Nonuniversal Critical Exponents Research Articles

Percolative proton transport in hexagonal boron nitride membranes with edge-functionalization.

Two-dimensional layered materials have been used as matrices to study the structure and dynamics of trapped water and ions. Here, we demonstrate unique features of proton transport in layered hexagonal boron nitride membranes with edge-functionalization subject to hydration. The hydration-independent interlayer spacing indicates the absence of water intercalation between the h-BN sheets. An 18-fold increase in water sorption is observed upon amine functionalization of h-BN sheet edges. A 7-orders of magnitude increase in proton conductivity is observed with less than 5% water loading attributable to edge-conduction channels. The extremely low percolation threshold and non-universal critical exponents (2.90 ≤ α ≤ 4.43), are clear signatures of transport along the functionalized edges. Anomalous thickness dependence of conductivity is observed and its plausible origin is discussed.

Entropy production at criticality in a nonequilibrium Potts model

Understanding nonequilibrium systems and the consequences of irreversibility for the system’s behavior as compared to the equilibrium case, is a fundamental question in statistical physics. Here, we investigate two types of nonequilibrium phase transitions, a second-order and an infinite-order phase transition, in a prototypical q-state vector Potts model which is driven out of equilibrium by coupling the spins to heat baths at two different temperatures. We discuss the behavior of the quantities that are typically considered in the vicinity of (equilibrium) phase transitions, like the specific heat, and moreover investigate the behavior of the entropy production (EP), which directly quantifies the irreversibility of the process. For the second-order phase transition, we show that the universality class remains the same as in equilibrium. Further, the derivative of the EP rate with respect to the temperature diverges with a power-law at the critical point, but displays a non-universal critical exponent, which depends on the temperature difference, i.e., the strength of the driving. For the infinite-order transition, the derivative of the EP exhibits a maximum in the disordered phase, similar to the specific heat. However, in contrast to the specific heat, whose maximum is independent of the strength of the driving, the maximum of the derivative of the EP grows with increasing temperature difference. We also consider entropy fluctuations and find that their skewness increases with the driving strength, in both cases, in the vicinity of the second-order transition, as well as around the infinite-order transition.

Quantum Lifshitz criticality in a frustrated two-dimensional XY model

Antiferromagnetic quantum spin systems can exhibit a transition between collinear and spiral ground states, driven by frustration. Classically this is a smooth crossover and the crossover point is termed a Lifshitz point. Quantum fluctuations change the nature of the transition. In particular it has been argued previously that in the two-dimensional (2D) case a spin liquid (SL) state is developed in the vicinity of the Lifshitz point, termed a Lifshitz SL. In the present work, using a field theory approach, we solve the Lifshitz quantum phase transition problem for the 2D frustrated XY-model. Specifically, we show that, unlike the SU(2) symmetric Lifshitz case, in the XY-model the SL exists only at the critical point. At zero temperature we calculate nonuniversal critical exponents in the Neel and in the spin spiral state and relate these to properties of the SL. We also solve the transition problem at a finite temperature and discuss the role of topological excitations.

Loss of coherence among coupled oscillators: From defect states to phase turbulence.

Synchronization of a large ensemble of identical phase oscillators with a nonlocal kernel and a phase lag parameter α is investigated for the classical Kuramoto-Sakaguchi model on a ring. We demonstrate, for low enough coupling radius r and α below π/2, a phase transition between coherence and phase turbulence via so-called defect states, which arise at the early stage of the transition. The defect states are a novel object resulting from the concatenation of two or more uniformly twisted waves with different wavenumbers. Upon further increase of α, defects lose their stability and give rise to spatiotemporal intermittency, resulting eventually in developed phase turbulence. Simulations close to the thermodynamic limit indicate that this phase transition is characterized by nonuniversal critical exponents.

Non-universal critical exponents in earthquake complex networks

The problem of universality of critical exponents in complex networks is studied based on networks built from seismic data sets. Using two data sets corresponding to Chilean seismicity (northern zone, including the 2014 Mw=8.2 earthquake in Iquique; and central zone without major earthquakes), directed networks for each set are constructed. Connectivity and betweenness centrality distributions are calculated and found to be scale-free, with respective exponents γ and δ. The expected relation between both characteristic exponents, δ>(γ+1)∕2, is verified for both data sets. However, unlike the expectation for certain scale-free analytical complex networks, the value of δ is found to be non-universal.

