Critical Exponents Research Articles

Positive solutions for nonhomogenous Choquard equations involving doubly critical exponents

This paper is concerned with nonhomogenous Choquard equations involving doubly critical exponents. First, we use a fixed point theorem to prove the existence of positive solutions to the above equation under certain assumptions. Then, on the positive solution of the equation, we get the regularity in a bounded domain with estimating the ‐bounds by Moser's iterative technique.

Out-of-equilibrium scaling of the energy density along the critical relaxational flow after a quench of the temperature.

We study the out-of-equilibrium behavior of statistical systems along critical relaxational flows arising from instantaneous quenches of the temperature T to the critical point T_{c}, starting from equilibrium conditions at time t=0. In the case of soft quenches, i.e., when the initial temperature T is assumed sufficiently close to T_{c} (to keep the system within the critical regime), the critical modes develop an out-of-equilibrium finite-size-scaling (FSS) behavior in terms of the rescaled time variable Θ=t/L^{z}, where t is the time interval after quenching, L is the size of the system, and z is the dynamic exponent associated with the dynamics. However, the realization of this picture is less clear when considering the energy density, whose equilibrium scaling behavior (corresponding to the starting point of the relaxational flow) is generally dominated by a temperature-dependent regular background term or mixing with the identity operator. These issues are investigated by numerical analyses within the three-dimensional lattice N-vector models, for N=3 and 4, which provide examples of critical behaviors with negative values of the specific-heat critical exponent α, implying that also the critical behavior of the specific heat gets hidden by the background term. The results show that, after subtraction of its asymptotic critical value at T_{c}, the energy density develops an asymptotic out-of-equilibrium FSS in terms of Θ as well, whose scaling function appears singular in the small-Θ limit.

Log-Normal Waiting Time Widths Characterize Dynamics

Many astronomical phenomena, including Fast Radio Bursts and Soft Gamma Repeaters, consist of brief distinct aperiodic events. The intervals between these events vary randomly, but there are periods of greater activity, with shorter mean intervals, and of lesser activity, with longer mean intervals. A single dimensionless parameter, the width of a log-normal function fitted to the distribution of waiting times between events, quantifies the variability of the activity. This parameter describes its dynamics in analogy to the critical exponents and universality classes of renormalization group theory. If the distribution of event strengths is a power law, the width of the log-normal fit is independent of the detection threshold and is a robust measure of the dynamics of the phenomenon.

Calorimetric, optical and dielectric measurements on two Schiff's based liquid crystalline materials exhibiting multiple phase transitions.

We report high-resolution calorimetric, optical and dielectric studies on two Schiff's based liquid crystalline materials, 4O.5 and 5O.5, which exhibit multiple phase transitions. The study goes beyond the commonly studied isotropic-to-nematic (I-N) and nematic-to-smectic A (N-SmA) phase transitions and explores higher-order smectic phase transitions. The critical exponent, α values, for different phase transitions has been explored. For the I-N, N-SmA, SmA-SmC, SmC-SmF, SmF-CrG and SmB-CrG transitions, α values close to 0.5 and amplitude ratios A-/A+ ~ 1.6 and D-/D+ ~ 1 indicate a first-order nature. Notably, for the SmA-SmB transition, although α (≈ 0.64) is larger than the tricritical value of 0.5, A-/A+ ~ 1.2 and D-/D+ ~ 1, and the study reports an order parameter critical exponent (β) of 0.26 ± 0.002, which supports a tricritical nature for this transition. Importantly, the ratios of A-/A+ and D-/D+ are found to have similar values across all three-measurement methods, indicating consistency and agreement among these methods.

Quantum tricriticality in a generalized quantum Rabi system

Quantum tricriticality, a unique form of high-order criticality, is expected to exhibit fascinating features including unconventional critical exponents and universal scaling laws. However, a quantum tricritical point (QTCP) is much harder to access, and the corresponding phenomena at tricriticality have rarely been investigated. In this study, we explore a tricritical quantum Rabi model, which incorporates a non-trivial parameter to adjust the coupling ratio between a cavity and a three-level atom. The QTCP emerges at the intersection of first- and second-order superradiant phase transitions according to Landau theory. By using finite-frequency scaling analysis on quantum fluctuations and the average photon number, universal critical exponents differentiate the QTCP from the second-order critical point. Our results indicate that the phase transition at the tricritical point goes beyond the conventional second-order phase transition. Our work explores an interesting direction in the generalization of the well-known Rabi model for the study of higher-order critical points due to its high control and tunability.

