Order Parameter Exponent Research Articles

Phase separation in ordered polar active fluids: A completely different universality class from that of equilibrium fluids.

It has recently been shown that large collections of self-propelled entities (a.k.a. "flocks" or, more technically, polar-ordered active fluids), all spontaneously moving in the same direction, can undergo phase separation in the presence of sufficiently strong attractive interactions (which can be caused by, e.g., autochemotaxis). At this transition, the system spontaneously separates into a high-density band and a low-density band, moving parallel to each other, and to the direction of mean flock motion, at different speeds. We show here that phase separation in ordered polar active fluids belongs to a new universality class, completely different from that of phase separation in equilibrium fluids. The upper critical dimension for this transition is d_{c}=5, in contrast to the well-known d_{c}=4 of equilibrium phase separation. With a dynamical renormalization group analysis, we obtain the large-distance, long-time scaling laws of the velocity and density fluctuations, which are characterized by universal critical correlation length and order parameter exponents ν_{⊥},ν_{∥}, and β, respectively. We calculate these to O(ε) in a d=5-ε expansion.

Quantum-critical properties of the one- and two-dimensional random transverse-field Ising model from large-scale quantum Monte Carlo simulations

We study the ferromagnetic transverse-field Ising model with quenched disorder at T = 0T=0 in one and two dimensions by means of stochastic series expansion quantum Monte Carlo simulations using a rigorous zero-temperature scheme. Using a sample-replication method and averaged Binder ratios, we determine the critical shift and width exponents \nu_\mathrm{s}νs and \nu_\mathrm{w}νw as well as unbiased critical points by finite-size scaling. Further, scaling of the disorder-averaged magnetisation at the critical point is used to determine the order-parameter critical exponent \betaβ and the critical exponent \nu_{\mathrm{av}}νav of the average correlation length. The dynamic scaling in the Griffiths phase is investigated by measuring the local susceptibility in the disordered phase and the dynamic exponent z'z′ is extracted. By applying various finite-size scaling protocols, we provide an extensive and comprehensive comparison between the different approaches on equal footing. The emphasis on effective zero-temperature simulations resolves several inconsistencies in existing literature.

Effect of CdSe/ZnS quantum dot dispersion on phase transitional behavior of 8OCB liquid crystal.

We report the effect on the phase transitional behavior of 8OCB liquid crystal (LC) doped with functionalized CdSe/ZnS quantum dots (QDs) in different concentrations. The temperature-dependent data of high-resolution optical birefringence and dielectric anisotropy are utilized to characterize the critical anomaly for both the isotropic-to-nematic (I-N) and nematic-to-smectic-A (N-SmA) phase transitions. The obtained results reveal that the order parameter exponent (β) for the pure LC is found to be β=0.249 and does not vary upon the inclusion of QDs in the pure matrix. It describes the weakly first-order characteristic of the I-N phase transition for all the studied samples, which falls within the limit of the tricritical hypothesis. Conversely, depending on the range of the N phase, we observed a nonuniversal nature of the specific heat capacity critical exponent (α^{'}) linked with the N-SmA phase transition for all the studied samples. A relative comparison was made amongst the α^{'} values extracted from both the anisotropy data, and further, a theoretical relationship is established with these exponent values. The coupling strength among the N and SmA order parameters is determined using the optical birefringence data and discussed from the perspective of mutual interaction between the LC-QDs ligands. The results signify that a strong ligand-ligand interaction between neighboring QDs effectively reduces the N range and slightly influences the N-SmA phase transition.

