Characterising chaos-hyperchaos transition using correlation dimension

Transition to hyperchaos is uaually studied by computing the spectrum of Lyapunov Exponents (LE). But such a procedure can be employed mainly when the equations governing the dynamical system are known. However, if the information available on the system is only through time series, the method becomes difficult to implement. We show that the transition to hyperchaos is followed by a sudden change in the topological structure of the underlying attractor. Our numerical results indicate that the transition to hyperchaos can be characterized accurately through the computation of correlation dimension (D ₂) from time series. We use two standard time delayed hyperchaotic systems as examples since, for such systems, D ₂ varies smoothly as a function of the time delay τ which can be used as the control parameter.     

Conductance distribution near the Anderson transition

Using a modification of the Shapiro approach, we introduce the two-parameter family of conductance distributions W(g), defined by simple differential equations, which are in the one-to-one correspondence with conductance distributions for quasi-one-dimensional systems of size L ^d–1 × L _z , characterizing by parameters L/ξ and L _z /L (ξ is the correlation length, d is the dimension of space). This family contains the Gaussian and log-normal distributions, typical for the metallic and localized phases. For a certain choice of parameters, we reproduce the results for the cumulants of conductance in the space dimension d = 2 + ? obtained in the framework of the σ-model approach. The universal property of distributions is existence of two asymptotic regimes, log-normal for small g and exponential for large g. In the metallic phase they refer to remote tails, in the critical region they determine practically all distribution, in the localized phase the former asymptotics forces out the latter. A singularity at g = 1, discovered in numerical experiments, is admissible in the framework of their calculational scheme, but related with a deficient definition of conductance. Apart of this singularity, the critical distribution for d = 3 is well described by the present theory. One-parameter scaling for the whole distribution takes place under condition, that two independent parameters characterizing this distribution are functions of the ratio L/ξ.     

Logarithmic Correction for the Susceptibility of the 4-Dimensional Weakly Self-Avoiding Walk: A Renormalisation Group Analysis

We prove that the susceptibility of the continuous-time weakly self-avoiding walk on \({\mathbb{Z}^d}\), in the critical dimension d = 4, has a logarithmic correction to mean-field scaling behaviour as the critical point is approached, with exponent \({\frac{1}{4}}\) for the logarithm. The susceptibility has been well understood previously for dimensions d ≥ 5 using the lace expansion, but the lace expansion does not apply when d = 4. The proof begins by rewriting the walk two-point function as the two-point function of a supersymmetric field theory. The field theory is then analysed via a rigorous renormalisation group method developed in a companion series of papers. By providing a setting where the methods of the companion papers are applied together, the proof also serves as an example of how to assemble the various ingredients of the general renormalisation group method in a coordinated manner.     

Gravitational phase transitions with an exclusion constraint in position space

We discuss the statistical mechanics of a system of self-gravitating particles with anexclusion constraint in position space in a space of dimension d. Theexclusion constraint puts an upper bound on the density of the system and can stabilize itagainst gravitational collapse. We plot the caloric curves giving the temperature as afunction of the energy and investigate the nature of phase transitions as a function ofthe size of the system and of the dimension of space in both microcanonical and canonicalensembles. We consider stable and metastable states and emphasize the importance of thelatter for systems with long-range interactions. For d ≤ 2, there is nophase transition. For d > 2, phase transitions can take place betweena “gaseous” phase unaffected by the exclusion constraint and a “condensed” phase dominatedby this constraint. The condensed configurations have a core-halo structure made of a“rocky core” surrounded by an “atmosphere”, similar to a giant gaseous planet. For largesystems there exist microcanonical and canonical first order phase transitions. Forintermediate systems, only canonical first order phase transitions are present. For smallsystems there is no phase transition at all. As a result, the phase diagram exhibits twocritical points, one in each ensemble. There also exist a region of negative specificheats and a situation of ensemble inequivalence for sufficiently large systems. We showthat a statistical equilibrium state exists for any values of energy and temperature inany dimension of space. This differs from the case of the self-gravitating Fermi gas forwhich there is no statistical equilibrium state at low energies and low temperatures whend ≥ 4. By a proper interpretation of the parameters, our results haveapplication for the chemotaxis of bacterial populations in biology described by ageneralized Keller-Segel model including an exclusion constraint in position space. Theyalso describe colloids at a fluid interface driven by attractive capillary interactionswhen there is an excluded volume around the particles. Connexions with two-dimensionalturbulence are also mentioned.     

