. Abstracting from regular lattices, one can identify sites
with neurons and consider quenched networks of neurons connected to a fixed
number G of other neurons, with G being drawn from a suitable distribution. The
neuronal activity is then propagated to G other neurons. The AAS model is hence
well suited for theoretical studies of nearly critical brain dynamics. We also
point out that the waitingtime distribution allows an avalanchefree
experimental access to criticality.
Authors: C. Gros
Journalref.: IEEE International Conference on
Humanized Computing and Communication, Laguna Hill 2019.
Given a certain complexity level, humanized agents may select from a wide
range of possible tasks, with each activity corresponding to a transient goal.
In general there will be no overarching credit assignment scheme allowing to
compare available options with respect to expected utilities. For this
situation we propose a task selection framework that is based on time
allocation via emotional stationarity (TAES). Emotions are argued to correspond
to abstract criteria, such as satisfaction, challenge and boredom, along which
activities that have been carried out can be evaluated. The resulting timeline
of experienced emotions is then compared with the `character' of the agent,
which is defined in terms of a preferred distribution of emotional states. The
longterm goal of the agent, to align experience with character, is achieved by
optimizing the frequency for selecting the individual tasks. Upon optimization,
the statistics of emotion experience becomes stationary.
Authors: L. Schneider, C. Gros
Journalref.: Royal Society Open Science 6, 190944 (2019).
Analyzing the timeline of US, UK, German and Dutch
music charts, we find that the evolution of album
lifetimes and of the size of weekly rank changes
provide evidence for an acceleration of cultural
processes. For most of the past five decades number
one albums needed more than a month to climb to the
top, nowadays an album is in contrast top ranked
either from the start, or not at all. Over the last
three decades, the number of toplisted albums increased as a
consequence from roughly a dozen per year to about 40.
The distribution of album lifetimes evolved during
the last decades from a lognormal distribution
to a powerlaw, a profound change. Presenting an
informationtheoretical approach to human activities,
we suggest that the fading relevance of personal
time horizons may be causing this phenomenon.
Furthermore we find that sales and airplay based
charts differ statistically and that the inclusion
of streaming affects chart diversity adversely.
We point out in addition that opinion dynamics may accelerate
not only in cultural domains, as found here, but also
in other settings, in particular in politics,
where it could have far reaching consequences.
Authors: Kim Koglin, Blucsu Sandor, Claudius Gros
Journalref.: PLOS One 14, e0217004 (2019).
Behavior is characterized by sequences of goaloriented conducts, such as
food uptake, socializing and resting. Classically, one would define for each
task a corresponding satisfaction level, with the agent engaging, at a given
time, in the activity having the lowest satisfaction level. Alternatively, one
may consider that the agent follows the overarching objective to generate
sequences of distinct activities. To achieve a balanced distribution of
activities would then be the primary goal, and not to master a specific task.
In this setting, the agent would show two types of behaviors, taskoriented,
and tasksearching phases, with the latter interseeding the former.
We study the emergence of autonomous task switching for the case of a
simulated robot arm. Grasping one of several moving objects corresponds in this
setting to a specific activity. Overall, the arm should follow a given object
temporarily and then move away, in order to search for a new target and
reengage. We show that this behavior can be generated robustly when modeling
the arm as an adaptive dynamical system. The dissipation function is in this
approach time dependent. The arm is in a dissipative state when searching for a
nearby object, dissipating energy on approach. Once close, the dissipation
function starts to increase, with the eventual sign change implying that the
arm will take up energy and wander off. The resulting explorative state ends
when the dissipation function becomes again negative and the arm selects a new
target. We believe that our approach may be generalized to generate
selforganized sequences of activities in general.
Authors: Frederike Kubandt, Michael Nowak, Tim Koglin,
Claudius Gros, Bulcsu Sandor
Journalref.: Adaptive Behavior 27, 285 (2019).
Which kind of complex behavior may arise from selforganizing principles? We
investigate this question for the case of snakelike robots composed of
passively coupled segments, with every segment containing two wheels actuated
separately by a single neuron. The robot is self organized both on the level of
the individual wheels and with respect to interwheel coordination, which
arises exclusively from the mechanical coupling of the individual wheels and
segments. For the individual wheel, the generating principle proposed results
in locomotive states that correspond to selforganized limit cycles of the
sensorimotor loop.
Our robot interacts with the environment by monitoring the state of its
actuators, that is via propriosensation. External sensors are absent. In a
structured environment the robot shows complex emergent behavior that includes
pushing movable blocks around, reversing direction when hitting a wall and
turning when climbing a slope. On flat grounds the robot wiggles in a
snakelike manner, when moving at higher velocities. We also investigate the
emergence of motor primitives, viz the route to locomotion, which is
characterized by a series of local and global bifurcations in terms of
dynamical system theory.
Authors: H. Wernecke, B. Sandor, C. Gros
Journalref.: Physics Reports 824, 1 (2019).
The time needed to exchange information in the physical world induces a delay
term when the respective system is modeled by differential equations. Time
delays are hence ubiquitous, being furthermore likely to induce instabilities
and with it various kinds of chaotic phases. Which are then the possible types
of time delays, induced chaotic states, and methods suitable to characterize
the resulting dynamics? This review presents an overview of the field that
includes an indepth discussion of the most important results, of the standard
numerical approaches and of several novel tests for identifying chaos. Special
emphasis is placed on a structured representation that is straightforward to
follow. Several educational examples are included in addition as entry points
to the rapidly developing field of time delay systems.
Authors: C. Gros
Journalref.: Acta Astronautica 157, 263 (2019).
Time is arguably the key limiting factor for interstellar exploration. At high
speeds, flyby missions to nearby stars by laser propelled wafersats taking
50100 years would be feasible. Directed energy launch systems could accelerate
on the other side also crafts weighing several tons to cruising speeds of the
order of 1000\,km/s (c/300). At these speeds, superconducting magnetic sails
would be able to decelerate the craft by transferring kinetic energy to the
protons of the interstellar medium. A tantalizing perspective, which would
allow interstellar probes to stop whenever time is not a limiting factor. Prime
candidates are in this respect Genesis probes, that is missions aiming to offer
terrestrial life new evolutionary pathways on potentially habitable but
hitherto barren exoplanets.
Genesis missions raise important ethical issues, in particular with regard to
planetary protection. Here we argue that exoplanetary and planetary protection
differ qualitatively as a result of the vastly different cruising times for
payload delivering probes, which are of the order of millennia for interstellar
probes, but only of years for solar system bodies. Furthermore we point out
that our galaxy may harbor a large number of habitable exoplanets, Mdwarf
planets, which could be sterile due to the presence of massive primordial
oxygen atmospheres. We believe that the prospect terrestrial life has in our
galaxy would shift on a fundamental level in case that the existence of this
type of habitable but sterile oxygen planets will be corroborated by future
research. It may also explain why our sun is not a M dwarf, the most common
star type, but a mediumsized Gclass star.
Authors: B. Sandor, M. Nowak, T. Koglin, L. Martin, C. Gros
Journalref.: Frontiers in Neurorobotics 12, 40 (2018).
Selforganized robots may develop attracting states within the sensorimotor
loop, that is within the phase space of neural activity, body, and
environmental variables. Fixpoints, limit cycles, and chaotic attractors
correspond in this setting to a nonmoving robot, to directed, and to irregular
locomotion respectively. Short higherorder control commands may hence be used
to kick the system from one selforganized attractor robustly into the basin of
attraction of a different attractor, a concept termed here as kick control. The
individual sensorimotor states serve in this context as highly compliant motor
primitives.
We study different implementations of kick control for the case of simulated
and realworld wheeled robots, for which the dynamics of the distinct wheels is
generated independently by local feedback loops. The feedback loops are
mediated by rateencoding neurons disposing exclusively of propriosensoric
inputs in terms of projections of the actual rotational angle of the wheel. The
changes of the neural activity are then transmitted into a rotational motion by
a simulated transmission rod akin to the transmission rods used for steam
locomotives.
