Finite-size Scaling Relations Research Articles

Biswas-Chatterjee-Sen Model on Solomon Networks with Two Three-Dimensional Lattices.

The Biswas-Chatterjee-Sen (BChS) model of opinion dynamics has been studied on three-dimensional Solomon networks by means of extensive Monte Carlo simulations. Finite-size scaling relations for different lattice sizes have been used in order to obtain the relevant quantities of the system in the thermodynamic limit. From the simulation data it is clear that the BChS model undergoes a second-order phase transition. At the transition point, the critical exponents describing the behavior of the order parameter, the corresponding order parameter susceptibility, and the correlation length, have been evaluated. From the values obtained for these critical exponents one can confidently conclude that the BChS model in three dimensions is in a different universality class to the respective model defined on one- and two-dimensional Solomon networks, as well as in a different universality class as the usual Ising model on the same networks.

Mass media and its impact on opinion dynamics of the nonlinear q-voter model

With the success of general conceptual frameworks of statistical physics, many scholars have tried to apply these concepts to other interdisciplinary fields, such as socio-politics, economics, biology, medicine, and many more. In this work, we study the effect of mass media on opinion evolution based on the nonlinear q-voter by means with probability p a voter adopts the mass media opinion whenever a q-sized agent in the population is not in unanimous agreement. We perform analytical and numerical calculations for some quantities of macroscopic parameters of the model such as order parameter (representing an average of public opinion), consensus (relaxation) time, and exit probability, and obtain the agreement results. We find the power-law relations for some quantities of the model. (1) The probability threshold pt, i.e a probability that makes the system reaches a homogeneous state, follows the power-law relation pt∼qγ with the q-sized agent, where γ=−1.00±0.01 is the best fitting parameter. The probability threshold pt also eliminates the coexistence two ordered states of the model. (2) The relaxation time (the time needed by the system to reach consensus) τ with the population size N is obtained in the form of τ∼Nδ, where δ depends on the probability p and q-sized agent. We also approximate the separator point rs and the system’s scaling parameters by employing the standard finite-size scaling relation.

Renormalization group study of marginal ferromagnetism.

When studying the collective motion of biological groups, a useful theoretical framework is that of ferromagnetic systems, in which the alignment interactions are a surrogate of the effective imitation among the individuals. In this context, the experimental discovery of scale-free correlations of speed fluctuations in starling flocks poses a challenge to common statistical physics wisdom, as in the ordered phase of standard ferromagnetic models with O(n) symmetry, the modulus of the order parameter has finite correlation length. To make sense of this anomaly, a ferromagnetic theory has been proposed, where the bare confining potential has zero second derivative (i.e., it is marginal) along the modulus of the order parameter. The marginal model exhibits a zero-temperature critical point, where the modulus correlation length diverges, hence allowing us to boost both correlation and collective order by simply reducing the temperature. Here, we derive an effective field theory describing the marginal model close to the T=0 critical point and calculate the renormalization group equationsat one loop within a momentum shell approach. We discover a nontrivial scenario, as the cubic and quartic vertices do not vanish in the infrared limit, while the coupling constants effectively regulating the exponents ν and η have upper critical dimension d_{c}=2, so in three dimensions the critical exponents acquire their free values, ν=1/2 and η=0. This theoretical scenario is verified by a Monte Carlo study of the modulus susceptibility in three dimensions, where the standard finite-size scaling relations have to be adapted to the case of d>d_{c}. The numerical data fully confirm our theoretical results.

Time-resolved observation of a dynamical phase transition with atoms in a cavity

We present a dynamical, multilevel atom-cavity blockade effect and monitor its breakdown transition in time. As in the case of optical bistability, atoms initially impede transmission by detuning a cavity mode from the driving laser. The interacting system, however, eventually transitions into an uncoupled state via a critical runaway process, resulting in maximum transmission. These two extremes of transmission are macroscopic reflections of well-defined atomic states, and thus are interpreted as phases of a dynamical transition. By monitoring the output of the cavity, we make time-resolved measurements of the order parameter and that of the enhanced photon number fluctuations. Considering these results for different cavity driving intensities, we establish finite-size scaling relations that suggest such a runaway effect is in fact a genuine dynamical phase transition.