Noisy Hegselmann-Krause Systems: Phase Transition and the 2<Emphasis Type="Italic">R</Emphasis>-Conjecture

The classic Hegselmann-Krause (HK) model for opinion dynamics consists of a set of agents on the real line, each one instructed to move, at every time step, to the mass center of the agents within a fixed distance R. In this work, we investigate the effects of noise in the continuous-time version of the model as described by its mean-field Fokker-Planck equation. In the presence of a finite number of agents, the system exhibits a phase transition from order to disorder as the noise increases. We introduce an order parameter to track the phase transition and resolve the corresponding phase diagram. The system undergoes a phase transition for small R but none for larger R. Based on the stability analysis of the mean-field equation, we derive the existence of a forbidden zone for the disordered phase to emerge. We also provide a theoretical explanation for the well-known 2R conjecture, which states that, for a random initial distribution in a fixed interval, the final configuration consists of clusters separated by a distance of roughly 2R. Our theoretical analysis confirms previous simulations and predicts properties of the noisy HK model in higher dimension.     

Periodic Striped Ground States in Ising Models with Competing Interactions

We consider Ising models in two and three dimensions, with short range ferromagnetic and long range, power-law decaying, antiferromagnetic interactions. We let J be the ratio between the strength of the ferromagnetic to antiferromagnetic interactions. The competition between these two kinds of interactions induces the system to form domains of minus spins in a background of plus spins, or vice versa. If the decay exponent p of the long range interaction is larger than d + 1, with d the space dimension, this happens for all values of J smaller than a critical value J_c(p), beyond which the ground state is homogeneous. In this paper, we give a characterization of the infinite volume ground states of the system, for p > 2d and J in a left neighborhood of J_c(p). In particular, we prove that the quasi-one-dimensional states consisting of infinite stripes (d = 2) or slabs (d = 3), all of the same optimal width and orientation, and alternating magnetization, are infinite volume ground states. Our proof is based on localization bounds combined with reflection positivity.     

Metamagnetic effects in epitaxial BaFe<Subscript>1.8</Subscript>Cr<Subscript>0.2</Subscript>As<Subscript>2</Subscript> thin films

Epitaxial BaFe_1.8Cr_0.2As₂ thin films with the tetragonal c-axis perpendicular to the thin film surface were grown on (LaAlO₃)_0.3(Sr₂AlTaO₆)_0.7 (LSAT) single crystalline substrates using pulsed laser deposition (PLD). Resistive measurements indicate the existence of two transitions at temperatures of about 80 K and 40 K. The transition at 80 K is attributed to the structural transition from the high temperature tetragonal phase to the low temperature orthorhombic phase accompanied with the magnetic transition from a paramagnetic to an antiferromagnetic state as known for doped bulk systems. Below T ≈ 40 K the magnetization curves measured perpendicularly to the orthorhombic c-axis in fields up to 9 Tesla show two inflexion points indicating metamagnetic transitions.     

On the Absence of Ferromagnetism in Typical 2D Ferromagnets

We consider the Ising systems in d dimensions with nearest-neighbor ferromagnetic interactions and long-range repulsive (antiferromagnetic) interactions that decay with power s of the distance. The physical context of such models is discussed; primarily this is d = 2 and s = 3 where, at long distances, genuine magnetic interactions between genuine magnetic dipoles are of this form. We prove that when the power of decay lies above d and does not exceed d + 1, then for all temperatures the spontaneous magnetization is zero. In contrast, we also show that for powers exceeding d + 1 (with d ≥ 2) magnetic order can occur.     

The Feynman Propagator on Perturbations of Minkowski Space

The singular values squared of the random matrix product \({Y = {G_{r} G_{r-1}} \ldots G_{1} (G_{0} + A)}\), where each \({G_{j}}\) is a rectangular standard complex Gaussian matrix while A is non-random, are shown to be a determinantal point process with the correlation kernel given by a double contour integral. When all but finitely many eigenvalues of A*A are equal to bN, the kernel is shown to admit a well-defined hard edge scaling, in which case a critical value is established and a phase transition phenomenon is observed. More specifically, the limiting kernel in the subcritical regime of \({0 < b < 1}\) is independent of b, and is in fact the same as that known for the case b =  0 due to Kuijlaars and Zhang. The critical regime of b =  1 allows for a double scaling limit by choosing \({{b = (1 - \tau/\sqrt{N})^{-1}}}\), and for this the critical kernel and outlier phenomenon are established. In the simplest case r =  0, which is closely related to non-intersecting squared Bessel paths, a distribution corresponding to the finite shifted mean LUE is proven to be the scaling limit in the supercritical regime of \({b > 1}\) with two distinct scaling rates. Similar results also hold true for the random matrix product \({T_{r} T_{r-1} \ldots T_{1} (G_{0} + A)}\), with each \({T_{j}}\) being a truncated unitary matrix.     