We find that the selforganized attractor landscape may be morphed both by
higherlevel control signals, in the spirit of kick control, and by interacting
with the environment. Bumping against a wall destroys the limit cycle
corresponding to forward motion, with the consequence that the dynamical
variables are then attracted in phase space by the limit cycle corresponding to
backward moving. The robot, which does not dispose of any distance or contact
sensors, hence reverses direction autonomously.
Authors: P. Trapp, R. Echeveste, C. Gros
Journalref.: Scientific Reports 8, 8939 (2018).
Spontaneous brain activity is characterized in part by a balanced asynchronous
chaotic state. Cortical recordings show that excitatory (E) and inhibitory (I)
drivings in the EI balanced state are substantially larger than the overall
input. We show that such a state arises naturally in fully adapting networks
which are deterministic, autonomously active and not subject to stochastic
external or internal drivings. Temporary imbalances between excitatory and
inhibitory inputs lead to large but shortlived activity bursts that stabilize
irregular dynamics. We simulate autonomous networks of rateencoding neurons
for which all synaptic weights are plastic and subject to a Hebbian plasticity
rule, the flux rule, that can be derived from the stationarity principle of
statistical learning. Moreover, the average firing rate is regulated
individually via a standard homeostatic adaption of the bias of each neuron’s
inputoutput nonlinear function. Additionally, networks with and without
shortterm plasticity are considered. EI balance may arise only when the mean
excitatory and inhibitory weights are themselves balanced, modulo the overall
activity level. We show that synaptic weight balance, which has been considered
hitherto as given, naturally arises in autonomous neural networks when the here
considered selflimiting Hebbian synaptic plasticity rule is continuously
active.
Authors: C. Gros
Journalref.: Royal Society Open Science 5, 180167 (2018).
We point out that the Nobel prize production of the USA, the UK, Germany and
France has been in numbers that are large enough to allow for a reliable
analysis or the longterm historical developments. Nobel prizes are often
split, such that up to three awardees receive a corresponding fractional prize.
The historical trends for the fractional number of Nobelists per population are
surprisingly robust, indicating in particular that the maximum Nobel
productivity peaked in the 1970s for the US and around 1900 for both France and
Germany. The yearly success rates of these three countries are to date of the
order of 0.20.3 physics, chemistry and medicine laureates per 100 million
inhabitants, with the US value being a factor 2.4 down from the maximum
attained in the 1970s. The UK managed in contrast to retain during most of the
last century a rate of 0.91.0 science Nobel prizes per year and per 100
million inhabitants. For the USA one finds that the entire history of science
Noble prizes is described on a per capita basis to an astonishing accuracy by a
single large productivity boost decaying at a continuously accelerating rate
since its peak in 1972.
Authors: H. Wernecke, B. Sandor, C. Gros
Journalref.: Journal of Physics Communications 2, 095008 (2018).
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
onedimensional (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
nonadiabatic, even though there are up to four orders of magnitude difference
between the slow and the fast time scales.
Authors:M. Bijelic, R. Kaneko, C. Gros, R. Valenti
Journalref.: Physical Review B 97, 125142 (2018).
We investigate the competition between chargedensitywave (CDW) states and a
Coulomb interactiondriven topological Mott insulator (TMI) in the honeycomb
extended Hubbard model. For the spinful model with onsite (U) and
nextnearestneighbor (V2) Coulomb interactions at half filling we find two
peculiar sixsublattice chargedensitywave (CDW) insulating states by using
variational Monte Carlo simulation as well as the HartreeFock approximation.
We observe that conventional ordered states always win with respect to the TMI.
The ground state is given in the largeV2 region by a CDW characterized by a
220200 (001122) charge configuration for smaller (larger) U, where 0, 1 and 2
denote essentially empty, singly and doublyoccupied sites. Within the
001122type CDW phase, we find a magnetic transition driven by an emergent
coupleddimer antiferromagnet on an effective square lattice of singlyoccupied
sites. Possible realizations of the states found are discussed.
Authors: C. Gros
Journalref.: European Physical Journal B 90, 223 (2017).
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 sociopolitical instabilities may develop
once the aggregate time scale for the evolution of the political values of the
electorate falls below 715 months.
Authors: C. Gros
Journalref.: Journal of Physics Communications 1, 045007 (2017).
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 braking 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 cm^{3}, 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) = 0.081A_{R}log^{3}(I/(βI_{c})), where
A_{R}=πR^{2} is the bare sail area,
I the current and β=v/c. The critical current I_{c}
is 1.55 10^{6} 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 braking would involve daunting mass requirements of the
order of 10^{3} tons. A low speed mission to the Trappist1 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 10^{4} years
imply however that a mission to the Trappist1 system would be viable only for
mission concepts for which time constrains are not relevant.
Authors: H. Wernecke, B. Sandor, C. Gros
Journalref.: Scientific Reports 7, 1087 (2017).
Implementation:
DynamicalSystems.jl (Julia)
For a chaotic system pairs of initially closeby 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 interorbital 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 01 indicator
for chaos based on the crossdistance 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 crosscorrelation 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
01 manner from the properties of pairs of trajectories.
Authors: C. Gros
Journalref.: Astrophysics and Space Science 361, 324 (2016).
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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 34 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 lowcost robotic microcraft equipped with a
onboard gene laboratory for the in situ synthesis of the microbes.
We review
here our current understanding of the processes determining the timescales
shaping the geoevolution of an earthlike 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.
Authors: R. Echeveste, C. Gros
Journalref.: 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 nonuniform network, composed exclusively of
excitatory neurons, we find a rich repertoire of selfinduced 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
networkwide distribution of natural frequencies is however generated
autonomously through selfconsistent 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.
Authors: L. Martin, B. Sandor, C. Gros
Journalref.: 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 shortterm synaptic plasticity (STSP) may
generate selforganized motor patterns. We simulated sphereshaped 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 STSPrules.
We find that a wide palette of motion patterns are
generated through the interaction of STSP, robot, and environment (closedloop
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.
Authors: R. Kaneko, L.F. Tocchio, R. Valenti, C. Gros
Journalref.: 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
onsite U and nearest neighbor V Coulomb interactions at 3/4 filling (n=3/2)
and (ii) the triangular lattice with onsite U, nearest neighbor V, and
nextnearest neighbor V′ Coulomb interactions at 3/8 filling (n=3/4). We
consider various approaches including meanfield 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 nearestneighbor 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 fourspin ringexchange 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 onedimensional
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.
Authors: R. Kaneko, L.F. Tocchio, R. Valenti, F. Becca, C. Gros
Journalref.: Physical Review B 93, 125127 (2016).
We show that JastrowSlater wave functions, in which a densitydensity Jastrow
factor is applied onto an uncorrelated fermionic state, may possess longrange
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.
Authors: B. Sandor, T. Jahn, L. Martin, C. Gros
Journalref.: 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 selforganized 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 backandforth
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 backandforth motion of the robot.
Authors: B. Sandor, C. Gros
Journalref.: 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 2ddimensional 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.
Authors: R. Echeveste, S. Eckmann, C. Gros
Journalref.: Entropy 17, 3838 (2015).
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 selflimiting 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 nonGaussian 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 nonlinear 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.
Authors: R. Echeveste, C. Gros
Journalref.: Neural Computation March 27, 672 (2015).
We present an effective model for timingdependent 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 anticausal 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 timedependent to ratedependent plasticity
occurring for both correlated and uncorrelated spike patterns.
Authors: R. Echeveste, C. Gros
Journalref.: 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, selflimiting, 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 firingrate 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 byproduct of the
synaptic flux minimization. This selflimiting 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.
Authors: C. Gros, M. Linkerhand, V. Walther
Journalref.: Artificial Neural Networks and Machine Learning
 ICANN 2014 , S. Wermter et al. (Eds),
pp. 6572. 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
continuoustime autonomous model networks. We find both first and secondorder
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.
Authors: L.F. Tocchio, C. Gros, R. Valenti, F. Becca
Journalref.: 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 spinliquid and magnetic phases with different
pitch vectors describing both collinear and coplanar (spiral) order. For
relatively large values of the onsite interaction U/t'≥10 and substantial
frustration, i.e., 0.3≤t/t'≤0.8, the spinliquid 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 interchain hoppings from small to intermediate frustration.