On finite-size spiky strings in AdS3\xd7 S3\xd7 T4 with mixed fluxes

We discuss finite-size corrections to the spiky strings in AdS space which is dual to the long \U0001d4a9 = 4 SYM operators of the form Tr( {varDelta}_{+}^{J_1}{phi}_1{varDelta}_{+}^{J_2}{phi}_2dots {varDelta}_{+}^{J_n}{phi}_n ). We express the finite-size dispersion relation in terms of Lambert W-function. We further establish the finite-size scaling relation between energy and angular momentum of the spiky string in presence of mixed fluxes in terms of W-function. We comment on the solution in pure NS-NS background as well.

Simulation of the Critical Adsorption of Semi-Flexible Polymers**Supported by the National Natural Science Foundation of China under Grant Nos 11674277, 11704210 and 21574117.

The critical adsorption of semi-flexible polymer chains on attractive surfaces is studied using Monte Carlo simulations. The results reveal that the critical adsorption point of a free polymer chain is the same as that of an end-grafted one. For the end-grafted polymer, we find that the finite-size scaling relation and the maximum fluctuation of adsorbed monomers are equivalent in estimating the critical adsorption point. The effect of chain stiffness on the critical adsorption is also investigated. The surface attraction strength for the critical adsorption of semi-flexible polymer chain decreases exponentially with an increase in the chain stiffness; In other words, lower adsorption energy is needed to adsorb a stiffer polymer chain. The result is explained from the viewpoint of the free energy profile for the adsorption.

Condensation of eigen microstate in statistical ensemble and phase transition

In a statistical ensemble with $M$ microstates, we introduce an $M \times M$ correlation matrix with the correlations between microstates as its elements. Using eigenvectors of the correlation matrix, we can define eigen microstates of the ensemble. The normalized eigenvalue by $M$ represents the weight factor in the ensemble of the corresponding eigen microstate. In the limit $M \to \infty$, weight factors go to zero in the ensemble without localization of microstate. The finite limit of weight factor when $M \to \infty$ indicates a condensation of the corresponding eigen microstate. This indicates a phase transition with new phase characterized by the condensed eigen microstate. We propose a finite-size scaling relation of weight factors near critical point, which can be used to identify the phase transition and its universality class of general complex systems. The condensation of eigen microstate and the finite-size scaling relation of weight factors have been confirmed by the Monte Carlo data of one-dimensional and two-dimensional Ising models.

Two-dimensional Ising model on random lattices with constant coordination number.

We study the two-dimensional Ising model on networks with quenched topological (connectivity) disorder. In particular, we construct random lattices of constant coordination number and perform large-scale Monte Carlo simulations in order to obtain critical exponents using finite-size scaling relations. We find disorder-dependent effective critical exponents, similar to diluted models, showing thus no clear universal behavior. Considering the very recent results for the two-dimensional Ising model on proximity graphs and the coordination number correlation analysis suggested by Barghathi and Vojta [Phys. Rev. Lett. 113, 120602 (2014)PRLTAO0031-900710.1103/PhysRevLett.113.120602], our results indicate that the planarity and connectedness of the lattice play an important role on deciding whether the phase transition is stable against quenched topological disorder.

