Finite-size Scaling Hypothesis Research Articles

Complexity of fermionic states

How much information does a fermionic state contain? To address this fundamental question, we define the complexity of a particle-conserving many-fermion state as the entropy of its Fock space probability distribution, minimized over all Fock representations. The complexity characterizes the minimum computational and physical resources required to represent the state and store the information obtained from it by measurements. Alternatively, the complexity can be regarded as a Fock space entanglement measure describing the intrinsic many-particle entanglement in the state. We establish a universal lower bound for the complexity in terms of the single-particle correlation matrix eigenvalues and formulate a finite-size complexity scaling hypothesis. Remarkably, numerical studies on interacting lattice models suggest a general model-independent complexity hierarchy: ground states are exponentially less complex than average excited states, which, in turn, are exponentially less complex than generic states in the Fock space. Our work has fundamental implications on how much information is encoded in fermionic states. Published by the American Physical Society 2024

Opinion Dynamics Systems via Biswas–Chatterjee–Sen Model on Solomon Networks

The critical properties of a discrete version of opinion dynamics systems, based on the Biswas–Chatterjee–Sen model defined on Solomon networks with both nearest and random neighbors, are investigated through extensive computer simulations. By employing Monte Carlo algorithms on SNs of different sizes, the magnetic-like variables of the model are computed as a function of the noise parameter. Using the finite-size scaling hypothesis, it is observed that the model undergoes a second-order phase transition. The critical transition noise and the respective ratios of the usual critical exponents are computed in the limit of infinite-size networks. The results strongly indicate that the discrete Biswas–Chatterjee–Sen model is in a different universality class from the other lattices and networks, but in the same universality class as the Ising and majority-vote models on the same Solomon networks.

Opinion Dynamics Systems on Barabási–Albert Networks: Biswas–Chatterjee–Sen Model

A discrete version of opinion dynamics systems, based on the Biswas-Chatterjee-Sen (BChS) model, has been studied on Barabási-Albert networks (BANs). In this model, depending on a pre-defined noise parameter, the mutual affinities can assign either positive or negative values. By employing extensive computer simulations with Monte Carlo algorithms, allied with finite-size scaling hypothesis, second-order phase transitions have been observed. The corresponding critical noise and the usual ratios of the critical exponents have been computed, in the thermodynamic limit, as a function of the average connectivity. The effective dimension of the system, defined through a hyper-scaling relation, is close to one, and it turns out to be connectivity-independent. The results also indicate that the discrete BChS model has a similar behavior on directed Barabási-Albert networks (DBANs), as well as on Erdös-Rènyi random graphs (ERRGs) and directed ERRGs random graphs (DERRGs). However, unlike the model on ERRGs and DERRGs, which has the same critical behavior for the average connectivity going to infinity, the model on BANs is in a different universality class to its DBANs counterpart in the whole range of the studied connectivities.

True scale-free networks hidden by finite size effects

We analyze about 200 naturally occurring networks with distinct dynamical origins to formally test whether the commonly assumed hypothesis of an underlying scale-free structure is generally viable. This has recently been questioned on the basis of statistical testing of the validity of power law distributions of network degrees. Specifically, we analyze by finite size scaling analysis the datasets of real networks to check whether the purported departures from power law behavior are due to the finiteness of sample size. We find that a large number of the networks follows a finite size scaling hypothesis without any self-tuning. This is the case of biological protein interaction networks, technological computer and hyperlink networks, and informational networks in general. Marked deviations appear in other cases, especially involving infrastructure and transportation but also in social networks. We conclude that underlying scale invariance properties of many naturally occurring networks are extant features often clouded by finite size effects due to the nature of the sample data.

