Length Exponent Research Articles

Walking behavior induced by PT -symmetry breaking in a non-Hermitian XY model with clock anisotropy

A quantum system governed by a non-Hermitian Hamiltonian may exhibit zero-temperature phase transitions that are driven by interactions, just as its Hermitian counterpart, raising the fundamental question of how non-Hermiticity affects quantum criticality. In this context, we consider a non-Hermitian system consisting of an XY model with a complex-valued four-state clock interaction that may or may not have parity-time-reversal (PT) symmetry. When the PT symmetry is broken, and time-evolution becomes nonunitary, a scaling behavior similar to the Berezinskii-Kosterlitz-Thouless phase transition ensues, but in a highly unconventional way, as the line of fixed points is absent. From the analysis of the d-dimensional renormalization group equations, we obtain that the unconventional behavior in the PT broken regime follows from the collision of two fixed points in the d→2 limit, leading to walking behavior or pseudocriticality. For d=2+1 the near critical behavior is characterized by a correlation length exponent ν=3/8, a value smaller than the mean-field one. These results are in sharp contrast with the PT-symmetric case where only one fixed point arises for 2<d<4 and in d=1+1 three lines of fixed points occur with a continuously varying critical exponent ν. Published by the American Physical Society 2024

Characteristic Analysis of Vertical Tidal Profile Parameters at Tidal Current Energy Site

Many mathematical models have been proposed to estimate vertical tidal current profiles. However, as previous studies have shown that tidal current energy sites have different characteristics in their vertical tidal current profiles, it is necessary to estimate the profiles from field-measured data for practical purposes. In this study, we measured layered tidal currents over two months using an acoustic Doppler current profiler (ADCP) to analyze the characteristics of vertical tidal current profiles at the Jangjuk Strait, a candidate site for tidal current energy. As a result, the power law parameter α and bed roughness β were estimated as 4.51–12.41 and 0.38–0.42, respectively. Additionally, the maximum roughness length representing seabed roughness in the logarithmic profile was estimated as 0.221 m, and the estimated friction velocity was 0.038–0.194 m/s. Furthermore, a high correlation was observed between the depth-averaged tidal current velocity and friction velocities at all sites during flood and ebb tide conditions. A high correlation was also found between the bed roughness, roughness length, and power law exponent at relatively deeper sites. Tidal current energy sites display distinct characteristics compared to other sea areas. Therefore, it is essential to account for field conditions when conducting numerical modeling and design.

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.

Relation between critical exponent of the conductivity and the morphological exponents of percolation theory.

A central unsolved problem in percolation theory over the past five decades has been whether there is a direct relationship between the critical exponents that characterize the power-law behavior of the transport properties near the percolation threshold, particularly the effective electrical conductivity σ_{e}, and the exponents that describe the morphology of percolation clusters. The problem is also relevant to the relation between the static exponents of percolation clusters and the critical dynamics of spin waves in dilute ferromagnets, the elasticity of gels and composite solids, hopping conductivity in semiconductors, solute transport in porous media, and many others. We propose an approach to address the problem by showing that the contributions to σ_{e} can be decomposed into several groups representing the structure of percolation networks, including their mass and tortuosity, as well as constrictivity that describes the fluctuations in the driving potential gradient along the transport paths. The decomposition leads to a relationship between the critical exponent t of σ_{e} and other percolation exponents in d dimensions, t/ν=(d-D_{bb})+2(D_{op}-1)+d_{C}, where ν, D_{bb}, D_{op}, and d_{C} are, respectively, the correlation length exponent, the fractal dimensions of the backbones and the optimal paths, and the exponent that characterizes the constrictivity. Numerical simulations in two and three dimensions, as well as analytical results in d=1 and d=6, the upper critical dimension of percolation, validate the relationship. We, therefore, believe that the solution to the 50-year-old problem has been derived.

Surface integrity analysis and inspection for nanochannel sidewalls using the self-affine fractal model-based statistical quality control for the atomic force microscopy (AFM)-based nanomachining process

The atomic force microscopy (AFM) technology is a promising method for nanofabrication due to the high tunability of this affordable platform. The quality inspection and control significantly impact the manufacturing effectiveness for realizing the functionality of the achieved nanochannel. Particularly, the surface characteristics of nanochannel sidewalls, which play a significant role in determining the quality of the nanomachined products, can not be accurately captured using conventional surface integrity metrics (e.g., surface roughness). Therefore, it is necessary to propose a method to quantitatively characterize the surface morphology and detect the abnormal parts/regions of the nanochannel sidewall. This paper presents a statistical process control approach derived from the self-affine fractal model to detect the sidewall surface anomalies. It evaluates changes in the self-affine fractal model parameters (standard deviation, correlation length, and roughness exponent), which can be used to signify the changes on the sidewall surface; the statistical distributions of these parameters are derived and used to develop control charts to allow inspection of the sidewall morphology. The approach was tested on the AFM-based nanomachined samples. The results suggest that the presented approach can effectively reflect the abnormal regions on the machined parts, which opens up a new avenue toward guiding the quality control and rework for process improvement for AFM-based nanomachining.