Authors: L.F. Tocchio, C. Gros, X.F. Zhang, S. Eggert
Journalref.: 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 longrange 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.
Authors: R. Rüger, L.F. Tocchio, R. Valenti, C. Gros
Journalref.: 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 onsite
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 manybody approaches.
Authors: G.A. Luduena, M.D. Behzad, C. Gros
Journalref.: 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{ACTR} 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{ACTR} 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.
Authors: D. Markovic, C. Gros
Journalref.: Physics Reports 536, 4174 (2014).
[100+]
Power laws 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 noncritical steady state
characterized by heavytailed distributions. The third line of modeling
proposes a noncritical state which is selforganizing, 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 realworld 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 selforganized criticality, the
interaction between theoretical development and experimental observations has
been very fruitful, leading to a series of novel concepts and insights.
Authors: C. Gros
Journalref.: M. Prokopenko (ed.), Guided SelfOrganization: Inception,
5366, 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 selforganizing 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.
Authors: G.A. Luduena, H. Meixner, G. Kaczor, C. Gros
Journalref.: European Physical Journal B 86, 348 (2013).
We performed a largescale crawl of the World Wide Web, covering 6.9 Million
domains and 57 Million subdomains, including all hightraffic 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
outdegree, the local clustering coefficient, the firstneighbor indegree 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 scalefree 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 indegree and the Alexa traffic rank, a popular index
for the traffic volume, finding nontrivial powerlaw 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.
Authors: M. Linkerhand, C. Gros
Journalref.: 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
informationtheoretical 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.
Authors: C. Gros, D. Markovic
Journalref.: 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 scaleinvariant 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 noncritical 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.
Authors: L.F. Tocchio, H. Lee, H.O. Jeschke, R.Valentí, C. Gros
Journalref.: 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 GutzwillerJastrow
wavefunctions with backflow corrections. We find good agreement,
apart from the location of the MottHubbard 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 pseudogap state of the highT_{c} superconductors.
Furthermore, we investigate the stability of the pseudogap 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.
Authors: L.F. Tocchio, H. Feldner, F. Becca, R. Valentí, C. Gros
Journalref.: Physical Review B 87, 035143 (2013).
We study the competition between magnetic and spinliquid
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
chargetransfer salts. By using a variational approach that includes
spiral magnetic order, we provide solid evidence that a spinliquid
phase is stabilized in the stronglycorrelated 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 HartreeFock 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 chargetransfer salts.
Authors: D. Markovic, A. Schuelein, C. Gros
Journalref.: 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 onestep routing memory. Investigating
the respective cycle length distributions for complete graphs
we find log corrections to powerlaw
scaling for the mean cycle length, as a function
of the number of vertices, and a subpolynomial growth
for the overall number of cycles.
When observing experimentally a realworld 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 scaleinvariant, but may show powerlaw
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.
Authors: M. Linkerhand, C. Gros
Journalref.:
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 timeaverage firing rate. For the case of
sparse encoding, when the target firingrated distribution is
bimodal, we observe the occurrence of spontaneous quasiperiodic
adaptive oscillations resulting from fast transition between two
quasistationary attractors. We interpret this behavior as
selforganized stochastic tipping, with noise driving the escape
from the quasistationary attractors.
Authors: G.A. Luduena, C. Gros
Journalref.: 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 onetoone 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
selforganizing, 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 selforganization 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
feedforward neural network which follows a local output minimization
(antiHebbian) 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.
Authors: C. Gros
Journalref.: Philosophy and Theory of Artificial Intelligence, 187198;
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 shortterm utility functions, in the context
of an environment where information, computing power and
time constitute scarce resources.
Authors: C. Gros
Journalref.: 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 sublinear 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.
Authors: L.F. Tocchio, F. Becca, C. Gros
Journalref.: Physical Review B 86, 035102 (2012).
The underlying Fermi surface is a key concept for stronglyinteracting
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 twodimensional tt' Hubbard model
and find a nontrivial 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
metalinsulator transition, from a weakly interacting metal at
small coupling to a resonating valencebond superconductor at
intermediate coupling. A violation of the Luttinger theorem
is detected at low hole dopings.
Authors: D. Markovic, C. Gros
Journalref.: 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 autonomousstate 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 nonsynaptic plasticity on the default dynamical
state of recurrent neural networks. The nonsynaptic 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 selforganized information processing and critical brain dynamics.
Authors: C. Gros, G. Kaczor, D. Markovic
Journalref.: 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 252633 Million publicly available data files
on the Internet corresponding to an overall storage volume of
284675 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 realworld 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
lognormal distribution for multimedia files, for which time is
a defining qualia.
Authors: L.F. Tocchio, F. Becca, C. Gros
Journalref.: 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 SlaterJastrow state, usually considered in this context.
We provide evidence that, within this approach, it is possible to
have a satisfactory connection with the strongcoupling 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 Mottinsulator 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 particlehole excitations
á la Feynman over the groundstate wave function.
Authors: A. Di Ciolo, L.F. Tocchio, C. Gros
Journalref.: 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 singleparticle
(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.
Authors: D. Markovic, C. Gros
Journalref.: 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
steadystate dynamical state. We consider polyhomeostasis
for individual and networks of firingrate neurons,
adapting to achieve target distributions of firing
rates maximizing information entropy. We show that any
finite polyhomeostatic adaption rate destroys all attractors
in Hopfieldlike network setups, leading to intermittently
bursting behavior and selforganized chaos. The importance
of polyhomeostasis to adapting behavior in general is discussed.
Authors: C. Gros
Journalref.: Cognitive Computation 2, 78 (2010).
The primary tasks of a cognitive system is to survive
and to maximize a lifelong utility function, like the
number of offsprings. A direct computational maximization
of lifelong utility is however not possible in complex
environments, especially in the context, of realworld
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.
Authors: L.F. Tocchio, F. Becca, C. Gros
Journalref.: Physical Review B 81, 205109 (2010).
We investigate the nature of the interactiondriven
MottHubbard transition of the halffilled t_{1}t_{2}
Hubbard model in one dimension, using a fullfledged
variational Monte Carlo approach including a
distancedependent Jastrow factor and backflow
correlations. We present data for the evolution
of the magnetic properties across the MottHubbard
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 MottHubbard transition in the insulating
state, with a firstorder reorganization when
crossing into the conducting state.
Authors: C. Gros, G. Kaczor
Journalref.: Logic Journal IGPL 18, 686 (2010).
The human brain is autonomously active, being characterized by a
selfsustained neural activity which would be present even in the
absence of external sensory stimuli. Here we study the
interrelation between the selfsustained 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 timeseries 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 semistable 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 nonlinear
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.
Authors: L.F. Tocchio, A. Parola, C. Gros, F. Becca
Journalref.: Physical Review B 80, 064419 (2009).
The twodimensional Hubbard model on the anisotropic
triangular lattice, with two different hopping
amplitudes t and t', is relevant to describe the
lowenergy physics of κ(ET)_{2}X,
a family of organic salts. The groundstate
properties of this model are studied by using
Monte Carlo techniques, on the basis of a
recent definition of backflow correlations for
stronglycorrelated lattice systems. The results
show that there is no magnetic order for
reasonably large values of the electronelectron
interaction U and frustrating ratio
t'/t = 0.85, suitable to describe the nonmagnetic
compound with X=Cu_{2}(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=Cu_{2}(SCN)_{2},
Cu[N(CN)_{2}]Cl, or Cu[N(CN)_{2}]Br.
Authors: D. Markovic, C. Gros
Journalref.: 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 longterm 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 nonzero information centrality
to be extensive/subextensive 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.
Authors: C. Gros
Journalref.: Cognitive Computation 1, 77 (2009).
The human brain is autonomously active. To understand
the functional role of this selfsustained 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. selfsustained neural activity in the framework of
reservoir computing. The main emphasis will be then
on two paradigmal neural network architectures showing
continuously ongoing transientstate dynamics:
saddle point networks and networks of attractor relics.
