Griffiths Phases Research Articles

Nuclear magnetic resonance studies in a model transverse field Ising system

The suppression of ferroquadrupolar order in TmVO4 in a magnetic field is well-described by the transverse field Ising model, enabling detailed studies of critical dynamics near the quantum phase transition. We describe nuclear magnetic resonance measurements in pure and Y-doped single crystals. The non-Kramers nature of the ground state doublet leads to a unique form of the hyperfine coupling that exclusively probes the transverse field susceptibility. Our results show that this quantity diverges at the critical field, in contrast to the mean-field prediction. Furthermore, we find evidence for quantum critical fluctuations present near Tm-rich regions in Y-doped crystals at levels beyond which long-range order is suppressed, suggesting the presence of quantum Griffiths phases.

From asynchronous states to Griffiths phases and back: Structural heterogeneity and homeostasis in excitatory-inhibitory networks

Balanced neural networks, in which excitatory and inhibitory inputs compensate each other on average, give rise to a dynamical phase dominated by fluctuations called an asynchronous state, crucial for brain functioning. However, structural disorder, which is inherent to random networks, can hinder such an excitation-inhibition balance. Indeed, structural and synaptic heterogeneities can generate extended regions in phase space akin to critical points, called Griffiths phases, with dynamical features very different from those of asynchronous states. Here we study a simple neural-network model with tunable levels of heterogeneity able to display these two types of dynamical regimes, i.e., asynchronous states and Griffiths phases, putting them together within a single phase diagram. Using this simple model, we are able to emphasize the crucial role played by synaptic plasticity and homeostasis to reestablish balance in intrinsically heterogeneous networks. Overall, we shed light onto how diverse dynamical regimes, each with different functional advantages, can emerge from a given network as a result of self-organizing homeostatic mechanisms. Published by the American Physical Society 2024

"Heterogeneity and Criticality in Neural Networks: A Computational Model of the Griffiths Phase"

This article presents a computational exploration into the dynamics of neural networks, focusing on the manifestation of heterogeneity and critical behavior akin to the Griffiths phase observed in condensed matter physics. Drawing inspiration from the rich tapestry of neural activities within the human neocortex, we develop a simplified model to investigate how networks with diverse propensities for critical behavior can exhibit complex, adaptive functionalities. Utilizing an Erdős–Rényi random graph as the foundation, we introduce a proportion of nodes designated as "critical," possessing a higher likelihood of activation, to simulate the nuanced interplay between different regions of a theoretical brain network. Through iterative simulations, we examine the activation patterns over time, paying close attention to the influence of critical nodes on the overall network dynamics. The model integrates fundamental principles from network theory and neuroscience, offering insights into the potential mechanisms underlying the brain's remarkable ability to process information efficiently, adapt to new stimuli, and maintain robustness in the face of perturbations. Our findings highlight the importance of heterogeneity and criticality in enhancing the computational capabilities of neural networks, providing a conceptual bridge between theoretical physics and cognitive neuroscience. This work contributes to a deeper understanding of the brain's operational principles and opens avenues for further research into the applications of criticality and Griffiths phases in artificial intelligence and neural network design.

Numerical evidence of a universal critical behavior of two-dimensional and three-dimensional random quantum clock and Potts models.

The random quantum q-state clock and Potts models are studied in two and three dimensions. The existence of Griffiths phases is tested in the two-dimensional case with q=6 by sampling the integrated probability distribution of local susceptibilities of the equivalent McCoy-Wu three-dimensional classical models with Monte Carlo simulations. For the random Potts model, numerical evidence of the existence of Griffiths phases is given and the finite-size effects are analyzed. For the clock model, the data also suggest the existence of a Griffiths phase but with much larger finite-size effects. The critical point of the random quantum clock model is then studied with the Strong-Disorder Renormalization Group. Evidence is given that, at strong enough disorder, this critical behavior is governed by the same infinite-disorder fixed point as the Potts model, for all the number of states q considered. At weak disorder, our renormalization group method becomes unstable and does not allow us to make conclusions.