On the Dimension of the Singular Set of Solutions to the Navier–Stokes Equations

In this paper we prove that if a suitable weak solution u of the Navier–Stokes equations is an element of \({L^w(0,T;L^s(\mathbb{R}^3))}\), where 1 ≤ 2/w + 3/s ≤ 3/2 and 3 < w, s < ∞, then the box-counting dimension of the set of space-time singularities is no greater than max{w, s}(2/w + 3/s ? 1). We also show that if \({\nabla u \in L^w(0,T;L^s(\Omega))}\) with 2 < s ≤ w < ∞, then the Hausdorff dimension of the singular set is bounded by w(2/w + 3/s ? 2). In this way we link continuously the bounds on the dimension of the singular set that follow from the partial regularity theory of Caffarelli, Kohn, &; Nirenberg (Commun. Pure Appl. Math. 35:771–831, 1982) to the regularity conditions of Serrin (Arch. Ration. Mech. Anal. 9:187–191, 1962) and Beirão da Veiga (Chin. Ann. Math. Ser. B 16(4):407–412, 1995).     

Scaling and complex avalanche dynamics in the Abelian sandpile model

We study the two-dimensional Abelian Sandpile Model on a squarelattice of linear size L. We introduce the notion of avalanche’sfine structure and compare the behavior of avalanches and waves oftoppling. We show that according to the degree of complexity inthe fine structure of avalanches, which is a direct consequence ofthe intricate superposition of the boundaries of successive waves,avalanches fall into two different categories. We propose scalingansätz for these avalanche types and verify them numerically.We find that while the first type of avalanches (α) has a simplescaling behavior, the second complex type (β) is characterized by anavalanche-size dependent scaling exponent. In particular, we define an exponent γto characterize the conditional probability distribution functions for these typesof avalanches and show that γ _α = 0.42, while 0.7 ≤ γ _β ≤ 1.0depending on the avalanche size. This distinction provides aframework within which one can understand the lack of aconsistent scaling behavior in this model, and directly addresses thelong-standing puzzle of finite-size scaling in the Abelian sandpile model.     

Finite-size scaling from the self-consistent theory of localization

Accepting the validity of Vollhardt and Wölfle’s self-consistent theory of localization, we derive the finite-size scaling procedure used for studying the critical behavior in the d-dimensional case and based on the consideration of auxiliary quasi-1D systems. The obtained scaling functions for d = 2 and d = 3 are in good agreement with numerical results: it signifies the absence of substantial contradictions with the Vollhardt and Wölfle theory on the level of raw data. The results ν = 1.3–1.6, usually obtained at d = 3 for the critical exponentν of the correlation length, are explained by the fact that dependence L + L ₀ with L ₀ > 0 (L is the transversal size of the system) is interpreted as L ^1/ν with ν > 1. The modified scaling relations are derived for dimensions d ≥ 4; this demonstrates the incorrectness of the conventional treatment of data for d = 4 and d = 5, but establishes the constructive procedure for such a treatment. The consequences for other finite-size scaling variants are discussed.     

Spectral Networks and Fenchel–Nielsen Coordinates

It is known that spectral networks naturally induce certain coordinate systems on moduli spaces of flat SL(K)-connections on surfaces, previously studied by Fock and Goncharov. We give a self-contained account of this story in the case K = 2 and explain how it can be extended to incorporate the complexified Fenchel–Nielsen coordinates. As we review, the key ingredient in the story is a procedure for passing between moduli of flat SL(2)-connections on C (equipped with a little extra structure) and moduli of equivariant GL(1)-connections over a covering \({\Sigma \to C}\); taking holonomies of the equivariant GL(1)-connections then gives the desired coordinate systems on moduli of SL(2)-connections. There are two special types of spectral network, related to ideal triangulations and pants decompositions of C; these two types of network lead to Fock–Goncharov and complexified Fenchel–Nielsen coordinate systems, respectively.     