Selfactive 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 Hebbianstyle 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
setup. The system performs online, and on its own,
a nonlinear independent component analysis of
the sensory data stream, all the time being
continuously and autonomously active. This
process maps the independent components of
the sensory input onto the attractor relics,
which acquire in this way a semantic meaning.
Authors: C. Gros
Journalref.: (Book Chapter)
Handbook of Research on Synthetic Emotions and Sociable Robotics: New
Applications in Affective Computing and Artificial Intelligence,
J. Vallverdu, D. Casacuberta (Eds). IGIGlobal (2009).
All selfactive living beings need to solve the motivational problem:
The question what to do at any moment of their live. For humans and
nonhuman 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 metalearning. This intrinsic
relation between emotions, metalearning 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).
Authors: I. Opahle, H. C. Kandpal, Y. Zhang, C. Gros,
R. Valentí
Journalref.: 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 densityfunctional 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 lowtemperature structural distortion
is found to have only a minor influence on them. Analysis of
the Fermisurface 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.
Authors: H. Das, T. SahaDasgupta, C. Gros, R. Valentí
Journalref.: Physical Review B 77, 224437 (2008).
Using firstprinciples electronic structure calculations based on the
Nth order muffin tin orbital (NMTO)downfolding technique,
we derived the lowenergy spin model for CuTe_{2}O_{5}.
Our study reveals
that this compound is a 2D coupled spindimer system with
the strongest CuCu interaction mediated by two OTeO 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.
Authors: G. Kaczor, C. Gros
Journalref.: 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 realworld graphs.
Authors: C. Gros
Journalref.: Springer (2008/2010/2013/2015).
[100+]
We are living in an ever more complex world, an epoch where human actions
can accordingly acquire farreaching 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 "commonssense"
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 SmallWorld Networks;
 Chaos, bifurcations and diffusion;
 Complexity and Information Theory;
 Random Boolean Networks;
 Cellular Automata and SelfOrganized 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.
Authors: B. Edegger, V.N. Muthukumar, C. Gros
Journalref.: 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.
Authors: C. Gros
Journalref.: New Journal of Physics 9, 109 (2007).
We investigate dynamical systems characterized by a
time series of distinct semistable 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 transientstate 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 continuoustime neural network model,
that a single local updating rule for online learning
allows simultaneously (i) for information storage via
unsupervised Hebbiantype 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 timereversal symmetry and link symmetry
are discussed.
Authors: B. Edegger, C. Gros, V.N. Muthukumar
Journalref.: Physical Review B 74 165109 (2006).
We present a variational Monte Carlo (VMC) study of spontaneous Fermi
surface symmetry breaking in the tJ 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 ydirections.
However, the best variational state remains a pure superconducting
state with dwave 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 halffilled limit, revealing some limitations of
mean field theories. VMC and complementary RMFT calculations
also confirm the SBMFT predictions that manybody 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.
Authors: C. Gros, B. Edegger, V.N. Muthukumar, P.W. Anderson
Journalref.: 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.
Authors: B. Edegger, V.N. Muthukumar, C. Gros, P.W. Anderson
Journalref.: Physical Review Letters 96, 207002 (2006).
We study the electronic structure of a strongly correlated
dwave superconducting state. Combining a renormalized mean field
theory with direct calculation of matrix elements, we obtain results
for the nodal Fermi velocity, v_{F},
the Fermi wave vector, k_{F},
and momentum distribution, n_{k}, as a function of hole
doping in a Gutzwiller projected dwave superconductor.
We calculate the energy dispersion, E_{k}, and spectral weight,
of the GutzwillerBogoliubov 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.
Authors: C. Gros
Journalref.: KI 2005, Springer Lecture Notes in Artificial
Intelligence 3698, 366 (2005).
Several guiding principles for thought processes are proposed
and a neuralnetworktype model implementing these principles is presented
and studied. We suggest to consider thinking within
an associative network builtup 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 winnerstakeall neural network with sparse
coding and an additional coupling to local reservoirs.
We show that this network is capable to generate autonomously a
selfsustained timeseries 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.
Authors: B. Edegger, N. Fukushima, C. Gros, V.N. Muthukumar
Journalref.: 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.
Authors: N. Fukushima, B. Edegger, V.N. Muthukumar, C. Gros
Journalref.: 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 MonteCarlo (VMC). We also report VMC
results for projected superconducting states that show novel oscillations
in the hole density near the reservoir site.
Authors: C. Gros
Journalref.: JBIS 58, 108 (2005).
See also: peregrinus interstellar
The 1950 lunchtable 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 milkyway. This conclusion rest fundamentally on
the adhoc 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.
Authors: T. SahaDasgupta, R. Valentí, F. Capraro, C. Gros
Journalref.: Physical Review Letters 95, 107201 (2005).
Following the recent discussion on the nature of the interactions in the
tubular system Na_{2}V_{3}O_{7}, we present a
detailed abinitio 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 edgesharing
VV hopping interactions are of the same order of magnitude as the
cornersharing 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 weaklycoupled partially frustrated
ninesite rings with the geometry of a spindiamond necklace.
We calculate the susceptibility by exact diagonalization and obtain
good agreement with experimental observations.
Authors: C. Gros, K. Hamacher, W. Wenzel
Journalref.: Europhys. Lett. 69, 616 (2005).
We investigate the momentum distribution function near the
MottHubbard transition in the onedimensional
t_{1}t_{2} Hubbard model
(the zigzag Hubbard chain), with the densitymatrix
renormalizationgroup technique. We show that for strong
interactions the MottHubbard transition occurs between the
metallicphase 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 MottHubbard transition and the commensurateincommensurate
transition in the insulating spingapped state in their respective
groundstate momentum distribution functions.
Authors: C. Gros, G.Y. Chitov
Journalref.: Europhys. Lett. 69, 447 (2005).
We present microscopic estimates for the spinspin and
spinspeudospin interactions of the quarterfilled ladder compound
NaV_{2}O_{5}, obtained by exactly diagonalizing
appropriate clusters of
the underlying generalized Hubbard Hamiltonian. We present evidence
for a substantial interladder spinpseudospin 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
NaV_{2}O_{5} and deduce the absence of a soft
antiferroelectric mode.
Authors: G.Y. Chitov, C. Gros
Journalref.: Low Temp. Phys. 31, 722 (2005).
We study the 2D Ising Model on a square lattice with additional
nonequal diagonal nextnearest 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 nonequal diagonal couplings, are predicted.
In a particular region of interactions, corresponding to the Ising
model's superantiferromagnetic (SAF) ground state, we also
consider a spinpseudospin model comprised of the quantum Ising
model coupled to XY spin chains. The spinpseudospin model's
spinSAF transition into the phase with coexistent the SAF Ising
(pseudospin) longrange order and the spin gap, has the properties
analogous to the reported earlier by us (condmat/0310494) for a
simpler coupled model. Along with destruction of the quantum
critical point of the transverse Ising model, the phase digram
of the spinpseudospin model can also demonstrate a reentrance.
A detailed study of the latter is presented. The mechanism of
the reentrance, due to interplay of interactions in the
coupled model, and the conditions of its appearance are established.
Authors: G.Y. Chitov, C. Gros
Journalref.: J. Phys.: Cond. Matt. 16, L415 (2004).
We propose a mechanism for the observed stacking charge order in the
quarterfilled ladder compound NaV_{2}O_{5}.
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 nextnearest planes with the 4fold degenerate superantiferroelectric
inplane order.
Authors: K. Louis, C. Gros
Journalref.: 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 intrachain interactions)
for different fillings.
Authors: T. SahaDasgupta, R. Valentí, H. Rosner, C. Gros
Journalref.: 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 electronphonon coupling. Based on these
results, we propose a microscopic explanation of the behavior of TiOCl near the
phase transition to a spingapped system.
Authors: G.Y. Chitov, C. Gros
Journalref.: Physical Review B 69, 104423 (2004).