Differences in the critical dynamics underlying the human and fruit-fly connectome

Previous simulation studies on human connectomes suggested, that critical dynamics emerge subcrititcally in the so called Griffiths Phases. %This is the consequence of the strong heterogeneity of the graphs. Now we investigate this on the largest available brain network, the $21.662$ node fruit-fly connectome, using the Kuramoto synchronization model. As this graph is less heterogeneous, lacking modular structure and exhibit high topological dimension, we expect a difference from the previous results. Indeed, the synchronization transition is mean-field like, and the width of the transition region is larger than in random graphs, but much smaller than as for the KKI-18 human connectome. This demonstrates the effect of modular structure and dimension on the dynamics, providing a basis for better understanding the complex critical dynamics of humans.

Criticality and Griffiths phases in random games with quenched disorder.

The perceived risk and reward for a given situation can vary depending on resource availability, accumulated wealth, and other extrinsic factors such as individual backgrounds. Based on this general aspect of everyday life, here we use evolutionary game theory to model a scenario with randomly perturbed payoffs in a prisoner's dilemma game. The perception diversity is modeled by adding a zero-average random noise in the payoff entries and a Monte Carlo simulation is used to obtain the population dynamics. This payoff heterogeneity can promote and maintain cooperation in a competitive scenario where only defectors would survive otherwise. In this work, we give a step further, understanding the role of heterogeneity by investigating the effects of quenched disorder in the critical properties of random games. We observe that payoff fluctuations induce a very slow dynamic, making the cooperation decay behave as power laws with varying exponents, instead of the usual exponential decay after the critical point, showing the emergence of a Griffiths phase. We also find a symmetric Griffiths phase near the defector's extinction point when fluctuations are present, indicating that Griffiths phases may be frequent in evolutionary game dynamics and play a role in the coexistence of different strategies.

Modelling on the very large-scale connectome

In this review, we discuss critical dynamics of simple nonequilibrium models on large connectomes, obtained by diffusion MRI, representing the white matter of the human brain. In the first chapter, we overview graph theoretical and topological analysis of these networks, pointing out that universality allows selecting a representative network, the KKI-18, which has been used for dynamical simulation. The critical and sub-critical behaviour of simple, two- or three-state threshold models is discussed with special emphasis on rare-region effects leading to robust Griffiths phases (GP). Numerical results of synchronization phenomena, studied by the Kuramoto model, are also shown, leading to a continuous analog of the GP, termed frustrated synchronization. The models presented here exhibit dynamical scaling behaviour with exponents in agreement with brain experimental data if local homeostasis is provided.

Heterogeneous excitable systems exhibit Griffiths phases below hybrid phase transitions

In $d > 2$ dimensional, homogeneous threshold models discontinuous transition occur, but the mean-field solution provides $1/t$ power-law activity decay and other power-laws, thus it is called mixed-order or hybrid type. It has recently been shown that the introduction of quenched disorder rounds the discontinuity and second order phase transition and Griffiths phases appear. Here we provide numerical evidence, that even in case of high graph dimensional hierarchical modular networks the Griffiths phase of the $K=2$ threshold model is present below the hybrid phase transition. This is due to the fragmentation of the activity propagation by modules, which are connected via single links. This provides a widespread mechanism in case of threshold type of heterogeneous systems, modeling the brain or epidemics for the occurrence of dynamical criticality in extended Griffiths phase parameter spaces. We investigate this in synthetic modular networks with and without inhibitory links as well as in the presence of refractory states.

Quantum annealing speedup of embedded problems via suppression of Griffiths singularities

Optimal parameter setting for applications problems embedded into hardware graphs is key to practical quantum annealers (QA). Embedding chains typically crop up as harmful Griffiths phases, but can be used as a resource as we show here: to balance out singularities in the logical problem changing its universality class. Smart choice of embedding parameters reduces annealing times for random Ising chain from $O(exp[c\sqrt N])$ to $O(N^2)$. Dramatic reduction in time-to-solution for QA is confirmed by numerics, for which we developed a custom integrator to overcome convergence issues.

Quantum criticality in Ising chains with random hyperuniform couplings

We study quantum phase transitions in transverse-field Ising spin chains in which the couplings are random but hyperuniform, in the sense that their large-scale fluctuations are suppressed. We construct a one-parameter family of disorder models in which long-wavelength fluctuations are increasingly suppressed as a parameter $\alpha$ is tuned. For $\alpha = 0$, one recovers the familiar infinite-randomness critical point. For $0 < \alpha < 1$, we find a line of infinite-randomness critical points with continuously varying critical exponents; however, the Griffiths phases that flank the critical point at $\alpha = 0$ are absent at any $\alpha > 0$. When $\alpha > 1$, randomness is a dangerously irrelevant perturbation at the clean Ising critical point, leading to a state we call the critical Ising insulator. In this state, thermodynamics and equilibrium correlation functions behave as in the clean system. However, all finite-energy excitations are localized, thermal transport vanishes, and autocorrelation functions remain finite in the long-time limit. We characterize this line of hyperuniform critical points using a combination of perturbation theory, renormalization-group methods, and exact diagonalization.