Long-range quenched bond disorder in the bidimensional Potts model

We study the bidimensional q-Potts model with long-range bond correlated disorder. Similarly to Chatelain (2014 Phys. Rev. E 89 032105), we implement a disorder bimodal distribution by coupling the Potts model to auxiliary spin-variables, which are correlated with a power-law decaying function. The universal behaviour of different observables, especially the thermal and the order-parameter critical exponents, are computed by Monte-Carlo techniques for -Potts models for different values of the power-law decaying exponent a. On the basis of our conclusions, which are in agreement with previous theoretical and numerical results for q = 1 and q = 2, we can conjecture the phase diagram for . In particular, we establish that the system is driven to a fixed point at finite or infinite long-range disorder depending on the values of q and a. Finally, we discuss the role of the higher cumulants of the disorder distribution. This is done by drawing the auxiliary spin-variables from different statistical models. While the main features of the phase diagram depend only on the first and second cumulant, we argue, for the infinite disorder fixed point, that certain universal effects are affected by the higher cumulants of the disorder distribution.

Dynamic correlations in the conserved Manna sandpile.

We study dynamic correlations for current and mass, as well as the associated power spectra, in the one-dimensional conserved Manna sandpile. We show that, in the thermodynamic limit, the variance of cumulative bond current up to time T grows subdiffusively as T^{1/2-μ} with the exponent μ≥0 depending on the density regimes considered and, likewise, the power spectra of current and mass at low frequency f varies as f^{1/2+μ} and f^{-3/2+μ}, respectively. Our theory predicts that, far from criticality, μ=0 and, near criticality, μ=(β+1)/2ν_{⊥}z>0 with β,ν_{⊥}, and z being the order parameter, correlation length, and dynamic exponents, respectively. The anomalous suppression of fluctuations near criticality signifies a "dynamic hyperuniformity," characterized by a set of fluctuation relations, in which current, mass, and tagged-particle displacement fluctuations are shown to have a precise quantitative relationship with the density-dependent activity (or its derivative). In particular, the relation, D_{s}(ρ[over ¯])=a(ρ[over ¯])/ρ[over ¯], between the self-diffusion coefficient D_{s}(ρ[over ¯]), activity a(ρ[over ¯]) and density ρ[over ¯] explains a previous simulation observation [Eur. Phys. J. B 72, 441 (2009)10.1140/epjb/e2009-00367-0] that, near criticality, the self-diffusion coefficient in the Manna sandpile has the same scaling behavior as the activity.

Study of Nanocrystallization Kinetics in Fe 73.5 Cu 1 Nb 3 Si 13.5 B 9 Finemet Type Alloy by Differential Thermal Analysis and Using Different Models

The study of the crystallization processes in the FINEMET type nanocrystalline amorphous alloy is interesting not only from the fundamental aspect of establishing reaction mechanism of crystal nucleation and growth, but also from a technological point of view. The process and nature of crystallization phase constitution of nanocrystalline amorphous alloy of composition Fe 73.5 Cu 1 Nb 3 Si 13.5 B 9 prepared by rapid quenching method is investigated in the present study. The amorphous nature of the alloy has been verified by x-ray diffraction (XRD). The differential thermal analysis (DTA) experiments were performed at different continuous heating rates of 10, 20, 30, 40 and 50 0 C/min. Two different crystalline phases are observed. The crystallization temperatures, the volume fraction of crystallizations and enthalpies of two different crystalline phases of the alloy have been determined from DTA traces. The dependence of on-set crystallization temperature (T x ) on the heating rate of different phases have been used for the determination of different crystallization parameters such as, the activation energy of crystallization, the order parameter or Avrami exponent (n). The results of crystallization were discussed on the basis of different models such as Kissinger’s approach and modification for non-isothermal crystallization of Matusita in addition to Kolmogorov, Johnson, Mehl, Avrami and Ozawa.

Superconductivity and thermal fluctuation of Y1-x Mx: 123 systems with (M = Pr,Ca) & (0.00 < x < 0.50): Comparative study