We study the spinpseudospin Hamiltonian of the Ising Model
in Transverse Field (IMTF) for pseudospins,
coupled to the XYspins on a triangular lattice.
This model appears from analyses of the quarterfilled
ladder compound NaV_{2}O_{5}, and pseudospins represent
its charge degrees of freedom. In the molecularfield
approximation we find that the model possesses two phases:
chargedisordered without spin gap;
and a lowtemperature phase containing both
the antiferroelectric (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 interladder spinpseudospin coupling,
responsible for the spin gap generation,
also destroys the IMTF quantum critical point,
resulting in the exponential behavior of T_{c}
in the region of Ising's coupling where
the IMTF is always disordered. We conclude that our
meanfield results give a qualitatively correct description
of the phase transition in NaV_{2}O_{5},
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.
Authors: K. Louis, C. Gros
Journalref.: Physical Review B 70, R100410 (2004).
In this paper we develop a clustervariant of the Stochastic
Series expansion method (SCSE). For certain systems with
longerrange 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=0conductance for a linear chain with a (diagonal)
next nearest neighbor interaction.
Authors: K. Louis, C. Gros
Journalref.: Physical Review B 68, 184424 (2003).
Recently, the stochastic series expansion (SSE) has been proposed
as a powerful MCmethod, which allows simulations at low T for
quantumspin systems. We show that the SSE allows to compute the
magnetic conductance for various onedimensional spin systems
without further approximations. We consider various modifications
of the anisotropic Heisenberg chain. We recover the KaneFisher
scaling for one impurity in a Luttingerliquid and study the
influence of noninteracting leads for the conductance of an
interacting system.
Authors: J. Jensen, P. Lemmens, C. Gros
Journalref.: Europhys. Lett. 64, 689 (2003).
Raman lightscattering experiments in the antiferromagnetic
phase of the
Cu_{2}Te_{2}O_{5}(Br_{1x}
Cl_{x})_{2}
compounds are analyzed in terms of a dimerized spin model for the
tetrahedral Cuclusters. 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 DzyaloshinskiiMoriya 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 spin1/2 order perpendicular to each other, when the composition x
is larger than about 0.25.
Authors: K. Pozgajcic, C. Gros
Journalref.: Physical Review B 68, 085106 (2003).
We study the ionic Hubbard model at temperature T=0
within the meanfield approximation and show that the
charge gap does not close completely at the ionicband insulator
to antiferromagnetic insulator transition, contrary to previous expectations.
Furthermore, we find a new intermediate phase for onsite repulsions
U>U_{c} for different lattices and calculate the phase diagram
for the ionic Hubbard model with alternating U,
corresponding to a CuO lattice.
Authors: R. Valentí, T. SahaDasgupta, C. Gros, H. Rosner
Journalref.: Physical Review B 67, 245110 (2003).
Motivated by recent discussion on possible quantum
critical behavior in the coupled Cutetrahedra system
Cu_{2}Te_{2}O_{5}Br_{2},
we present a comparative ab initio study of the electronic
properties of
Cu_{2}Te_{2}O_{5}Br_{2},
and the isostructural
Cu_{2}Te_{2}O_{5}Cl_{2},
A detailed investigation of the coppercopper interaction pathes
reveals that the halogenions play an important
role in the intertetrahedral couplings via X_{4}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_4rings.
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
Journalref.: Physical Review B 67, 174405 (2003).
We present a comprehensive study of the coupled tetrahedracompound
Cu_{2}Te_{2}O_{5}Br_{2},
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 tetrahedralsinglet
sets the energy scale for the magnetic ordering temperature
T_{N}. This energy is determined experimentally.
The ordering temperature T_{N} has an
inverselog dependence on the coupling parameters near
quantum criticality.
Authors: K. Louis, C. Gros
Journalref.: Physical Review B 67, 224410 (2003).
A current of magnetic moments will flow in the
spin1/2 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 Heisenbergmodel
in one dimension. We setup 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
onedimensional Heisenberg compound Sr_{2}Cu_{O}.
Authors: P. Lemmens, G. Güntherodt, C. Gros
Journalref.: Physics Reports, 375 , 1103 (2003).
[100+]
We review recent progress in magnetic light scattering
in one and twodimensional quantum spin systems.
We give an overview of lowdimensional transitionmetal
oxides of current interest, such as spinPeierls,
spindimer, 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.
Authors: C. Gros, P. Lemmens, K.Y. Choi, G. Güntherodt,
M. Baenitz, H.H. Otto
Journalref.: Europhys. Lett. 60, 276 (2002).
The study of quantum phase transitions, which are zerotemperature phase
transitions between distinct states of matter, is of current interest
in research since it allows for a description of
lowtemperature properties based on universal relations.
Here we show that the crystal green dioptase
Cu_{6}Si_{6}O_{18} . 6H_{2}O,
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.
Authors: F. Capraro and C. Gros
Journalref.: Euro. Phys. J. B 29, 35 (2002).
Using the density matrix renormalization group technique, we evaluate the
lowenergy spectrum (ground state and first excited states) of the anisotropic
antiferromagnetic spinonehalf 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 spingap. We find that only a
staggered transversal field opens a substantial gap.
Authors: J.V. Alvarez and Claudius Gros
Journalref.: Physical Review B 66, 094403 (2002).
We discuss zerofrequency transport properties of various spin1/2 chains. We
show, that a careful analysis of Quantum MonteCarlo (QMC) data on the
imaginary axis allows to distinguish between intrinsic ballistic and diffusive
transport. We determine the Drude weight, currentrelaxation lifetime and the
meanfree path for integrable and a nonintegrable quantumspin chain. We
discuss, in addition, some phenomenological relations between various
transportcoefficients and thermal response functions.
Authors: R. Valentí, T. SahaDasgupta and C. Gros
Journalref.: Physical Review B 66 , 054426 (2002).
We investigate by means of ab initio electronic structure
analysis and Quantum Monte Carlo calculations the scenario
where longerranged magnetic interactions dominate over
shortranged interactions in the physical description of
compounds. This question is discussed, in particular, for the
case of CaCuGe_{2}O_{6}, which shows a spinsinglet
behavior induced by third nearest neighbor copper pairs.
Authors: J.V. Alvarez and C. Gros
Journalref.: 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 nondiverging prefactor kappa^{(th)}(T)
peaks is in general substantially lower than the temperature at which
the corresponding specific heat c_{V}(T) is maximal. We show that this effect has
farreaching consequences for the magnetic meanfree path lambda extracted by
analyzing recent experiments with the microscopic theory results.
Authors: K. Hamacher, C. Gros and W. Wenzel
Journalref.: Physical Review Letters 88, 217203 (2002).
(selected for publication in the
Virtual Journal of Nanoscale Science & Technology)
Using the nextnearest neighbor (zigzag) Hubbard chain as an one
dimemensional model, we investigate the influence of interactions on
the position of the Fermi wavevectors with the densitymatrix
renormalizationgroup technique (DMRG). For suitable choices of the
hopping parameters we observe that electronelectron 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.
Authors: J.V. Alvarez and C. Gros
Journalref.: Physical Review Letters 88, 077203 (2002).
A technique to determine accurately transport properties
of integrable and nonintegrable quantumspin chains at
finite temperatures by Quantum MonteCarlo 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 nonintegrable system with longerranged
interactions is presented. We estimate, in addition,
the effect of the nonintegrability on the transport
properties and compare with recent experiments on
onedimensional quantumspin chains.
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
Journalref.: Physical Review Letters 87, 227201 (2001).
Thermodynamic experiments as well as Raman scattering have been
used to study the magnetic instabilities in the spintetrahedra
systems
Cu_{2}Te_{2}O_{5}X_{2},
X=Cl and Br. While the phase
transition observed in the Cl system at T_{o}=18.2 K
is consistent with 3D AF ordering, the phase transition at
T_{o}=11.3 K in the Br system has several unusual features.
We propose an explanation in terms of weakly coupled tetrahedra
with a singlettriplet gap and low lying singlets.
Authors: K. Louis, J.V. Alvarez and C. Gros
Journalref.: Physical Review B 64 , 113 106 (2001).