Anomalous Behavior in an Effective Model of Graphene with Coulomb Interactions

We analyze by exact Renormalization Group (RG) methods the infrared properties of an effective model of graphene, in which two-dimensional massless Dirac fermions propagating with a velocity smaller than the speed of light interact with a three-dimensional quantum electromagnetic field. The fermionic correlation functions are written as series in the running coupling constants, with finite coefficients that admit explicit bounds at all orders. The implementation of Ward Identities in the RG scheme implies that the effective charges tend to a line of fixed points. At small momenta, the quasi-particle weight tends to zero and the effective Fermi velocity tends to a finite value. These limits are approached with a power law behavior characterized by non-universal critical exponents.

Biased percolation on scale-free networks

Biased (degree-dependent) percolation was recently shown to provide strategies for turning robust networks fragile and vice versa. Here, we present more detailed results for biased edge percolation on scale-free networks. We assume a network in which the probability for an edge between nodes i and j to be retained is proportional to (k(i)k(j)(-alpha) with k(i) and k(j) the degrees of the nodes. We discuss two methods of network reconstruction, sequential and simultaneous, and investigate their properties by analytical and numerical means. The system is examined away from the percolation transition, where the size of the giant cluster is obtained, and close to the transition, where nonuniversal critical exponents are extracted using the generating-functions method. The theory is found to agree quite well with simulations. By presenting an extension of the Fortuin-Kasteleyn construction, we find that biased percolation is well-described by the q-->1 limit of the q -state Potts model with inhomogeneous couplings.

Tunnelling and percolation in lattices and the continuum

In a system of conducting particles embedded in a continuous insulating matrix each particle is electrically connected to all others by tunnelling. This scenario is not compatible with percolation connectivity where two particles are either locally connected or disconnected and the global connectivity can be made only via a tortuous path through other particles that are locally connected along this path. It is therefore quite a surprise that a well-defined critical percolation behaviour is observed experimentally in composites where the above scenario prevails. In this paper we explain this paradox by considering the exponential decay of the tunnelling probability, on the one hand, and the very special features of the corresponding distribution function of the interparticle separations, on the other hand. It is concluded that the experimentally observed behaviour is associated with the network of particles that have neighbours within a separation of the order of the tunnelling decay constant. The percolation threshold is determined then by the particle concentration necessary for that network to form while the percolation non-universal critical exponent of the conductivity is associated with the resistor value distribution in that particular network.

Poly(ethylene terephthalate)/expanded graphite conductive composites: Structure, properties, and transport behavior

AbstractPoly(ethylene terephthalate)/expanded graphite conductive composites were prepared by the melt‐blending method. The relationships between the preparation methods, microstructures, and conductivity properties of the composites were studied with scanning electron microscopy and conductivity measurements. The results showed that the composites presented a low percolation threshold and strong anisotropic conductivity. The epoxy resin had a strong intercalation effect on the expanded graphite that led to the easy formation of the conductive network. With classical statistical percolation theory, the conductivity behaviors of the composites were investigated. The results indicated that the nonuniversal critical exponent should be attributed to the anisotropy of conductivity, the tunneling conduction, and the particular structure. In addition, preliminary studies on the crystallization and dynamic mechanical behavior of the composites were performed. © 2008 Wiley Periodicals, Inc. J Appl Polym Sci 2008

Study of electrical properties of piezoresistive pastes and determination of the electrical transport

Thick-film resistors (TFRs) consist of a percolating network of conducting oxide nanoparticles dispersed in an insulating glassy matrix, whose resistive properties are dominated by quantum tunneling across insulating layers separating adjacent conducting grains. Tunneling processes are at the origin of the high sensitivity of the TFRs resistances to applied strains. In this work, we aim to define the electrical transport law between metallic nanoparticles in piezoresistive pastes. We have measured transport and piezoresistive response for different RuO 2-based TFRs as a function of metallic concentration x and RuO 2 grain sizes. The study reveals that the conductivity is shown to vanish as x approaches a critical concentration x c by following a power law with non-universal critical exponents, while the piezoresistivity diverges at the same critical concentration. We argue that non-universality and diverging piezoresistivity have the same origin and arise from the highly fluctuation inter-grain tunneling distances determined by the segregated microstructure of TFRs.

Monte Carlo simulation for square planar model with a small fourfold symmetry-breaking field

Monte Carlo (MC) simulations on the square planar rotator model with small fourfold symmetry-breaking field ${h}_{4}$ points out the existence of a Berezinskii-Kosterlitz-Thouless phase at intermediate temperatures between the low-temperature ferromagnetic phase and the high-temperature paramagnetic phase. This result contrasts with the expectation of a single order-disorer phase transition characterized by nonuniversal critical exponents for any ${h}_{4}$ as suggested by a renormalization group analysis and confirmed by MC simulations for intermediate and large symmetry-breaking fields.