Emergence of order from chaos through a continuous phase transition in a turbulent reactive flow system.

As the Reynolds number is increased, a laminar fluid flow becomes turbulent, and the range of time and length scales associated with the flow increases. Yet, in a turbulent reactive flow system, as we increase the Reynolds number, we observe the emergence of a single dominant timescale in the acoustic pressure fluctuations, as indicated by its loss of multifractality. Such emergence of order from chaos is intriguing. We perform experiments in a turbulent reactive flow system consisting of flame, acoustic, and hydrodynamic subsystems interacting nonlinearly. We study the evolution of short-time correlated dynamics between the acoustic field and the flame in the spatiotemporal domain of the system. The order parameter, defined as the fraction of the correlated dynamics, increases gradually from zero to one. We find that the susceptibility of the order parameter, correlation length, and correlation time diverge at a critical point between chaos and order. Our results show that the observed emergence of order from chaos is a continuous phase transition. Moreover, we provide experimental evidence that the critical exponents characterizing this transition fall in the universality class of directed percolation. Our paper demonstrates how a real-world complex, nonequilibrium turbulent reactive flow system exhibits universal behavior near a critical point.

Effect of shape anisotropy on percolation of aligned and overlapping objects on lattices.

We investigate the percolation transition of aligned, overlapping, anisotropic shapes on lattices. Using the recently proposed lattice version of excluded volume theory, we show that shape-anisotropy leads to some intriguing consequences regarding the percolation behavior of anisotropic shapes. We consider a prototypical anisotropic shape-rectangle-on a square lattice and show that, for rectangles of width unity (sticks), the percolation threshold is a monotonically decreasing function of the stick length, whereas, for rectangles of width greater than two, it is a monotonically increasing function. Interestingly, for rectangles of width two, the percolation threshold is independent of its length. We show that this independence of threshold on the length of a side holds for d-dimensional hypercubiods as well as for specific integer values for the lengths of the remaining sides. The limiting case of the length of the rectangles going to infinity shows that the limiting threshold value is finite and depends upon the width of the rectangle. This "continuum" limit with the lattice spacing tending to zero only along a subset of the possible directions in d dimensions results in a "semicontinuum" percolation system. We show that similar results hold for other anisotropic shapes and lattices in different dimensions. The critical properties of the aligned and overlapping rectangles are evaluated using Monte Carlo simulations. We find that the threshold values given by the lattice-excluded volume theory are in good agreement with the simulation results, especially for larger rectangles. We verify the isotropy of the percolation threshold and also compare our results with models where rectangles of mixed orientation are allowed. Our simulation results show that alignment increases the percolation threshold. The calculation of critical exponents places the model in the standard percolation universality class. Our results show that shape anisotropy of the aligned, overlapping percolating units has a marked influence on the percolation properties, especially when a subset of the dimensions of the percolation units is made to diverge.

Considerations on the relaxation time in shear-driven jamming.

We study the jamming transition in a model of elastic particles under shear at zero temperature, with a focus on the relaxation time τ_{1}. This relaxation time is from two-step simulations where the first step is the ordinary shearing simulation and the second step is the relaxation of the energy after stopping the shearing. τ_{1} is determined from the final exponential decay of the energy. Such relaxations are done with many different starting configurations generated by a long shearing simulation in which the shear variable γ slowly increases. We study the correlations of both τ_{1}, determined from the decay, and the pressure, p_{1}, from the starting configurations as a function of the difference in γ. We find that the correlations of p_{1} are longer lived than the ones of τ_{1} and find that the reason for this is that the individual τ_{1} is controlled both by p_{1} of the starting configuration and a random contribution which depends on the relaxation path length-the average distance moved by the particles during the relaxation. We further conclude that it is γ_{τ}, determined from the correlations of τ_{1}, which is the relevant one when the aim is to generate data that may be used for determining the critical exponent that characterizes the jamming transition.