We derive the oneloop renormalization equations for the shift in the
Fermiwavevectors for onedimensional interacting models with four Fermipoints
(two left and two right movers) and two Fermi velocities v_1 and v_2. We find
the shift to be proportional to (v_1v_2)U^2, where U is the HubbardU. Our
results apply to the Hubbard ladder and to the t_1t_2 Hubbard model. The
Fermisea with fewer particles tends to empty. The stability of a saddle point
due to shifts of the Fermienergy and the shift of the Fermiwavevector at the
MottHubbard transition are discussed.
Authors: R. Valentí, T. SahaDasgupta, J.V. Alvarez, K. Pozgajcic
and C. Gros
Journalref.: Physical Review Letters 86, 5381 (2001).
We determine the electronic structure of γLiV_{2}O_{5}, which has two
inequivalent vanadium ions, V(1) and V(2), via densityfunctional 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 onedimensional spin1/2 Heisenberg
antiferromagnet and give estimates of the basic hopping matrix elements.
Comparison with the most studied αNaV_{2}O_{5} is presented.
Authors: R. Valentí, C. Gros and W. Brenig
Journalref.: Physical Review B 62, 14 164 (2000).
We present a unified account of magnetic exchange and Raman scattering in the
quasionedimensional transitionmetal oxide NaV_{2}O_{5}.
Based on a
clustermodel approach explicit expressions for the exchange integral and the
Ramanoperator are given. It is demonstrated that a combination of the
electronicstructure and the DzyaloshinskiiMoriya interaction, allowed by
symmetry in this material, are responsible for the finite Raman crosssection
giving rise to both, one and twomagnon scattering amplitudes.
Authors: C. Gros, R. Valentí, J. V. Alvarez,
K. Hamacher and W. Wenzel
Journalref.: Physical Review B 62, R14 617 (2000).
Recent experimental evidence suggest the existence of three distinct
Vvalence states (V^{+4}, V^{+4.5} and V^{+5})
in the lowtemperature phase of
NaV_{2}O_{5} in apparent discrepancy with the
observed spingap. We investigate a
novel spin cluster model, consisting of weakly coupled, frustrated fourspin
clusters aligned along the crystallographic baxis 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 clusteroperator theory. We find a spingap 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 lowtemperature regime of
NaV_{2}O_{5}.
Authors: Debanand Sa and C. Gros
Journalref.: Euro. Phys. J. B 18, 421 (2000).
An effective intra and interladder chargespin hamiltonian
for the quarterfilled ladder
compound NaV_{2}O_{5} has been derived
by using the standard canonical transformation method.
In the derivation, it is clear that a finite intersite Coulomb repulsion is
needed to get a meaningful result otherwise the
perturbation becomes illdefined. 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 intraladder exchange may
become ferromagnetic for the case of zigzag charge ordering
in a purely electronic model. We estimate the magnitude of
the effective interrung Coulomb repulsion in a ladder and find it to
be about oneorder of magnitude too small in
order to stabilize chargeordering.
Authors: J.V. Alvarez and C. Gros
Journalref.: Euro. Phys. J. B 15, 641 (2000).
We present an efficient way to compute diagonal and offdiagonal
npoint correlation functions for quantum spinsystems 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.
Authors: Debanand Sa, R. Valentí and C. Gros
Journalref.: Euro. Phys. J. B 14, 301 (2000).
We develop a generalized GinzburgLandau 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 hightemperature phase. The nonzero components of the SHG
susceptibility in the ordered phase are derived from the symmetries of the
susceptibility tensor in the hightemperature 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
Cr_{2}O_{3}
as well as for the ferroelectromagnet YMnO_{3}.
Authors: R. Werner, C. Gros and M. Braden
Journalref.: 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 Peierlsactive phonon
modes to the spin chains of CuGeO_{3}.
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 spinphonon coupling. The caxis anomaly of the thermal expansion is
explained. We give the values of the coupling constants in an effective
onedimensional Hamiltonian.
Authors: C. Gros and R. Valentí
Journalref.: Physical Review Letters 82, 976 (1999).
We consider the effects of charge ordering in
NaV_{2}O_{5} (below T_{SP}) 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 'zigzag' and 'inline' patterns of charge ordering. Only the zigzag
ordering is consistent with the experimental results regarding (i) the unusual
intensity modulation observed in magnetic neutron scattering, (ii) the
reduction in the intraladder exchange constant below T_{SP},
and (iii) the
magnon dispersion along a. We estimate the interladder exchange constant to be
1.01meV=11.7K for T>T_{SP}.
Authors: R. Valentí, C. Gros and V. N. Muthukumar
Journalref.: 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 (nontime invariant) effects:
A boundarycondition 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.
Authors: C. Gros and W. Wenzel
Journalref.: Euro. Phys. J. B 8, 569 (1999).
An iterative procedure for the explicit construction of the nontrivial
subspace of all symmetryadapted configurations with nonzero weight
in the groundstate of the
infinitedimensional Hubbard model is developed on the basis of a
symmetrized representation of the transition operators on a sequence of
BetheLattices of finite depth. The relation ship between
these operators and the well known mapping of the infinitedimensional
Hubbard model onto an effective impurity problem coupled to a
(selfconsistent) bath on noninteracting electrons is given.
As an application we calculate the properties of various Hubbard stars
and give estimates for the halffilled Hubbard model with up to 0.1% accuracy.
Authors: R. Werner and C. Gros
Journalref.: Physical Review B 58, R14 677 (1998).
We reconsider the Cross and Fischer approach to spinPeierls transitions. We
show that a soft phonon occurs only if Omega_0<2.2 T_{SP}.
For CuGeO_{3} this
condition is not fulfilled and the calculated temperature dependence of the
Peierlsactive phonon modes is in excellent agreement with experiment. A
central peak of a width ~0.2 meV is predicted at T_{SP}.
Good agreement is found
between theory and experiment for the pretransitional Peierlsfluctuations.
Finally, we consider the problem of quantum criticality in
CuGeO_{3}.
Authors: H. Smolinski, C. Gros, W. Weber, U. Peuchert,
G. Roth, M. Weiden, C. Geibel
Journalref.: Physical Review Letters 80, 5164 (1998).
[100+]
A new Xray diffraction study of the onedimensional spinPeierls
compound αNaV_{2}O_{5}
reveals a centrosymmetric (Pmmn) crystal
structure with one type of V site, contrary to the previously
postulated noncentrosymmetric P2_1mn structure with two types of
V sites (V^{+4} and V^{+5}).
Density functional calculations indicate
that NaV_{2}O_{5}
is a quarterfilled ladder compound with the spins
carried by VOV molecular orbitals on the rungs of the ladder. Estimates
of the chargetransfer gap and the exchange coupling agree well
with experiment and explain the insulating behavior of
NaV_{2}O_{5} and its magnetic properties.
Authors: R. Werner and C. Gros
Journalref.: Physical Review B 57, 2897 (1998).
We present a theory for the spinPeierls transition in
CuGeO_{3}.
We map the elementary excitations of the dimerized chain (solitons)
on an effective Ising model. Interchain coupling (or phonons) then
introduce a linear binding potential between a pair of soliton and
antisoliton, leading to a finite transition temperature. We evaluate,
as a function of temperature, the order parameter, the singlettriplet
gap, the specific heat, and the susceptibility and compare with
experimental data on CuGeO_{3}. We find that
CuGeO_{3} is close to a
firstorder 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 antisolitons
in dimerized spin chains.
Authors: P. Lemmens, M. Fischer, G. Güntherodt, C. Gros,
P. G. J. van Dongen, M. Weiden, W. Richter,
C. Geibel, F. Steglich
Journalref.: Physical Review B 55, 15076 (1997).