We report here a comparative study between the effects of Pr and Ca in place of Y on superconductivity and thermal fluctuation of the Y1-x Mx: 123 system with M = Pr and Ca and (0.00 < x < 0.50). Furthermore, some remarks on the hole carrier per Cu ions in high Tc superconductors are revaluated. It is found that numerical values of hole carrier per Cu ion based on M.R. Presland et al., Physica C 176, 95 (1991) disagree with those deduced from thermoelectric power (TEP) and some modifications to the M.R. Presland formula are necessary. On the other hand, the orthorhombic distortion is decreased, while the c-parameter is increased as x increases in Y1-x Mx: 123, but the rate of decrease/increase is dependent on the chosen M. The oxygen content has gradually decreased for the Ca series, whereas it is nearly unchanged (y = 6.94) for the Pr series. The doping density and the doping distance L are almost constant with increasing Pr up to 0.30 and then sharply decrease to Pr = 0.40. In contrast, a gradual sharp decrease in them is obtained for the Ca series. The spacing between Cu ions has gradually increased for the Pr series, but it has decreased for the Ca series. The critical temperature Tc is decreased as x increases for both series, but the rate of decrease is higher for Ca than for Pr as well as for hole carriers per Cu ion. The order parameter exponent (OPD) is two dimensional (2D) in and above the mean field region (MFR) as x increases up to 0.20 for both series. But with increasing x above 0.20, the OPD remains (2D) for the Ca series and becomes zero dimensional (0D) for the Pr series. While in the critical field region (CFR), the OPD is three dimensional (3D) for both series, but it becomes (2D) above x = 0.20 for the Pr series. The upper critical fields and critical currents are sharply increased for the Ca series up to x = 0.50, while they are slightly decreased by Pr up to 0.20 followed by an increase at 0.40. Our results are discussed in terms of the balance between the whole carriers generated/lost by substitution with those lost/introduced by oxygen vacancies in the Cu–O2 planes. As compared to Pr, Ca makes Y: 123 more compact and has higher interlayer coupling, higher Ginzburg-Landau parameter and lower anisotropy, which as possible highlights the present work.

Percolation transitions in growing networks under achlioptas processes: Analytic solutions

Networks are ubiquitous in diverse real-world systems. Many empirical networks grow as the number of nodes increases with time. Percolation transitions in growing random networks can be of infinite order. However, when the growth of large clusters is suppressed under some effects, e.g., the Achlioptas process, the transition type changes to the second order. However, analytical results for the critical behavior, such as the transition point, critical exponents, and scaling relations are rare. Here, we derived them explicitly as a function of a control parameter $m$ representing the suppression strength using the scaling ansatz. We then confirmed the results by solving the rate equation and performing numerical simulations. Our results clearly show that the transition point approaches unity and the order-parameter exponent $\beta$ approaches zero algebraically as $m \to \infty$, whereas they approach these values exponentially for a static network. Moreover, the upper critical dimension becomes $d_u=4$ for growing networks, whereas it is $d_u=2$ for static ones.

Calculation of Order Parameter and Critical Exponents of the Spin Glass in the Frame of Edwards-Anderson Model

It is well known that critical phenomena occur in condensed matter under certain conditions, when an abrupt change in its properties occurs. In the vicinity of critical points, various phenomena may arise. The critical region can be described by a set of state parameters (order parameters), which allow one to obtain information about the anomalous behavior of thermodynamic averages, internal processes, and the nature of the objects of study. The abnormal nonlinear behavior of state parameters is described by critical exponents. In this article, we considered spin glass on the example of the Edwards-Anderson model. For the simulation, the replica-exchange Monte-Carlo method was used. Critical exponents were obtained to describe the behavior of the model in the critical region.

Effect Melting Time on the Excess Conductivity and Critical Parameters of BSCCO Cooper Oxide System