Quantum scaling for the metal–insulator transition in a two-dimensional electron system

The quantum phase transition observed experimentally in two-dimensional (2D) electron systems has been a subject of theoretical and experimental studies for almost 30 years. We suggest Gaussian approximation to the mean-field theory of the second-order phase transition to explain the experimental data. Our approach explains self-consistently the universal value of the critical exponent 3/2 (found after scaling measured resistivities on both sides of the transition as a function of temperature) as the result of the divergence of the correlation length when the electron density approaches the critical value. We also provide numerical evidence for the stretched exponential temperature dependence of the metallic phase’s resistivities in a wide range of temperatures and show that it leads to correct qualitative results. Finally, we interpret the phase diagram on the density-temperature plane exhibiting the quantum critical point, quantum critical trajectory and two crossover lines. Our research presents a theoretical description of the seminal experimental results.

Phase properties of the mean-field Ising model with three-spin interaction

The equilibrium and phase properties of the Ising model with three-spin interaction and an external field are studied within the framework of mean-field approximation. The thermodynamic properties of the model reveals two coexistence curves, signifying two distinct second-order phase transitions, dependent on the domain of the interaction parameter. The critical exponents of the magnetic order parameter are calculated in all directions of the phase space and show their agreement with the mean-field universality class.

Environmentally-friendly calcite scale mitigation: encapsulation of CDs@ MS composite within membranes framework for nanofiltration

Scale formation caused by the precipitation of sparingly soluble salts such as CaCO3 is a major challenge in membrane-based desalination technologies, leading to flux decline and increased energy costs. Surface modification using nanoscale coatings is a promising approach to mitigate membrane fouling. In this work, we demonstrate the antiscalant properties of a MoS2-carbon quantum dot (CDs@MS-NF) composite with a nanoflower-like structure coating for suppressing calcite scale formation. The CDs@ MS-NF coating was synthesized via a solvothermal method and applied to a polyamide reverse osmosis membrane surface using an in-situ growth technique. Characterization revealed uniform coverage of MoS2 nanosheets decorated with CDs@MS-NF on the membrane surface. Static fouling tests simulating calcium sulfate scaling conditions were conducted to evaluate the antifouling performance. The CDs@MS-NF coated membrane exhibited a flux reduction of only 11 %, compared to 34 % for the unmodified membrane after 24 hours. Further analyses showed a calcite nucleation induction time of 5.2 hours and a critical scaling threshold of 1.8 times the solubility for the CDs@ MS-NF coated membrane, indicating significant inhibition of scale formation. We hypothesize that the CDs@ MS-NF nanocomposite layer alters surface charge and wettability while also catalyzing redox reactions to hinder calcite crystal growth. This work demonstrates that multifunctional two-dimensional nanocomposite coatings can effectively mitigate membrane scaling through multiple synergistic mechanisms.

Scaling study of the magnetic entropy change in room-temperature ferromagnet Cr3Te4

The intrinsic room-temperature ferromagnetism, stability in air, and easy tunability make chromium telluride Cr3Te4 a material with great promise for application in spintronics. This study investigates the magnetic entropy change (ΔSM) associated with critical exponents of Cr3Te4 single crystal. Temperature-dependent ΔSM suggests a drastic entropy change caused by the transition, the parameters of which show field-dependent power-law behaviors. Fitting the ΔSM(T,H) parameters yields critical exponents α=0.056(9), β=0.365(4), γ=1.212(3), δ=4.318(1), and Δ=1.562(5) for H//ab, while α=−0.077(3), β=0.369(6), γ=1.338(1), δ=4.620(6), and Δ=1.689(2) for H//c. The universality principle allows the ΔSM(T,H) to be scaled onto a single universal curve independent of external field. The direction-dependent differing critical exponents imply anisotropic magnetic couplings in Cr3Te4, which reveal a magnetic interaction of the 3D-Ising type for H//ab while a 3D-XY one for H//c. This work is beneficial for understanding the intrinsic and anisotropic room-temperature ferromagnetism in CrxTey family.