The effect of inchain and offchain substitutions on 1D spin
fluctuations in the spinPeierls compound
CuGeO_{3} has been studied
using Raman scattering in order to understand the interplay
between defect induced states, enhanced spinspin correlations and
the ground state of low dimensional systems. Inchain and offchain
substitutions quench the spinPeierls state and induce
3D antiferromagnetic order at T≤ 5 K. Consequently a
suppression of a 1D gapinduced mode as well as a constant intensity
of a spinon continuum are observed at low temperatures. A 3D
twomagnon density of states now gradually extends to higher
temperatures T≤ 60K compared with pure CuGeO_{3}. This effect
is more pronounced in the case of offchain 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 spinspin 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.
Authors: C. Gros, W. Wenzel, A. Fledderjohann, P. Lemmens, M. Fischer,
G. Güntherodt, M. Weiden, C. Geibel, F. Steglich
Journalref.: Physical Review B 55, 15048 (1997).
In a magnetic substance the gap in the Raman spectrum, Delta_{R},
is approximatively twice the value of the neutron
scattering gap, Delta_{S}, if the the magnetic excitations (magnons)
are only weakly interacting.
But for CuGeO_{3} the experimentally observed ratio
Delta_{R}/Delta_{S}
is approximatively 1.491.78, indicating
attractive magnonmagnon interactions in the quasi1D SpinPeierls
compound CuGe_{3}.
We present numerical estimates for Delta_{R/}Delta_{S}
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 twomagnon bound states in the
SpinPeierls phase. We discuss in detail the numerical method used,
the dependence of the results on the model parameters and
a novel matrixelement effect due to the dimerization of the
Ramanoperator in the SpinPeierls phase.
Authors: V.N. Muthukumar, C. Gros, R. Valentí, M. Weiden,
C. Geibel, F. Steglich, P. Lemmens, M. Fischer, G. Güntherodt
Journalref.: Physical Review B, 55, 5944 (1997).
We present a mean field solution of the antiferromagnetic
Heisenberg chain with nearest (J_{1}) and
next to nearest neighbor (J_{2}) interactions.
This solution provides a way to estimate the effects of
frustration. We calculate the temperaturedependent spinwave velocity,
v_{s}(T) and discuss the possibility to determine the
magnitude of frustration J_{2}/J_{1} present in
quasi1D compounds from measurements of v_{s}(T).
We compute the thermodynamic susceptibility at finite temperatures and
compare it with the observed susceptibility of the spinPeierls compound
CuGeO_{3}. We also use the method to study the twomagnon Raman
continuum observed in CuGeO_{3} above the spinPeierls transition.
Authors: A. Fledderjohann, C. Gros
Journalref.: Europhys. Lett. 37, 189 (1997).
We study numerically the dimerized Heisenberg model with frustration
appropriate for the quasi1D spinPeierls compound CuGeO_{3}.
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 spinwave velocity v_{s} is renormalized by the n.n.n.
frustration term α as v_{s}=pi/2 J(1b alpha),
with b~1.12
Authors: V.N. Muthukumar, C. Gros, W. Wenzel, R. Valentí,
P. Lemmens, B. Eisener, G. Güntherodt, M. Weiden, C. Geibel, F. Steglich
Journalref.: Physical Review B, 54, R9635 (1996).
We present experimental data for the Raman intensity in the spinPeierls
compound CuGeO_{3} and theoretical
calculations from a onedimensional 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 1Dtheory in the homogeneous
phase and evidence for a novel dimerization of the Raman operator
in the spinPeierls state. Finally we present
evidence for a coupling between the interchain exchange,
the spinPeierls order parameter and the magnetic excitations along the chains.
Authors: J. Richter, A. Voigt, S.E. Krüger, C. Gros
Journalref.: 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=J_{1} ∑_{i=1}^{N}
S_{0}⋅S_{i} +
J_{2} H_{R}{S_{i}}, with
J_{1}, J_{2} ≥ 0 and i=1,...,N.
In extension to a previous publication we consider a more general
H_{R}{S_{i}}
describing the properties of the spins surrounding the central
spin S_{0}. The Heisenberg star may be considered as an essential
structure element of a lattice with frustration (namely a spin
embedded in a magnetic matrix H_{R}) or, alternatively,
as a magnetic system H_{R} with a perturbation by an extra spin.
We present some general features of the eigenvalues,
the eigenfunctions as well as the spin correlation
⟨S_{0}⋅S_{i}⟩ of the model. For
H_{R} being a linear chain, a square lattice or a LiebMattis type system
we present the ground state properties of the model in dependence on the
frustration parameter α=J_{2}/J_{1}.
Furthermore the thermodynamic properties are calculated for
H_{R} being a LiebMattis antiferromagnet.
Authors: C. Gros
Journalref.: 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 finitesize corrections can be be reduced
systematically by a grandcanonical 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 groundstate properties of the 2D Hubbard model
and an evaluation of the specific heat for the 1D and 2D Hubbard model.
Authors: V.N. Muthukumar, R. Valentí, C. Gros
Journalref.: Physical Review B, 54, 433 (1996).
A microscopic model of nonreciprocal optical effects in antiferromagnets
is developed by considering the case of Cr_{2}O_{3}
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 spinorbit 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 nonreciprocal
second harmonic generation and gyrotropic birefringence.
Authors: V.N. Muthukumar, R. Valentí, C. Gros
Journalref.: 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 nonreciprocal
optical effects in centrosymmetric Cr_{2}O_{3}. It is
shown that light can couple directly to the antiferromagnetic
order parameter. This coupling is mediated by the
spinorbit 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 Cr^{3+} ion.
Authors: C. Gros, W. Wenzel, J. Richter
Journalref.: Europhys. Lett. 32, 747 (1995).
We present a spin1/2 bilayer model for the quantum orderdisorder
transition which (i) can be solved by meanfield
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,
groundstate energy, order parameter and for the gap.
We show that the critical coupling decreases linearly with frustration.
Authors: C. Gros, R. Valentí
Journalref.: Journal of Low Temperature Physics 99, 509 (1995).
We consider projected wave functions for the twodimensional tJ model.
For various wave functions, including correlated
Fermiliquid and Luttingertype wave functions,
we present the static chargecharge and spinspin structure factors.
Comparisons with recent results from a hightemperature expansion
by Putikka et al. indicates spincharge separation at small length scales.
Authors: C. Gros
Journalref.: Physical Review B 50, 7295 (1994).
We consider the Hubbard model on the infinitedimensional
Bethe lattice and construct a systematic series of selfconsistent
approximations to the oneparticle Green's function,
G^{(n)}(omega), n=2,3,... .
The first n1 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 HubbardIII approximation.
We present analytic and numerical results for the MottHubbard transition at
half filling for n=2,3,4.
Authors: J. Richter, S.E. Krüger, A. Voigt, C. Gros
Journalref.: Europhysics Letters 28, 363 (1994).
We present a class of exactly solvable quantum spin models
which consist of two Heisenbergsubsystems coupled via a
longrange LiebMattis 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
LiebMattis antiferromagnet for the subsystemHamiltonians and present
(i) the complete groundstate phase diagram
and (ii) the full thermodynamic phase diagram.
We find a novel phase which exhibits order from disorder phenomena.
Authors: C. Gros, W. Wenzel, R. Valentí, G. Hülsenbeck, J. Stolze
Journalref.: Europhysics Letters 27, 299 (1994).
In view of a recent controversy we investigate the MottHubbard
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 Hubbardlike cluster.
ii) We evaluate the selfenergy numerically for
n=0,1,2 and compare with a series of selfconsistent equationofmotion
solutions.
iii) We find the gap to open continously at the critical
U_{c}~2.5t^{*}.
iv) A lowenergy theory for the MottHubbard transition is developed
and relations between critical exponents are presented.
Authors: C. Gros, R. Valentí
Journalref.: Ann. Phys. 3, 460 (1994).
We calculate Fermisurface properties of the Cuprate superconductors
within the threeband Hubbard model, using a cluster expansion for the
proper selfenergy. The Fermisurface topology is in agreement with
angularresolved photoemission data for dopings ~20%. We discuss
possible violations of the Luttinger sumrule for smaller dopings
and the role of vanHove singularities in the density of states
of the ZhangRice singlets. We calculate the shift in the
chemical potential upon doping and find quantitative agreement with
recent experiments.