Size-dependent structural, magnetic, and optical properties of MnCo2O4 nanocrystallites

Finite-size (d = 5.4–112 nm) and surface effects on the structural, optical, and magnetic properties of ferrimagnetic inverse-spinel MnCo2O4 are reported. For d ≥ 87 nm, partial tetragonal distortion of the inverse spinel-lattice was observed. The Curie temperature TC of MnCo2O4 nanostructures, as determined by dc-magnetic susceptibility (χ) measurements, follows a finite-size scaling relation TC(d) = TC(∞)[1−(ξ0/d)λ] with a shift exponent λ = 0.75 ± 0.15 and microscopic correlation length ξ0 = 1.4 ± 0.3 nm, which is consistent with the mean field theory. For T > TC, χ(T) fits Néel's expression for the two-sublattice model with antiferromagnetic molecular field (exchange) constants NBB ∼ 85.16 (JBB ∼ 2.94 × 10−22 J), NAB ∼ 110.96 (JAB ∼ 1.91 × 10−22 J), and NAA ∼ 43.8 (JAA ∼ 1.13 × 10−22 J) and asymptotic Curie temperature Ta ∼ 717.63 K. The optical energy bandgap Eg, evaluated from the Kubelka-Munk function ([F(R∞)ℏω]2 = C2(ℏω - Eg)) is blueshifted to 2.4 eV (d ∼ 5.4 nm) from 1.73 eV (d ∼ 112 nm) due to the quantum confinement and non-stoichiometry. The role of tetragonal distortion and grain-size-effects in the intensity of crystal field transitions and variation in the magnetic ordering are further discussed and compared with Co3O4 nanostructures.

Irreversible work versus fidelity susceptibility for infinitesimal quenches

We compare the irreversible work produced in an infinitesimal sudden quench of a quantum system at zero temperature with its ground state fidelity susceptibility, giving an explicit relation between the two quantities. We find that the former is proportional to the latter but for an extra term appearing in the irreversible work which includes also contributions from the excited states. We calculate explicitly the two quantities in the case of the quantum Ising chain, showing that at criticality they exhibit different scaling behaviors. The irreversible work, rescaled by square of the quench’s amplitude, exhibits a divergence slower than that of the fidelity susceptibility. As a consequence, the two quantities obey also different finite-size scaling relations.

Finite-size scaling law in single-crystalline Fe3O4 hollow nanostructures

Single-crystalline Fe3O4 hollow nanostructures (nanoring and nanotube) have been successfully synthesized by a hydrothermal method along with a heat treatment process. The temperature dependences of the magnetization of the hollow nanostructures were measured under a high vacuum ([Formula: see text] Torr) from 300[Formula: see text]K to 900[Formula: see text]K. The Curie temperatures of the nanoring and nanotube samples were found to decrease with decreasing the mean wall thickness. The Curie temperatures of the hollow magnetite nanostructures follow a finite-size scaling relation with the scaling exponent [Formula: see text]. By comparison with those of the zero-dimensional Fe3O4 particles and two-dimensional Fe3O4 films, we show that the scaling relation for our hollow nanostructures is in better agreement with the quasi-two-dimensional finite-size scaling law.

Tuning the magnetic properties of multisegmented Ni/Cu electrodeposited nanowires with controllable Ni lengths

The fabrication of segmented Ni/Cu nanowires (NWs), with tunable structural and magnetic properties, is reported. A potentiostatic electrodeposition method with a single electrolytic bath has been used to fabricate multisegmented Ni/Cu NWs inside a highly hexagonally ordered anodic nanoporous alumina membrane, with diameters of 50 nm and Ni segment lengths (LNi) tuned from 10 nm up to 140 nm. The x-ray diffraction results evidenced a strong dependence of the Ni NWs crystallographic face-centered-cubic (fcc) texture along the [220] direction on the aspect ratio of the NWs. The magnetic behavior of the multisegmented Ni/Cu NW arrays, as a function of the magnetic field and temperature, is also studied and correlated with their structural and morphological properties. Micromagnetic simulations, together with the experimental results, showed a dominant antiferromagnetic coupling between Ni segments along the wire length for small low aspect-ratio magnetic segments. When increasing the Ni segments’ length, the magnetic interactions between these along the wire became stronger, favouring a ferromagnetic coupling. The Curie temperature of the NWs was also found to strongly depend on the Ni magnetic segment length. Particularly the Curie temperature was found to be reduced 75 K for the 20 nm Ni segments, following the finite-size scaling relation with ξ0 = 8.1 Å and γ = 0.48. These results emphasize the advantages of using a template assisted method to electrodeposit multilayer NWs, as it allows an easy tailor of the respective morphological, chemical, structural and magnetic properties.