Griffiths physics in an ultracold Bose gas

Coupled $XY$ model systems consisting of three-dimensional (3D) systems with disordered interlayer physics are of significant theoretical interest. We realize a set of coupled quasi-2D layers of $^{87}\mathrm{Rb}$ in the presence of disordered interlayer coupling. This is achieved with our high bandwidth arbitrary optical lattice to obviate restrictions on the dimensionality of disorder with speckle-generated optical fields. We identify phase crossover regions compatible with the existence of a pair of intermediate Griffiths phases between a thermal state and the emergence of bulk 3D superfluidity.

Disorder in Quantum Many-Body Systems

Impurities, defects, and other types of imperfections are ubiquitous in realistic quantum many-body systems and essentially unavoidable in solid state materials. Often, such random disorder is viewed purely negatively as it is believed to prevent interesting new quantum states of matter from forming and to smear out sharp features associated with the phase transitions between them. However, disorder is also responsible for a variety of interesting novel phenomena that do not have clean counterparts. These include Anderson localization of single-particle wave functions, many-body localization in isolated many-body systems, exotic quantum critical points, and glassy ground-state phases. This brief review focuses on two separate but related subtopics in this field. First, we review under what conditions different types of randomness affect the stability of symmetry-broken low-temperature phases in quantum many-body systems and the stability of the corresponding phase transitions. Second, we discuss the fate of quantum phase transitions that are destabilized by disorder as well as the unconventional quantum Griffiths phases that emerge in their vicinity.

Griffiths phases in infinite-dimensional, non-hierarchical modular networks

Griffiths phases (GPs), generated by the heterogeneities on modular networks, have recently been suggested to provide a mechanism, rid of fine parameter tuning, to explain the critical behavior of complex systems. One conjectured requirement for systems with modular structures was that the network of modules must be hierarchically organized and possess finite dimension. We investigate the dynamical behavior of an activity spreading model, evolving on heterogeneous random networks with highly modular structure and organized non-hierarchically. We observe that loosely coupled modules act as effective rare-regions, slowing down the extinction of activation. As a consequence, we find extended control parameter regions with continuously changing dynamical exponents for single network realizations, preserved after finite size analyses, as in a real GP. The avalanche size distributions of spreading events exhibit robust power-law tails. Our findings relax the requirement of hierarchical organization of the modular structure, which can help to rationalize the criticality of modular systems in the framework of GPs.

Extinction phase transitions in a model of ecological and evolutionary dynamics

We study the non-equilibrium phase transition between survival and extinction of spatially extended biological populations using an agent-based model. We especially focus on the effects of global temporal fluctuations of the environmental conditions, i.e., temporal disorder. Using large-scale Monte-Carlo simulations of up to $3\times 10^7$ organisms and $10^5$ generations, we find the extinction transition in time-independent environments to be in the well-known directed percolation universality class. In contrast, temporal disorder leads to a highly unusual extinction transition characterized by logarithmically slow population decay and enormous fluctuations even for large populations. The simulations provide strong evidence for this transition to be of exotic infinite-noise type, as recently predicted by a renormalization group theory. The transition is accompanied by temporal Griffiths phases featuring a power-law dependence of the life time on the population size.

Emergent infinite-randomness fixed points from the extensive random bipartitions of the spin-1 Affleck-Kennedy-Lieb-Tasaki topological state

Quantum entanglement under an extensive bipartition can reveal the critical boundary theory of a topological phase in the parameter space. In this study we demonstrate that the infinite-randomness fixed point for spin-1/2 degrees of freedom can emerge from an extensive random bipartition of the spin-1 Affleck-Kennedy-Lieb-Tasaki chain. The nested entanglement entropy of the ground state of the reduced density matrix exhibits a logarithmic scaling with an effective central charge $\tilde{c} = 0.72 \pm 0.02 \approx \ln 2$. We further discuss, in the language of bulk quantum entanglement, how to understand all phase boundaries and the surrounding Griffiths phases for the antiferromagnetic Heisenberg spin-1 chain with quenched disorder and dimerization.