Dynamical crossover in invasion percolation

The dynamical properties of the invasion percolation on the square lattice are investigated with an emphasis on the geometrical properties on the growing cluster of infected sites. The exterior frontier of this cluster forms a critical loop ensemble (CLE), whose length (l), the radius (r) and also roughness (w) fulfill the finite-size scaling hypothesis. The dynamical fractal dimension of the CLE defined as the exponent of the scaling relation between l and r is estimated to be Df = 1.76 ± 0.04. By studying the autocorrelation functions of these quantities we show importantly that there is a crossover between two time regimes, in which these functions change behavior from power-law at the small times, to exponential decay at long times. In the vicinity of this crossover time, these functions are estimated by log-normal functions. We also show that the increments of the considered statistical quantities, which are related to the random forces governing the dynamics of the observables undergo an anticorrelation/correlation transition at the time that the crossover takes place.

Absence of localization in disordered two-dimensional electron gas at weak magnetic field and strong spin-orbit coupling

The one-parameter scaling theory of localization predicts that all states in a disordered two-dimensional system with broken time reversal symmetry are localized even in the presence of strong spin-orbit coupling. While at constant strong magnetic fields this paradigm fails (recall the quantum Hall effect), it is believed to hold at weak magnetic fields. Here we explore the nature of quantum states at weak magnetic field and strongly fluctuating spin-orbit coupling, employing highly accurate numerical procedure based on level spacing distribution and transfer matrix technique combined with one parameter finite-size scaling hypothesis. Remarkably, the metallic phase, (known to exist at zero magnetic field), persists also at finite (albeit weak) magnetic fields, and eventually crosses over into a critical phase, which has already been confirmed at high magnetic fields. A schematic phase diagram drawn in the energy-magnetic field plane elucidates the occurrence of localized, metallic and critical phases. In addition, it is shown that nearest-level statistics is determined solely by the symmetry parameter β and follows the Wigner surmise irrespective of whether states are metallic or critical.

Statistical investigation of the cross sections of wave clusters in the three-dimensional Bak-Tang-Wiesenfeld model.

We consider the three-dimensional (3D) Bak-Tang-Wiesenfeld model in a cubic lattice. Along with analyzing the 3D problem, the geometrical structure of the two-dimensional (2D) cross section of waves is investigated. By analyzing the statistical observables defined in the cross sections, it is shown that the model in that plane (named as 2D-induced model) is in the critical state and fulfills the finite-size scaling hypothesis. The analysis of the critical loops that are interfaces of the 2D-induced model is of special importance in this paper. Most importantly, we see that their fractal dimension is D(f)=1.387±0.005, which is compatible with the fractal dimension of the external perimeter of geometrical spin clusters of 2D critical Ising model. Some hyperscaling relations between the exponents of the model are proposed and numerically confirmed. We then address the problem of conformal invariance of the mentioned domain walls using Schramm-Lowener evolution (SLE). We found that they are described by SLE with the diffusivity parameter κ=2.8±0.2, nearly consistent with observed fractal dimension.

Finite size scaling in minimal walking technicolor

We compare observables to the finite size scaling hypothesis in SU(2) lattice gauge theory with two Dirac fermions in the adjoint representation. The fits that we obtain yield an estimate of the anomalous mass dimension that is consistent with four loop perturbation theory: $\gamma = 0.51 \pm 0.16$, with the error due to systematic uncertainties in the finite size scaling analysis. The result is somewhat larger than one Schr\"odinger functional study (by 1.3$\sigma$) but consistent with another.

Finite size scaling in BTW like sandpile models

Lattice statistical models of equilibrium critical phenomena generally obey finite size scaling (FSS) ansatz. However, the critical behavior of the prototypical BTW sandpile model demonstrating self-organized criticality at out of equilibrium is described by a peculiar multiscaling behaviour. FSS hypothesis is verified here on two versions (RSM1 and RSM2) of a rotational sandpile model (RSM) with broken mirror symmetry. In these models, sand grains flow only in the forward direction and in a specific rotational direction from an active site after toppling. The toppling rules are such that RSM1 will have less randomness whereas RSM2 will have more randomness with respect to RSM. RSM1 is expected to be more closer to BTW whereas RSM2 is expected to be more closer to Manna's stochastic model. Both RSM1 and RSM2 are found to belong to the same universality class of RSM. The scaling functions of RSM1 and RSM2 are also found to obey usual FSS hypothesis at out of equilibrium instead of multiscaling as in BTW.