Determination of critical linear lattice size for the four dimensional Ising model on the Creutz cellular automaton

The four-dimensional Ising model is simulated on the Creutz cellular automaton (CCA) near the infinite-lattice critical temperature for the lattice with the linear dimension 4 ⩽ L ⩽ 22. The temperature dependence of Binder parameter (gL) are analyzed for the lattice with the linear dimension 4 ⩽ L ⩽ 22. In this study conducted highly detailed, two different types of behavior were determined as a result of varying linear lattice dimension. The infinite lattice critical temperatures are obtained to be Tc = 6.6845 ± 0.0005 in interval 4 ⩽ L ⩽ 12 and Tc = 6.6807 ± 0.0024 in interval 14 ⩽ L ⩽ 22. The finite and infinite lattice critical exponents for the order parameter, the magnetic susceptibility and the specific heat are computed from the results of simulations by using finite-size scaling relations. Critical linear lattice size have been identified as L = 14.

Finite-size scaling, dynamic fluctuations, and hyperscaling relation in the Kuramoto model.

We revisit the Kuramoto model to explore the finite-size scaling (FSS) of the order parameter and its dynamic fluctuations near the onset of the synchronization transition, paying particular attention to effects induced by the randomness of the intrinsic frequencies of oscillators. For a population of size N, we study two ways of sampling the intrinsic frequencies according to the same given unimodal distribution g(ω). In the "random" case, frequencies are generated independently in accordance with g(ω), which gives rise to oscillator number fluctuation within any given frequency interval. In the "regular" case, the N frequencies are generated in a deterministic manner that minimizes the oscillator number fluctuations, leading to quasiuniformly spaced frequencies in the population. We find that the two samplings yield substantially different finite-size properties with clearly distinct scaling exponents. Moreover, the hyperscaling relation between the order parameter and its fluctuations is valid in the regular case, but it is violated in the random case. In this last case, a self-consistent mean-field theory that completely ignores dynamic fluctuations correctly predicts the FSS exponent of the order parameter but not its critical amplitude.

The finite-size scaling study of four-dimensional Ising model in the presence of external magnetic field

The four-dimensional ferromagnetic Ising model in external magnetic field is simulated on the Creutz cellular automaton algorithm using finite-size lattices with linear dimension 4 ≤ L ≤ 8. The critical temperature value of infinite lattice, Tcχ(∞)=6,680(1) obtained for h = 0 agrees well with the values Tc(∞)≈ 6.68 obtained previously using different methods. Moreover, h = 0.00025 in our work also agrees with all the results obtained from h = 0 in the literature. However, there are no works for h ≠ 0 in the literature. The value of the field critical exponent (δ = 3.0136(3)) is in good agreement with δ = 3 which is obtained from scaling law of Widom. In spite of the finite-size scaling relations of |ML(t)| and χL(t) for 0 ≤ h ≤ 0.001 are verified; however, in the cases of 0.0025 ≤ h ≤ 0.1 they are not verified.

Contact changes near jamming.

We probe the onset and effect of contact changes in soft harmonic particle packings which are sheared quasistatically. We find that the first contact changes are the creation or breaking of contacts on a single particle. We characterize the critical strain, statistics of breaking versus making a contact, and ratio of shear modulus before and after such events, and explain their finite size scaling relations. For large systems at finite pressure, the critical strain vanishes but the ratio of shear modulus before and after a contact change approaches one: linear response remains relevant in large systems. For finite systems close to jamming the critical strain also vanishes, but here linear response already breaks down after a single contact change.