We report here the effect of melting time (tm = 2.5–4.5 min) on the conductivity and dimensionality of BSCCO cooper oxide system. It is found that the mean field and crossover temperatures (Tcmf, T01, and T02) are increased by increasing tm up to 3.5 min, followed by a decrease with further increase of tm up to 4.5 min. The logarithmic plots of excess conductivity (∆σ) and reduced temperature (Є) reveal three regions of different exponents corresponding to two crossover temperatures in the slope of each plot. Interestingly, the crossover occurs from three dimensional (3D) to zero dimensional (0D/SW) in the mean field region and from 0D/SW to two dimensional (2D) in the critical field region, for the samples melted at tm = 2.5, 4, and 4.5 min, while it occurs from (3D) to one dimensional (1D) and from (1D) to (2D) for the sample melted at tm = 3.5 min. On the other hand, we have estimated several physical parameters such as order parameter exponents (λ), interlayer coupling (K), c-axis coherence length (ξc (0)), anisotropy (γ), Ginsburg number (Gi), critical magnetic fields (Hc(0), Hc1(0), and Hc2(0)), and critical current (Jc (0)) for all samples. It is found that λ1, λ3, K, ξc (0), Gi, and γ are increased by increasing tm up to 3.5 min, followed by a decrease with further increase of tm up to 4.5 min as well as Tc, Tcmf, and T0 behaviors. But the vice is versa for the behaviors of λ2, κ, Hc(0), Hc1(0), Hc2(0), Jc(0), and NG. Moreover, it is observed that the behavior of critical fields and critical current against melting time is controlled by the order parameter exponent of the second region rather than the first and third regions. These results are discussed in terms of the correlation between the effects of melting time on the weak links and the flow of actual supercurrent in the considered system.

Order-parameter critical exponent of absorbing phase transitions in one-dimensional systems with two symmetric absorbing states.

Via extensive Monte Carlo simulations along with systematic analyses of corrections to scaling, we estimate the order parameter critical exponent β of absorbing phase transitions in systems with two symmetric absorbing states. The value of β was conjectured to be 13/14≈0.93 and Monte Carlo simulation studies in the literature have repeatedly reproduced values consistent with the conjecture. In this paper, we systematically estimate β by analyzing the effective exponent after finding how strong corrections to scaling are. We show that the widely accepted numerical value of β is not correct. Rather, we obtain β=1.020(5) from different models with two symmetric absorbing states.

Critical properties of the measurement-induced transition in random quantum circuits

We numerically study the measurement-driven quantum phase transition of Haar-random quantum circuits in $1+1$ dimensions. By analyzing the tripartite mutual information we are able to make a precise estimate of the critical measurement rate $p_c = 0.17(1)$. We extract estimates for the associated bulk critical exponents that are consistent with the values for percolation, as well as those for stabilizer circuits, but differ from previous estimates for the Haar-random case. Our estimates of the surface order parameter exponent appear different from that for stabilizer circuits or percolation, but we are unable to definitively rule out the scenario where all exponents in the three cases match. Moreover, in the Haar case the prefactor for the entanglement entropies $S_n$ depends strongly on the R\'enyi index $n$; for stabilizer circuits and percolation this dependence is absent. Results on stabilizer circuits are used to guide our study and identify measures with weak finite-size effects. We discuss how our numerical estimates constrain theories of the transition.

Field-induced anisotropy in the quasi-two-dimensional weakly anisotropic antiferromagnet [CuCl(pyz)2]BF4

We measured NMR and magnetic susceptibility for the quasi-two-dimensional, weakly $XY$-like, spin-$1/2$ square-lattice Heisenberg antiferromagnet $[\mathrm{CuCl}{(\mathrm{pyz})}_{2}]{\mathrm{BF}}_{4}$ (pyz = pyrazine $=\phantom{\rule{0.16em}{0ex}}{\mathrm{N}}_{2}{\mathrm{C}}_{4}{\mathrm{H}}_{4}$) near the critical temperature. The N\'eel temperature ${T}_{N}$ and the order-parameter critical exponent $\ensuremath{\beta}$ were obtained from the NMR line broadening as a function of temperature. As the applied field strength ($H\ensuremath{\parallel}c$) was increased, ${T}_{N}$ increased and $\ensuremath{\beta}$ decreased. This behavior indicates that the field effectively enhanced $XY$ anisotropy. The susceptibility as a function of temperature did not show a clear feature for ${T}_{N}$, but showed field-dependent minima below ${T}_{N}$ for both $H\ensuremath{\parallel}c$ and $H\ensuremath{\parallel}ab$, where minimum features disappeared for ${\ensuremath{\mu}}_{0}Hg2\phantom{\rule{0.16em}{0ex}}\mathrm{T}$.