Temperature Dependence of the Upper Critical Field in Disordered Hubbard Model with Attraction

We study disorder effects upon the temperature behavior of the upper critical magnetic field in an attractive Hubbard model within the generalized DMFT+Σ approach. We consider the wide range of attraction potentials U—from the weak coupling limit, where superconductivity is described by BCS model, up to the strong coupling limit, where superconducting transition is related to Bose–Einstein condensation (BEC) of compact Cooper pairs, formed at temperatures significantly higher than superconducting transition temperature, as well as the wide range of disorder—from weak to strong, when the system is in the vicinity of Anderson transition. The growth of coupling strength leads to the rapid growth of H_c2(T), especially at low temperatures. In BEC limit and in the region of BCS–BEC crossover H_c2(T), dependence becomes practically linear. Disordering also leads to the general growth of H_c2(T). In BCS limit of weak coupling increasing disorder lead both to the growth of the slope of the upper critical field in the vicinity of the transition point and to the increase of H_c2(T) in the low temperature region. In the limit of strong disorder in the vicinity of the Anderson transition localization corrections lead to the additional growth of H_c2(T) at low temperatures, so that the H_c2(T) dependence becomes concave. In BCS–BEC crossover region and in BEC limit disorder only slightly influences the slope of the upper critical field close to T _c . However, in the low temperature region H_c2 (T may significantly grow with disorder in the vicinity of the Anderson transition, where localization corrections notably increase H_c2 (T = 0) also making H_c2(T) dependence concave.     

Absorbing phase transition in energy exchange models

We study energy exchange models with dissipation (λ) and noise (of amplitude σ) and show that in presence of a threshold these models undergo an absorbing phase transition when either dissipation or noise strength or both are varied. Using Monte Carlo simulations we find that the behaviour along the critical line, which separates the active phase from the absorbing one, belongs to directed percolation (DP) universality class. We claim that the conserved version with λ = 1 and σ = 0 also shows a DP transition; the apparent non-DP behaviour observed earlier is an artifact of undershooting in the decay of activity density starting from a random initial condition.     

Finite-size scaling of correlation functions in finite systems

We propose the finite-size scaling of correlation functions in finite systems near their critical points.At a distance r in a ddimensional finite system of size L,the correlation function can be written as the product of|r|~(-(d-2+η))and a finite-size scaling function of the variables r/L and tL~(1/ν),where t=(T-T_c)/T_c,ηis the critical exponent of correlation function,andνis the critical exponent of correlation length.The correlation function only has a sigificant directional dependence when|r|is compariable to L.We then confirm this finite-size scaling by calculating the correlation functions of the two-dimensional Ising model and the bond percolation in two-dimensional lattices using Monte Carlo simulations.We can use the finite-size scaling of the correlation function to determine the critical point and the critical exponentη.     

Behavior of Fermi systems approaching the fermion condensation quantum phase transition from the disordered phase

The behavior of Fermi systems that approach the fermion condensation quantum phase transition (FCQPT) from the disordered phase is considered. We show that the quasiparticle effective mass M* diverges as M* ∝ 1/¦x?x_FC¦, where x is the system density and x_FC is the critical point at which FCQPT occurs. Such behavior is of general form and takes place in both three-dimensional (3D) and two-dimensional (2D) systems. Since the effective mass M* is finite, the system exhibits the Landau Fermi liquid behavior. At ¦x? x_FC¦/x_FC?1, the behavior can be viewed as highly correlated, because the effective mass is large and strongly depends on the density. In the case of electronic systems, the Wiedemann-Franz law is valid and the Kadowaki-Woods ratio is preserved. Beyond the region ¦x–x_FC¦/x_FC?1, the effective mass is approximately constant and the system becomes a conventional Landau Fermi liquid.     

Evolution of spin susceptibility from the Pauli to Curie-Weiss type in strongly correlated Fermi systems

The magnetic properties of strongly correlated Fermi systems are studied within the framework of the fermioncondensation model—phase transition associated with the rearrangement of the Landau quasiparticle distribution, resulting in the appearance of a plateau at T=0 exactly in the Fermi surface of the single-particle excitation spectrum. It is shown that the Curie-Weiss term ～T^?1 appears in the expression for the spin susceptibility χ_ac(T) of the system after the transition point at finite temperatures. The behavior of χ_ac(T, H) as a function of temperature and static magnetic field H in the region where the critical fermion-condensation temperature T _f is close to zero is discussed. The results are compared with the available experimental data.