Finite size effects in L1o-FePt nanoparticles

Finite size effects on the temperature dependence of the uniaxial magnetic anisotropy, longitudinal and transverse susceptibilities and specific heat are examined for L1o-ordered FePt nanoparticles using an atomistic model based on an effective classical spin Hamiltonian. At low temperatures below criticality, we study the intrinsic uniaxial magnetic anisotropy energy (MAE) K1 and its scaling with magnetization K1(T)∼Ms(T)δ and using Langevin dynamics simulations we show that the dependence of the exponent δ on the size L and aspect ratio of the grain arises from decomposition of the MAE into bulk and surface dependent terms. Monte Carlo simulations in the critical regime near the Curie temperature Tc, show that the temperature variation of the specific heat and longitudinal susceptibility is given by finite size scaling relations c=Lα/νc̃(L1/νϵ) and χ=Lγ/νχ̃(L1/νϵ), respectively, where ϵ=(T−Tc)/Tc is the reduced temperature, and the susceptibility scaling function χ̃ can be approximated by a Lorentzian. Our estimates of the critical exponents α,γ, and ν appear to be in agreement with the universality class of the 3D Ising model.

MAJORITY-VOTE MODEL WITH HETEROGENEOUS AGENTS ON SQUARE LATTICE

We study a nonequilibrium model with up-down symmetry and a noise parameter q known as majority-vote model (MVM) of [M. J. Oliveira, J. Stat. Phys.66, 273 (1992)] with heterogeneous agents on square lattice (SL). By Monte Carlo (MC) simulations and finite-size scaling relations, the critical exponents β∕ν, γ∕ν and 1∕ν and points qc and U* are obtained. After extensive simulations, we obtain β∕ν = 0.35(1), γ∕ν = 1.23(8) and 1∕ν = 1.05(5). The calculated values of the critical noise parameter and Binder cumulant are qc = 0.1589(4) and U* = 0.604(7). Within the error bars, the exponents obey the relation 2β∕ν + γ∕ν = 2 and the results presented here demonstrate that the MVM heterogeneous agents belongs to a different universality class than the nonequilibrium MVM with homogeneous agents on SL.

Equation of state of the fermionic two-dimensional Hubbard model

We present results for the equation of state of the two-dimensional Hubbard model on an isotropic square lattice as obtained from a controlled and numerically exact large-cluster dynamical mean field simulation. Our results are obtained for large but finite systems and are extrapolated to infinite system size using a known finite-size scaling relation, and are supplemented by reliable error bars accounting for all sources of errors. We establish the importance of examining the decay of spatial spin correlations to determine whether a sufficiently large cluster has been used and with this in mind we present the energy, entropy, double occupancy, and nearest-neighbor spin correlations extrapolated to the thermodynamic limit. We discuss the implications of these calculations on pseudogap physics of the 2D Hubbard model away from half filling, where we find a strong behavioral shift in energy below a temperature ${T}^{*}$ which becomes more pronounced for larger clusters. Finally, we provide reference calculations and tables for the equation of state for values of doping away from half filling which are of interest to cold-atom experiments.

MAJORITY-VOTE MODEL ON OPINION-DEPENDENT NETWORK

We study a nonequilibrium model with up–down symmetry and a noise parameter q known as majority-vote model (MVM) of Oliveira 1992 on opinion-dependent network or Stauffer–Hohnisch–Pittnauer (SHP) networks. By Monte Carlo (MC) simulations and finite-size scaling relations the critical exponents β∕ν, γ∕ν and 1∕ν and points qc and U* are obtained. After extensive simulations, we obtain β∕ν = 0.230(3), γ∕ν = 0.535(2) and 1∕ν = 0.475(8). The calculated values of the critical noise parameter and Binder cumulant are qc = 0.166(3) and U* = 0.288(3). Within the error bars, the exponents obey the relation 2β∕ν + γ∕ν = 1 and the results presented here demonstrate that the MVM belongs to a different universality class than the equilibrium Ising model on SHP networks, but to the same class as majority-vote models on some other networks.