Critical behavior of a one-dimensional contact process with time-uncorrelated disorder

In this work, the critical behavior of the one-dimensional contact process with time-uncorrelated disorder is investigated. We develop simulations on finite chains and explore the finite size scaling hypothesis to obtain estimates for the relevant parameters associated with static and dynamical critical quantities. We use an auto-adaptative technique that has been recently shown to provide reliable results for the standard contact process transition. We compare the main results with those derived from the usual short-time dynamics scaling. We found that, contrary to the behavior of the contact-process with quenched disorder which displays an infinite randomness critical point with activated scaling, the contact process with time-uncorrelated disorder belongs to the usual universality class of directed percolation.

Finite-size effects in the spherical model of finite thickness

A detailed analysis of the finite-size effects on the bulk critical behaviour of the d-dimensional mean spherical model confined to a film geometry with finite thickness L is reported. Along the finite direction different kinds of boundary conditions are applied: periodic (p), antiperiodic (a) and free surfaces with Dirichlet (D), Neumann (N) and a combination of Neumann and Dirichlet (ND) on both surfaces. A systematic method for the evaluation of the finite-size corrections to the free energy for the different types of boundary conditions is proposed. The free energy density and the equation for the spherical field are computed for arbitrary d. It is found, for 2 < d < 4, that the singular part of the free energy has the required finite-size scaling form at the bulk critical temperature only for (p) and (a). For the remaining boundary conditions the standard finite-size scaling hypothesis is not valid. At d = 3, the critical amplitude of the singular part of the free energy (related to the so-called Casimir amplitude) is estimated. We obtain Δ(p) = −2ζ(3)/(5π) = −0.153 051…, Δ(a) = 0.274 543… and Δ(ND) = 0.019 22…, implying a fluctuation-induced attraction between the surfaces for (p) and repulsion in the other two cases. For (D) and (N) we find a logarithmic dependence on L.

Critical behavior of the two-dimensional Anderson model with long-range correlateddisorder

We describe the critical behavior of the Anderson-like transition predicted to occur in the2D tight-binding model with isotropic scale-free long-range correlated disordercharacterized by a power-law spectral density. We explore the scale invarianceof the participation function relative fluctuation at the critical point to locatethe mobility edge as a function of the power-law spectral density exponentα2D. The statesnear the band center, which exhibit power-law localization for uncorrelated disorder, become delocalizedfor α2D>2. In addition, we consider the finite-size scaling hypothesis to estimate thecorrelation length critical exponent. We find that the critical exponentν dependson α2D, thus indicating that correlations in the disorder distribution are indeed relevant in thisregime, in agreement with the extended Harris criterion.

Critical wave-packet dynamics in the power-law bond disordered Anderson model

We investigate the wave-packet dynamics of the power-law bond disordered one-dimensional Anderson model with hopping amplitudes decreasing as ${H}_{nm}\ensuremath{\propto}{\ensuremath{\mid}n\ensuremath{-}m\ensuremath{\mid}}^{\ensuremath{-}\ensuremath{\alpha}}$. We consider the critical case $(\ensuremath{\alpha}=1)$. Using an exact diagonalization scheme on finite chains, we compute the participation moments of all stationary energy eigenstates as well as the spreading of an initially localized wave packet. The eigenstates multifractality is characterized by the set of fractal dimensions of the participation moments. The wave packet shows a diffusivelike spread developing a power-law tail and achieves a stationary nonuniform profile after reflecting at the chain boundaries. As a consequence, the time-dependent participation moments exhibit two distinct scaling regimes. We formulate a finite-size scaling hypothesis for the participation moments relating their scaling exponents to the ones governing the return probability and wave-function power-law decays.