Breaking-symmetry fields in two-dimensional planar models

The planar model in presence of symmetry-breaking fields is studied on square and triangular lattices by Monte Carlo simulation. In the square model a phase transition with non-universal critical exponents is found. In the triangular lattice a Berezinskii–Kosterlitz–Thouless phase is observed over a finite temperature region between the low-temperature ordered phase and the high-temperature paramagnetic phase.

Frustrated magnets in three dimensions: a nonperturbative approach

Frustrated magnets exhibit unusual critical behaviours: they display scaling lawsaccompanied by nonuniversal critical exponents. This suggests that these systemsgenerically undergo very weak first-order phase transitions. Moreover, the differentperturbative approaches used to investigate them are in conflict and fail to correctlyreproduce their behaviour. Using a nonperturbative approach we explain the mismatchbetween the different perturbative approaches and account for the nonuniversal scalingobserved.

Breaking-symmetry fields in two-dimensional planar models

Abstract The planar model in presence of symmetry-breaking fields is studied on square and triangular lattices by Monte Carlo simulation. In the square model a phase transition with non-universal critical exponents is found. In the triangular lattice a Berezinskii–Kosterlitz–Thouless phase is observed over a finite temperature region between the low-temperature ordered phase and the high-temperature paramagnetic phase.

Phase structure of (2+1)-dimensional compact lattice gauge theories and the transition from Mott insulator to fractionalized insulator

Large-scale Monte Carlo simulations are employed to study phase transitions in the three-dimensional compact Abelian Higgs model in adjoint representations of the matter field, labeled by an integer q, for $q=2,3,4,5.$ We also study various limiting cases of the model, such as the ${Z}_{q}$ lattice gauge theory, dual to the three-dimensional (3D) spin model, and the 3D $\mathrm{XY}$ spin model which is dual to the ${Z}_{q}$ lattice gauge theory in the limit $\stackrel{\ensuremath{\rightarrow}}{q}\ensuremath{\infty}.$ In addition, for benchmark purposes, we study the square lattice eight-vertex model, which is exactly solvable and features nonuniversal critical exponents. We have computed the first, second, and third moments of the action to locate the phase transition of the compact Abelian Higgs model in the parameter space $(\ensuremath{\beta},\ensuremath{\kappa}),$ where $\ensuremath{\beta}$ is the coupling constant of the matter term and $\ensuremath{\kappa}$ is the coupling constant of the gauge term. We have found that for $q=3,$ the three-dimensional compact Abelian Higgs model has a phase-transition line ${\ensuremath{\beta}}_{\mathrm{c}}(\ensuremath{\kappa})$ which is first order for $\ensuremath{\kappa}$ below a finite tricritical value ${\ensuremath{\kappa}}_{\mathrm{tri}}$ and second order above. The $\ensuremath{\beta}=\ensuremath{\infty}$ first order phase transition persists for finite $\ensuremath{\beta}$ and joins the second order phase transition at a tricritical point $({\ensuremath{\beta}}_{\mathrm{tri}},{\ensuremath{\kappa}}_{\mathrm{tri}})=(1.23\ifmmode\pm\else\textpm\fi{}0.03,1.73\ifmmode\pm\else\textpm\fi{}0.03).$ For all other integer $q>~2$ we have considered, the entire phase-transition line ${\ensuremath{\beta}}_{c}(\ensuremath{\kappa})$ is critical. We have used finite-size scaling of the second and third moments of the action to extract critical exponents $\ensuremath{\alpha}$ and $\ensuremath{\nu}$ without invoking hyperscaling, for the $\mathrm{XY}$ model, the ${Z}_{2}$ spin and lattice gauge models, as well as the compact Abelian Higgs model for $q=2$ and $q=3.$ In all cases, we have found that for practical system sizes, the third moment gives scaling of superior quality compared to the second moment. We have also computed the exponent ratio for the $q=2$ compact $U(1)$ Higgs model along the critical line, finding a continuously varying ratio $(1+\ensuremath{\alpha})/\ensuremath{\nu},$ as well as continuously varying $\ensuremath{\alpha}$ and $\ensuremath{\nu}$ as $\ensuremath{\kappa}$ is increased from $0.76$ to $\ensuremath{\infty},$ with the Ising universality class $(1+\ensuremath{\alpha})/\ensuremath{\nu}=1.763$ as a limiting case for $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\beta}}\ensuremath{\infty},\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\kappa}}0.761,$ and the $\mathrm{XY}$ universality class $(1+\ensuremath{\alpha})/\ensuremath{\nu}=1.467$ as a limiting case for $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\beta}}0.454,\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\kappa}}\ensuremath{\infty}.$ However, the critical line exhibits a remarkable resilience of ${Z}_{2}$ criticality as $\ensuremath{\beta}$ is reduced along the critical line. Thus, the three-dimensional compact Abelian Higgs model for $q=2$ appears to represent a fixed-line theory defining a new universality class. We relate these results to a recent microscopic description of zero-temperature quantum phase transitions within insulating phases of strongly correlated systems in two spatial dimensions, proposing the above to be the universality class of the zero-temperature quantum phase transition from a Mott-Hubbard insulator to a charge-fractionalized insulator in two spatial dimensions, which thus is that of the 3D Ising model for a considerable range of parameters.