Hamiltonian short-time critical dynamics of the three-dimensional XY model.

The short-time relaxation critical behavior of the XY model on a simple-cubic lattice is investigated within the scope of deterministic Hamiltonian dynamics. The Hamiltonian includes a first-neighbor interaction between planar vectors and a rotational kinetic term from which the motion equations are derived. The dynamical evolution from a fully ordered initial state is followed by employing a symplectic algorithm based on a high-order Trotter-Suzuki decomposition of the time-evolution operator. A finite-time scaling analysis is performed to provide accurate estimates of the critical energy density, the order-parameter relaxation exponent, and the dynamical critical exponent. The estimated critical exponents are consistent with prior theoretical and experimental values reported for the superfluid ^{4}He, extreme type-II superconducting, and Bose-Einstein condensation transitions.

Critical dynamics of the Kuramoto model on sparse random networks

We consider the Kuramoto model on sparse random networks such as the Erdős–Rényi graph or its combination with a regular two-dimensional lattice and study the dynamical scaling behavior of the model at the synchronization transition by large-scale, massively parallel numerical integration. By this method, we obtain an estimate of critical coupling strength more accurate than obtained earlier by finite-size scaling of the stationary order parameter. Our results confirm the compatibility of the correlation-size and the temporal correlation-length exponent with the mean-field universality class. However, the scaling of the order parameter exhibits corrections much stronger than those of the Kuramoto model with all-to-all coupling, making thereby an accurate estimate of the order-parameter exponent hard. We find furthermore that, as a qualitative difference to the model with all-to-all coupling, the effective critical exponents involving the order-parameter exponent, such as the effective decay exponent characterizing the critical desynchronization dynamics show a non-monotonic approach toward the asymptotic value. In the light of these results, the technique of finite-size scaling of limited size data for the Kuramoto model on sparse graphs has to be treated cautiously.

Nonuniversal and anomalous critical behavior of the contact process near an extended defect.

We consider the contact process near an extended surface defect, where the local control parameter deviates from the bulk one by an amount of λ(l)-λ(∞)=Al^{-s}, with l being the distance from the surface. We concentrate on the marginal situation s=1/ν_{⊥}, where ν_{⊥} is the critical exponent of the spatial correlation length, and study the local critical properties of the one-dimensional model by Monte Carlo simulations. The system exhibits a rich surface critical behavior. For weaker local activation rates A<A_{c}, the phase transition is continuous, having an order-parameter critical exponent, which varies continuously with A. For stronger local activation rates A>A_{c}, the phase transition is of mixed order: the surface order parameter is discontinuous; at the same time the temporal correlation length diverges algebraically as the critical point is approached, but with different exponents on the two sides of the transition. The mixed-order transition regime is analogous to that observed recently at a multiple junction and can be explained by the same type of scaling theory.

Dimensional Reduction in Quantum Dipolar Antiferromagnets.

We report ac susceptibility, specific heat, and neutron scattering measurements on a dipolar-coupled antiferromagnet LiYbF_{4}. For the thermal transition, the order-parameter critical exponent is found to be 0.20(1) and the specific-heat critical exponent -0.25(1). The exponents agree with the 2D XY/h_{4} universality class despite the lack of apparent two-dimensionality in the structure. The order-parameter exponent for the quantum phase transitions is found to be 0.35(1) corresponding to (2+1)D. These results are in line with those found for LiErF_{4} which has the same crystal structure, but largely different T_{N}, crystal field environment and hyperfine interactions. Our results therefore experimentally establish that the dimensional reduction is universal to quantum dipolar antiferromagnets on a distorted diamond lattice.