Dynamical finite-size scaling function of the four dimensional ising model for Creutz algorithm

Dynamical critical exponents for the four dimensional Ising model are computed on a cellular automaton from the relaxations of the time displaced correlation for the order parameter and the internal energy at the critical temperatures. The analysis of the data within the frame of the dynamical finite-size scaling hypothesis gives the linear dynamical critical exponents for the order parameter and the internal energy. In our calculation, the linear dynamical critical exponents are found z M = z H I = 2.04 at T c ( ∞ ) , z M = 1.99 , z H I = 2.00 at T c χ ( L ) . The values of z M = z H I = 2.04 obtained for T c ( ∞ ) verify the dynamical finite-size scaling, for which it was adopted four dimensional Ising model.

C-dependence of fractal dimension in two-dimensional quantum gravity

We numerically investigate the fractal structure of two-dimensional quantum gravity coupled to matter central charge c for −2 < c < 1. We reformulate Q-state Potts model into the model which can be identified as a weighted percolation cluster model. The c-dependence of the fractal dimension based on the finite size scaling hypothesis is measured and has excellent agreement with our theoretical predictions previously evaluated analytically.

Numerical study for the c-dependence of fractal dimension in two-dimensional quantum gravity

We numerically investigate the fractal structure of two-dimensional quantum gravity coupled to matter central charge c for −2⩽ c⩽1. We reformulate Q-state Potts model into the model which can be identified as a weighted percolation cluster model and can make continuous change of Q, which relates c, on the dynamically triangulated lattice. The c-dependence of the critical coupling is measured from the percolation probability and susceptibility. The c-dependence of the string susceptibility of the quantum surface is evaluated and has very good agreement with the theoretical predictions. The c-dependence of the fractal dimension based on the finite size scaling hypothesis is measured and has excellent agreement with one of the theoretical predictions previously proposed except for the region near c≈1.

Anomalous scaling in the Zhang model

We apply the moment analysis technique to analyze large scale simulations of the Zhang sandpile model. We find that this model shows different scaling behavior depending on the update mechanism used. With the standard parallel updating, the Zhang model violates the finite-size scaling hypothesis, and it also appears to be incompatible with the more general multifractal scaling form. This makes impossible its affiliation to any one of the known universality classes of sandpile models. With sequential updating, it shows scaling for the size and area distribution. The introduction of stochasticity into the toppling rules of the parallel Zhang model leads to a scaling behavior compatible with the Manna universality class.

Phenomenological renormalization group methods

Some renormalization group approaches have been proposed during the last few years which are close in spirit to the Nightingale phenomenological procedure. In essence, by exploiting the finite size scaling hypothesis, the approximate critical behavior of model on infinite lattice is obtained through the exact computation of some thermal quantities of the model on finite clusters. In this work some of these methods are reviewed, namely the mean field renormalization group, the effective field renormalization group and the finite size scaling renormalization group procedures. Although special emphasis is given to the mean field renormalization group (since it has been, up to now, much more used to study a wide variety of different systems) a discussion of their potentialities and interrelations to other methods is also presented.

Oscillatory behavior of critical amplitudes of the Gaussian model on a hierarchical structure.

We studied oscillatory behavior of critical amplitudes for the Gaussian model on a hierarchical structure presented by a modified Sierpinski gasket lattice. This model is known to display nonstandard critical behavior on the lattice under study. The leading singular behavior of the correlation length xi near the critical coupling K=K(c) is modulated by a function which is periodic in ln/ln(K(c)-K)/. We have also shown that the common finite-size scaling hypothesis, according to which for a finite system at criticality xi should be of the order of the size of the system, is not applicable in this case. As a consequence of this, the exact form of the leading singular behavior of xi differs from the one described earlier (which was based on the finite-size scaling assumption).

Localization in Fock Space: A Finite-Energy Scaling Hypothesis for Many-Particle Excitation Statistics

The concept of localization in Fock space is extended to the study of the many particle excitation statistics of interacting electrons in a two dimensional quantum dot. In addition, a finite size scaling hypothesis for Fock space localization, in which the excitation energy replaces the system size, is developed and tested by analyzing the spectral properties of the quantum dot. This scaling hypothesis, modeled after the usual Anderson transition scaling, fits the numerical data obtained for the interacting states in the dot. It therefore attests to the relevance of the Fock space localization scenario to the description of many particle excitation properties.