Temporal disorder does not forbid discontinuous absorbing phase transitions in low-dimensional systems.

Recent papers have shown that spatial (quenched) disorder can suppress discontinuous absorbing phase transitions. Conversely, the scenario for temporal disorder is still unknown. To shed some light in this direction, we investigate its effect in three different two-dimensional models which are known to exhibit discontinuous absorbing phase transitions. The temporal disorder is introduced by allowing the control parameter to be time dependent p→p(t), either varying as a uniform distribution with mean p[over ¯] and variance σ or as a bimodal distribution, fluctuating between a value p and a value p_{l}≪p. In contrast to spatial disorder, our numerical results strongly suggest that such uncorrelated temporal disorder does not forbid the existence of a discontinuous absorbing phase transition. We find that all cases are characterized by behaviors similar to their pure (without disorder) counterparts, including bistability around the coexistence point and common finite-size scaling behavior with the inverse of the system volume, as recently proposed [M. M. de Oliveira et al., Phys. Rev. E 92, 062126 (2015)PLEEE81539-375510.1103/PhysRevE.92.062126]. We also observe that temporal disorder does not induce temporal Griffiths phases around discontinuous phase transitions, at least not for d=2.

Griffiths phases and localization in hierarchical modular networks.

We study variants of hierarchical modular network models suggested by Kaiser and Hilgetag [ Front. in Neuroinform., 4 (2010) 8] to model functional brain connectivity, using extensive simulations and quenched mean-field theory (QMF), focusing on structures with a connection probability that decays exponentially with the level index. Such networks can be embedded in two-dimensional Euclidean space. We explore the dynamic behavior of the contact process (CP) and threshold models on networks of this kind, including hierarchical trees. While in the small-world networks originally proposed to model brain connectivity, the topological heterogeneities are not strong enough to induce deviations from mean-field behavior, we show that a Griffiths phase can emerge under reduced connection probabilities, approaching the percolation threshold. In this case the topological dimension of the networks is finite, and extended regions of bursty, power-law dynamics are observed. Localization in the steady state is also shown via QMF. We investigate the effects of link asymmetry and coupling disorder, and show that localization can occur even in small-world networks with high connectivity in case of link disorder.

Entanglement spectrum of a random partition: Connection with the localization transition

We study the entanglement spectrum of a translationally-invariant lattice system under a random partition, implemented by choosing each site to be in one subsystem with probability $p\in[0, 1]$. We apply this random partitioning to a translationally-invariant (i.e., clean) topological state, and argue on general grounds that the corresponding entanglement spectrum captures the universal behavior about its disorder-driven transition to a trivial localized phase. Specifically, as a function of the partitioning probability $p$, the entanglement Hamiltonian $H_{A}$ must go through a topological phase transition driven by the percolation of a random network of edge-states. As an example, we analytically derive the entanglement Hamiltonian for a one-dimensional topological superconductor under a random partition, and demonstrate that its phase diagram includes transitions between Griffiths phases.

Influence of disorder on the appearance of Griffiths phases in La0.8−x□xCa0.2MnO3 (x=0.15 and 0.2) compounds

The La0.8−x□xCa0.2MnO3 (x=0.15 and 0.2) compounds were synthesized by the solid state reaction method. The X-ray diffraction (XRD) and magnetic measurements were used to study their physical and chemical properties. The structural characterization has revealed that these samples are crystallized in the orthorhombic structure with Pnma space group. Then, the magnetic properties of these compounds are discussed in detail, building on the magnetization and the susceptibility. The temperature dependence of magnetic susceptibility at higher temperature confirms the presence of the Griffiths phase above the Curie temperature which proves the existence of ferromagnetic clusters in the paramagnetic domain.

Localization transition, Lifschitz tails, and rare-region effects in network models.

Effects of heterogeneity in the suspected-infected-susceptible model on networks are investigated using quenched mean-field theory. The emergence of localization is described by the distributions of the inverse participation ratio and compared with the rare-region effects appearing in simulations and in the Lifschitz tails. The latter, in the linear approximation, is related to the spectral density of the Laplacian matrix and to the time dependent order parameter. I show that these approximations indicate correctly Griffiths phases both on regular one-dimensional lattices and on small-world networks exhibiting purely topological disorder. I discuss the localization transition that occurs on scale-free networks at γ=3 degree exponent.