XYfrustrated systems: Continuous exponents in discontinuous phase transitions

XY frustrated magnets exhibit an unsual critical behavior: they display scaling laws accompanied by nonuniversal critical exponents and a negative anomalous dimension. This suggests that they undergo weak first order phase transitions. We show that all perturbative approaches that have been used to investigate XY frustrated magnets fail to reproduce these features. Using a nonperturbative approach based on the concept of effective average action, we are able to account for this nonuniversal scaling and to describe qualitatively and, to some extent, quantitatively the physics of these systems.

Dielectric Properties of Composites of Multi-Walled Carbon Nanotubes in a Resin Matrix

The dielectric properties of a MWNT/epoxy composite have been studied by impedance spectroscopy. The results were very attractive and showed that the addition of a small amount of MWNTs allows increasing the permittivity of the resin by many orders of magnitude. The insulator-to-conductor transition, which occurred when a few wt% of MWNTs was added to the matrix, is related to a percolation phenomenon with non-universal critical exponent. Two types of MWNTs treatment have been compared and the results demonstrate that MWNTs should be used in as-prepared state. The treatment alters their morphology, the aspect ratio can be reduced by a factor of 10, leading to a modification of the shape of the insulator-to-conductor transition curve and a decrease of the global improvement.

Kriticheskie pokazateli trekhmernoi slabo razbavlennoi zamorozhennoi modeli Izinga

We discuss universal and non-universal critical exponents of a three dimensional Ising system in the presence of weak quenched disorder. Both experimental, computational, and theoretical results are reviewed. Special attention is paid to the results obtained by the field theoretical renormalization group approach. Different renormalization schemes are considered putting emphasis on analysis of divergent series obtained.

Fluctuation-induced constraints on the observation of unbinding in a confined complex fluid.

An extensive study of the effect of fluctuations on the unbinding of an interface from a wall in a ternary system is presented. The framework upon which the analysis is based is a linear functional renormalization group scheme of the appropriate effective interface Hamiltonian. The interface model includes position-dependent gradient coefficients, and their presence is shown to be equivalent to modifications of the bare interface potential that are highly relevant in determining the renormalized critical behavior. We analyze the modified interface potential in a mean-field-like way for both bare critical and first-order unbinding transitions in order to highlight the key effects. We further perform a detailed study of the linearized renormalization group equations identifying three fluctuation regimes and recovering earlier predictions for nonuniversal critical exponents. The surface phase diagram changes dramatically under renormalization with, most notably, fluctuation-induced reentrant behavior. We show that in the revised phase diagram the unbound region is limited in extent indicating that the opportunity for observing an unbinding transition in a confined complex fluid is highly restricted.

Exact renormalization-group study of aperiodic Ising quantum chains and directed walks

We consider the Ising model and the directed walk on two-dimensional layered lattices and show that the two problems are inherently related: The zero-field thermodynamical properties of the Ising model are contained in the spectrum of the transfer matrix of the directed walk. The critical properties of the two models are connected to the scaling behavior of the eigenvalue spectrum of the transfer matrix which is studied exactly through renormalization for different self-similar distributions of the couplings. The models show very rich bulk and surface critical behaviors with nonuniversal critical exponents, coupling-dependent anisotropic scaling, first-order surface transition, and stretched exponential critical correlations. It is shown that all the nonuniversal critical exponents obtained for the aperiodic Ising models satisfy scaling relations and can be expressed as functions of varying surface magnetic exponents.