Critical behaviors of van der Waals itinerant ferromagnet Fe3.8GaTe2

The critical behavior of the van der Waals ferromagnet Fe3.8GaTe2 was systematically studied through measurements of isothermal magnetization, with the magnetic field applied along the c-axis. Fe3.8GaTe2 undergoes a non-continuous paramagnetic to ferromagnetic phase transition at the Curie temperature T c ∼ 355 K. A comprehensive analysis of isotherms around T c utilizing the modified Arrott diagram, the Kouvel–Fisher method, the Widom scaling law, and the critical isotherm analysis yielded the critical exponent of β= 0.411, γ = 1.246, and δ = 3.99. These critical exponents are found to be self-consistent and align well with the scaling equation at high magnetic fields, underscoring the reliability and intrinsic nature of these parameters. However, the low-field data deviates from the scaling relation, exhibiting a vertical trend when T < T c. This discrepancy suggests the occurrence of a first-order phase transition in the crystal under a low magnetic field when T< T c. Mössbauer spectra were employed to provide insights into the critical behaviors of magnetic moments at different sites, including (Fe1)A, (Fe1)B, and Fe2. The results are consistent with the isothermal magnetization analysis. Additionally, the renormalization group theory analysis reveals that the spin coupling within the Fe3.8GaTe2 crystal follows the three-dimensional Heisenberg ({d:n} = {3:3}) type with long-range magnetic and exchange interaction decays with a distance as J(r) ≈ r −4.80.

Pinning-depinning transitions in two classes of discrete elastic-string models in (2+1)-dimensions

The pinning-depinning phase transitions of interfaces for two classes of discrete elastic-string models are investigated numerically. In the (1+1)-dimensions, we revisit these two elastic-string models with slight modification to the growth rule, and compare the estimated values with the previous numerical and experimental results. For the (2+1)-dimensional case, we perform extensive simulations on pinning-depinning transitions in these discrete models with quenched disorder. For full comparisons in the physically relevant spatial dimensions, we also perform numerically two distinct universality classes, including the quenched Edwards–Wilkinson, and the quenched Kardar–Parisi–Zhang equations with and without external driving forces. The critical exponents of these systems in the presence of quenched disorder are numerically estimated. Our results show that the critical exponents satisfy scaling relations well, and these two discrete elastic-string models do not fall into the existing universality classes. In order to visually comparisons of these discrete systems with quenched disorder in the (2+1)-dimensional cases, we present surface morphologies with various external driving forces during the saturated time regimes. The relationships between surface morphologies, scaling exponents and correlation length are also revealed.

Critical dynamics within the real-time FRG approach

The Schwinger-Keldysh functional renormalization group developed by Y.-y. Tan [Real-time dynamics of the O(4) scalar theory within the fRG approach, ] is employed to investigate critical dynamics related to a second-order phase transition. The effective action of model A is expanded to the order of O(∂2) in the derivative expansion for the O(N) symmetry. By solving the fixed-point equations of effective potential and wave function, we obtain static and dynamic critical exponents for different values of the spatial dimension d and the field component number N. It is found that one has z≥2 in the whole range of 2≤d≤4 for the case of N=1, while in the case of N=4, the dynamic critical exponent turns to z&lt;2 when the dimension approach towards d=2. Published by the American Physical Society 2024

Finite-size and finite-time scaling for kinetic rough interfaces.

We consider discrete models of kinetic rough interfaces that exhibit space-time scale invariance in height-height correlation. We use the generic scaling theory of Ramasco et al. [Phys. Rev. Lett. 84, 2199 (2000)0031-900710.1103/PhysRevLett.84.2199] to confirm that the dynamical structure factor of the height profile can uniquely characterize the underlying dynamics. We apply both finite-size and finite-time scaling methods that systematically allow an estimation of the critical exponents and the scaling functions, eventually establishing the universality class accurately. The finite-size scaling analysis offers an alternative way to characterize the anomalous rough interfaces. As an illustration, we investigate a class of self-organized interface models in random media with extremal dynamics. The isotropic version shows a faceted pattern and belongs to the same universality class (as shown numerically) as the Sneppen model (version A). We also examine an anisotropic version of the Sneppen model and suggest that the model belongs to the universality class of the tensionless Kardar-Parisi-Zhang (tKPZ) equationin one dimension.

Overlap renormalization group transformations for disordered systems

We establish a renormalization group approach which is implemented on the degrees of freedom defined by the overlap of two replicas to determine the critical fixed point and to extract four critical exponents for the phase transition of the three-dimensional Edwards-Anderson model. In addition, we couple the overlap order parameter to a fictitious field and introduce it within the two-replica Hamiltonian of the system to study its explicit symmetry-breaking with the renormalization group. Overlap transformations do not require a renormalization of the random couplings of a system to extract the critical exponents associated with the relevant variables of the renormalization group. We conclude by discussing the applicability of such transformations in the study of any phase transition which is fully characterized by an overlap order parameter.