Authors: C. Gros, R. Valentí
Journalref.: Physical Review B 48, 418 (1993).
[100+]
The selfenergy of a translational invariant system of
interacting fermions may be expanded in diagrams contributing to the
selfenergy of finite clusters with open boundary conditions.
The exact solution of small clusters might therefore be used to
construct a systematic approximation to the selfenergy
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 oneband Hubbard Hamiltonian and apply it
to the threeband 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 ZhangRice singlets separated from the
band of triplet states by a manybody gap,
and (iii) creation of density of states
above the top of the oxygen band upon doping.
Authors: C. Gros, R. Valentí
Journalref.: Mod. Phys. Lett. 7, 3 (1993).
We study a variational formulation of the Luttingerliquid concept
in two dimensions. We show that a Luttingerliquid wavefunction
with an algebraic singularity at the Fermiedge is given by
a JastrowGutzwiller 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 Luttingerliquid state is energetically favoured
by the proejected kinetic energy in the context of the 2D
tJ model. We study and find coexistence of dwave
superconductivity and Luttingerliquid behaviour in twodimensional
projected wavefunctions. We then argue that generally, any twodimensional
dwave superconductor should be unstable against Luttingerliquid
type correlations along the (quasi1D) nodes of the dwave order
parameter, at temperatures small compared to the gap.
Authors: R. Valentí, C. Gros
Journalref.: Z. Phys. B 90, 161 (1993).
We study the (D+1)band Hubbard model on generalized
Ddimensional perovskite structures. We show that in the
limit of high dimensions the possible scaling behaviour is
uniquely determined via the bandstructure and that the model
without direct oxygenoxygen hopping necessarily scales to the
cluster limit. A 1/dimension expansion the leads to a tJ like
Hamiltonian and the ZhangRice analysis becomes rigorous.
The large dimension fixed point, in general, still remains
the cluster model even when a hopping term between the
n.n. oxygensites is included. Only for a unique ratio
of the oxygen onsite energies to the oxygenoxygen hopping
amplitude is a new fixed point possible, corresponding to a
heavyFermion Hamiltonian.
Authors: R. Valentí, C. Gros
Journalref.: 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 tJ model. For this, we
generalize to 2D a LuttingerJastrowGutzwillertype wave
function introduced by Hellberg and Mele for the 1D tJ 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 shortrange correlations only. In addition, we
provide rigorous lower bounds to the transition to the
fully phase separated state at larger ratios J/t.
Authors: C. Gros
Journalref.: 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 grandcanonical 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
Mottisulating behaviour for the momentum distribution function
at half filling, both in 1D and 2D.
Off halffilling the shape of the canonical Fermi surface
is strongly distored in 2D with respect to the grandcanonical
Fermisurface. The shape of the grandcanonical Fermisurface
obtained by this finitesize techinque reduces in the weakcoupling
limit exactly to that of the infinitelattice Fermisea.
Authors: R. Valentí, C. Gros, P.J. Hirschfeld, W. Stephan
Journalref.: Physical Review B 44, 13203 (1991).
We show that upper and lower bounds on the groundstate energy of
models describing correlated Fermi systems may be combined
to produce bounds on the groundstate magnetization
and chemical potential. Such bounds are obtainable through standard
variational techniques and through recently developed
methods involving exact diagonalization of finitesize 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 1n.
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 MottHubbard gap. Nevertheless, the tendency
towards largeU 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.
Authors: J. Richer, C. Gros, W. Weber
Journalref.: Physical Review B 44, 906 (1991).
We examine the J1J2, spin1/2 Heisenberg model
on a square lattice with 16 and 20 sites. We evaluate the groundstate
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
columnwiseordered singlets.
We find both types of correlations to be viable candidates
for the groundstate correlations in the thermodynamic limit.
Authors: M.D. Johnson, C. Gros
Journalref.: Physical Review B 43, 11207 (1991).
We present in detail a Green'sfunction approach for studying
chargedspin 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 realspace randomphaseapproximation 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 tJ model by a
selfretracingpath 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 tightbinding 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.
Authors: C. Gros, S.M. Girvin, G.S. Canright, M. D. Johnson
Journalref.: Physical Review B 43, 5883 (1991).
e examine the validity of vectormeanfield 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 hardcore bosons as fermions plus attached flux
tubes. We use a variational Monte Carlo method to test the
variational properties of the meanfield groundstate 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 vectormeanfield theory. We show that the renormalization
coefficients can be understood by an analysis of the
phase fluctuations (whose magnitude we estimate) of the longrange 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 manybody wave function. In addition,
a remarkable variational theorem is discovered for the groundstate wave
function of the VMFT.
Authors: G.J. Chen, R.J. Joynt, F.C. Zhang, C. Gros
Journalref.: 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
twodimensional tJ model at various hole densities.
We searched a parameter space which includes dwave and swave
superconductivity and spindensitywave ordering as well as the
projected Fermiliquid state. Coexistence of different orderings,
such as the s+id state and dwave spindensitywave state,
were found to be stable states. We find a phase diagram in the
densityt/J plane which has coexistence of antiferromagnetism
and superconductivity at very low hole concentrations and
superconductivity up to rather high values of densityabout 40%.
At intermediate concentrations, the timereversal
symmetrybreaking s+id state is found.
Authors: C. Gros
Journalref.: 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 twodimensional 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.
Authors: C. Gros, M.D. Johnson
Journalref.: Physica B 165 & 166, 985 (1990).
We present an exact mapping of the thermodynamical properties of the
tJ 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 tJ model
to a particlehole symmetric form. Identifying a symmetrized combination
of a hole an a doublyoccupied site with the charge carrier, we
prove that the thermodynamical properties of original and the
generalzed tJ model are identical.
Authors: C. Gros, A.H. MacDonald
Journalref.: 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
Authors: C. Gros, M.D. Johnson
Journalref.: Physical Review B 40, 9423 (1989).
We present a new Green'sfunction 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 tJ model the
spin and hole Green's functions are treated on an equal footing.
In the Ising limit, the BrinkmanRice 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.
Authors: C. Gros
Journalref.: 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 onsite 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 twodimensional 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 MonteCarlo 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
CuO superconductors. In particular, we examine the question
wether a quantum spin liquid is intrisically superconducting
or not.
Authors: C. Gros
Journalref.: 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
resonatingvalencebond (RVB) wave function for the Hubbard model on the
square lattice. For the halffilled case we find that the dwave RVB state and
the antiferromagnetic ordered state have the same energy within numerical
accuracy. At 10% doping we find dwave superconductivity, consistent with
previous studies. We show that the superconducting order parameter is
proportional to the number of holes, for small hole concentrations.
Authors: F.C. Zhang, C. Gros, T.M. Rice, H. Shiba
Journalref.: 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 spinspin 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 halffilling 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
dwave RVB state is lowest in energy. This state has a
superconducting orderparameter whose amplitude is linear in
the density of holes.
Authors: C. Gros, R. Joynt, T.M. Rice
Journalref.: 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 halffilled two dimensional band with
interactions given by a Hubbard model with large onsite Coulomb
interactions. We find that the paramagnetic normal state is
unstable to dwave pairing but stable against swave 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.
Authors: C. Gros, R. Joynt, T.M. Rice
Journalref.: Physical Review B 36, 381 (1987).
[100+]
A Monte Carlo method is used to calculate various properties of oneband
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
shortrange spinspin correlations for nearlyhalffilled bands, thus extending
previous work on the halffilled 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 almostlocalized 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 spinspin
coupling. We show that the almostlocalized 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 Fermiliquid ground state.
Authors: D. Baeriswyl, C. Gros, T.M. Rice
Journalref.: Physical Review B 35, 8391 (1987).
We derive partial sum rules for the intraband contributions
to the charge and spin conductivities for almostlocalized
Fermi liquids in a lattice. From this we conclude that
the l=1 Landau parameters have small values.
Authors: K. Seiler, C. Gros, T.M. Rice, K. Ueda, D. Vollhardt
Journalref.: 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 ^{3}He. 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 temperaturedependent
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.