Synchronization transition with coexistence of attractors in coupled discontinuous system

The studies of extended dynamics systems are relevant to the understanding of spatiotemporal patterns observed in diverse fields. One of the well-established models for such complex systems is the coupled map lattices, and several features of pattern formation including synchronization, unsynchronization, traveling waves and clustering synchronization are found. Among the above-mentioned patterns, chaotic synchronization has been intensively investigated in recent years. It has been demonstrated that two or more chaotic systems can be synchronized by linking them with mutual coupling or a common signal or some signals. Over the last decade, a number of theoretical methods have been presented to deal with this problem, such as the methods of master stability functions and eigenvalue analysis. While much effort has been devoted to the networks with different topological structures in continuous systems. The coupled discontinuous maps have been investigated with increasing interest in recent years, they showed that the complete synchronization in coupled discontinuous systems is more complicated than in coupled continuous systems. However, a similar problem of synchronization transition in coupled discontinuous systems is much less known.The synchronization transition in coupled discontinuous map lattices is studied. The average order parameter and maximal Lyapunov exponent are calculated to diagnose the synchronization of coupled piecewise maps. The results indicate that there exist the periodic clusters and the synchronization state, and a new transition style from periodic cluster states to complete synchronization states is found. The periodic cluster states consist of two kinds of periodic orbits: symmetric periodic orbits and asymmetric periodic orbits.Based on the pattern analysis, the common features of the patterns are constructed by the two periodic attractors, and the periodic orbit is an unstable periodic orbit of the isolate map. The discontinuities in a system can divide the phase space into individual zones of different dynamical features. The interactions between the local nonlinearity and the spatial coupling confine orbit into certain spaces and form a dynamic balance between two periodic clusters. The system can reach complete synchronization states when the balance is off. It is shown that synchronization transition of the coupled discontinuous map can exhibit the different processes, which depends on coupling strength. Four transition modes are found in coupled discontinuous map: 1) the transition, from the coexistence of chaotic synchronization and chaotic un-synchronization states to the coexistence of chaotic synchronization, chaotic un-synchronization, symmetric periodic orbits and asymmetric periodic orbits; 2) the transition from the coexistence of chaotic synchronization, chaotic un-synchronization, symmetric periodic orbits and asymmetric periodic orbits to the coexistence of chaotic synchronization, symmetric periodic orbits and asymmetric periodic orbits; 3) the transition from the coexistence of chaotic synchronization, symmetric periodic orbits and asymmetric periodic orbits to the coexistence of chaotic synchronization and symmetric periodic orbits; 4) the transition from the coexistence of chaotic synchronization and symmetric periodic orbits to the chaotic synchronization. Because the local dynamics has discontinuous points, the coupled system shows a riddle basin characteristic in the phase space, and the synchronization transition of coupled piecewise maps looks more complex than continuous system.

Absorbing phase transition in a conserved lattice gas model with next-nearest-neighbor hopping in one dimension.

The absorbing phase transition of the modified conserved lattice gas (m-CLG) model was investigated in one dimension. The m-CLG model was modified from the conserved lattice gas (CLG) model in such a way that each active particle hops to one of the nearest-neighbor and next-nearest-neighbor empty sites. The order parameter exponent, the dynamic exponent, and the correlation length exponent were estimated from the power-law behavior and finite-size scaling of the active particle densities. The exponents were found to differ considerably from those of the ordinary CLG model and were also distinct from those of the Manna model, suggesting that next-nearest-neighbor hopping is a relevant factor that alters the critical behavior in the one-dimensional CLG model.

Universality class of the conserved Manna model in one dimension.

The nonequilibrium absorbing phase transition of the discrete conserved Manna model was studied via Monte Carlo simulations on a one-dimensional chain, using the natural initial states with a sequential update. The critical density of the particles was found to be smaller than the recently reported value, and the order-parameter exponent was considerably different from the directed percolation (DP) value. The influence of quenched disorder was also studied on a diluted strip of L_{x}×L_{y} lattice sites with L_{x}≫L_{y}, and the results were compared with those of the contact process (CP). It was found that the Manna model and the CP exhibited distinctly different behaviors; the CP exhibited nonuniversal power-law decreases of active-site densities in the Griffith phase, whereas the Manna model showed a standard critical behavior. These results consistently suggest that the Manna model belongs to a universality class that is different from the DP class.