Investigation of the magnetocaloric effect and critical behavior of the perovskite La0.65Ca0.35−xBaxMnO3 (x=0, 0.1, 0.2, 0.3)

Polycrystalline samples of La[Formula: see text]Ca[Formula: see text]BaxMnO3 (x = 0, 0.1, 0.2, 0.3) were prepared using a conventional solid-state reaction method. The prepared samples demonstrated good single-phase quality, characterized by a space group of Pbnm. There was a significant increase in the relative cooling power (at 5[Formula: see text]T), with values as follows: RCP[Formula: see text][Formula: see text]J[Formula: see text]⋅[Formula: see text]kg[Formula: see text], RCP[Formula: see text][Formula: see text]J[Formula: see text]⋅[Formula: see text]kg[Formula: see text], RCP[Formula: see text][Formula: see text]J[Formula: see text]⋅[Formula: see text]kg[Formula: see text], and RCP[Formula: see text][Formula: see text]J[Formula: see text]⋅[Formula: see text]kg[Formula: see text]. It is noteworthy that the sample with a doping level of x = 0.2 underwent a transition from a first-order phase transition to a second-order phase transition. The critical behavior of the x = 0.2 and 0.3 samples was analyzed using modified Arrott plots and the Kouvel–Fisher method to determine critical exponents. The results obtained from both methods were in close agreement, indicating that the critical behavior of the x = 0.2 and 0.3 samples can be described by the mean field model.

Changes in microstructure, magnetocaloric effect and critical behavior of La0.8Sr0.2MnO3 manganites by K/Co doping

We prepared La0.8-xKxSr0.2Mn0.95Co0.05O3 (0 ⩽ x ⩽ 0.1) (LKSMCO) through the modified sol-gel method (S-G) and its structure. Then, the magnetic, magnetocaloric, and critical behavior were analyzed. LKSMCO was confirmed to be the rhombohedral structure within the R-3c space group (No.167) using transmission electron microscopy and X-ray diffraction. For validating the chemical composition of LKSMCO, X-ray photoelectron spectroscopy and EDS were applied, showing that K+ was successfully doped. Phase transition of LKSMCO from paramagnetic to ferromagnetic phase near Curie temperature (Tc) was confirmed with a Magnetic Property Measurement System (MPMS). Using normalization and Banerjee's criterion, LKSMCO was demonstrated to be the second-order magnetic phase transition (SOMPT). As shown from the applied magnetic field, the maximal magnetic entropy (−ΔSMmax) of LKSMCO near Tc was 3.87 J/(kg K) (x= 0.00), 4.28 J/(kg K) (x= 0.05) and 4.44 J/(kg K) (x= 0.10). We employed the Kouvel-Fisher approach (KF) and modified Arrott plots to confirm the values of γ, δ, and β. This indicates that critical exponents of LKSMCO were consistent with the Mean-filed model.

Quantized plastic deformation

In engineering crystal plasticity inelastic mechanisms correspond to tensorial zero-energy valleys in the space of macroscopic strains. The flat nature of such valleys is in contradiction with the fact that plastic slips, mimicking lattice-invariant shears, are inherently discrete. A reconciliation has recently been achieved in the mesoscopic tensorial model (MTM) of crystal plasticity, which introduces periodically modulated energy valleys while also capturing in a geometrically exact way the crystallographically-specific aspects of plastic slips. In this paper, we extend the MTM framework, which in its original form had the appearance of a discretized nonlinear elasticity theory, by explicitly introducing the concept of plastic deformation. The ensuing model contains a novel matrix-valued spin variable, representing the quantized plastic distortion, whose rate-independent evolution can be described by a discrete (quasi-)automaton. The proposed reformulation of the MTM leads to a considerable computational speedup associated with the use of a robust and efficient hybrid Gauss-Newton–Cauchy energy minimization algorithm. To illustrate the effectiveness of the new approach, we present a detailed case-study focusing on the aspects of crystal plasticity that are beyond reach for the classical continuum theory. Thus, we provide compelling evidence that the re-formulated MTM is fully adequate to deal with the intermittency of plastic response under quasi-static loading. In particular, our numerical experiments show that the statistics of dislocational avalanches, associated with plastic yield in 2D square crystals, exhibits a power-law tail with a critical exponent matching the value predicted by general theoretical considerations and also independently observed in discrete-dislocation-dynamics (DDD) simulations.