Influence of Anisotropy on the Study of the Critical Behavior of Spin Models by Machine Learning Methods

In this paper, we applied a deep neural network to study the issue of knowledge transferability between statistical mechanics models. The following computer experiment was conducted. A convolutional neural network was trained to solve the problem of binary classification of snapshots of the Ising model’s spin configuration on a two-dimensional lattice. During testing, snapshots of the Ising model spins on a lattice with diagonal ferromagnetic and antiferromagnetic connections were fed to the input of the neural network. Estimates of the probability of samples belonging to the paramagnetic phase were obtained from the outputs of the tested network. The analysis of these probabilities allowed us to estimate the critical temperature and the critical correlation length exponent. It turned out that at weak anisotropy the neural network satisfactorily predicts the transition point and the value of the correlation length exponent. Strong anisotropy leads to a noticeable deviation of the predicted values from the precisely known ones. Qualitatively, strong anisotropy is associated with the presence of oscillations of the correlation function above the Stefenson disorder temperature and further approach to the point of the fully frustrated case.

Multiscale revelation of asphalt morphology and adhesion performance evolution during stress relaxation process

The objective of this study is to investigate the evolution of asphalt morphology and adhesion properties during stress relaxation using atomic force microscopy (AFM) and molecular dynamics (MD) simulations. Changes in surface roughness, Derjaguin-Muller-Toporov (DMT) modulus, adhesion force, and force-distance curves were analyzed. Morphological and adhesion property evolution was assessed using correlation length and Lyapunov exponent. MD simulations provided insights into the internal stress-time relationship and free volume fraction of asphalt during relaxation. Results indicate that under constant tensile strain, the “bee structure” on the asphalt surface elongates, dark pits transform into light protrusions, microcracks heal, and surface roughness stabilizes after 10 h. Under compressive strain, similar changes occur but are more pronounced during tensile relaxation. The combined use of AFM and MD simulations offers a comprehensive understanding of the microscopic evolution mechanisms of asphalt under stress relaxation, providing valuable insights for optimizing asphalt pavement design and maintenance.

Percolation in a three-dimensional nonsymmetric multicolor loop model.

We conduct Monte Carlo simulations to analyze the percolation transition of a nonsymmetric loop model on a regular three-dimensional lattice. We calculate the critical exponents for the percolation transition of this model. The percolation transition occurs at a temperature that is close to, but not exactly, the thermal critical temperature. Our finite-size study on this model yields a correlation length exponent that agrees with that of the three-dimensional XY model with an error margin of 6%.

Quantum wetting transition in the cluster Ising model

The wetting transition in the one-dimensional cluster Ising model with opposite boundary fields is studied. Tuning one boundary field while fixing another leads to the occurrence of phase transitions, where the transition points depend on the cluster coupling. Furthermore, the phase diagram is divided into three regions with different phase transition. For weak and strong cluster coupling, the phase transition is continuous and belongs to the same universality of transverse Ising model with boundary fields. For intermediate cluster coupling, the phase transition is first order. In the strong cluster coupling region, the critical region becomes exponentially small and the asymptotic behavior is absent even for lattice size up to 104. A numerical method to solve the energy gap and the correlation length is proposed on an infinite long spin chain. With this method, one can get the critical behavior as close to the critical point as possible provided that the numerical accuracy is high enough. In the light of this method, we clearly show that there is a preasymptotic regime in which the apparent critical exponents depend on the cluster coupling. Moreover, we obtain the accurate energy gap exponent zν and the correlation length exponent ν in the asymptotic critical region.

Disorder-induced topological quantum phase transitions in multigap Euler semimetals

We study the effect of disorder in systems having a nontrivial Euler class. As these recently proposed multigap topological phases come about by braiding non-Abelian charged band nodes residing between different bands to induce stable pairs within isolated band subspaces, novel properties may be expected. Namely, a modified stability and critical phases under the unbraiding to metals can arise when the disorder preserves the underlying C2T or PT symmetry on average. Employing elaborate numerical computations, we verify the robustness of associated topology by evaluating the changes in the average densities of states and conductivities for different types of disorders. Upon performing a scaling analysis around the corresponding quantum critical points, we retrieve a universality for the localization length exponent of ν=1.4±0.1 for Euler-protected phases, relating to two-dimensional percolation models. We generically find that quenched disorder drives Euler semimetals into critical metallic phases. Finally, we show that magnetic disorder can also induce topological transitions to quantum anomalous Hall plaquettes with local Chern numbers determined by the initial value of the Euler invariant. Published by the American Physical Society 2024

Tip-to-base conduit widening remains consistent across cambial age and climates in Fagus sylvatica L.

Water transport, mechanical support and storage are the vital functions provided by the xylem. These functions are carried out by different cells, exhibiting significant anatomical variation not only within species but also within individual trees. In this study, we used a comprehensive dataset to investigate the consistency of predicted hydraulic vessel diameter widening values in relation to the distance from the tree apex, represented by the relationship Dh ∝ Lβ (where Dh is the hydraulic vessel diameter, L the distance from the stem apex and β the scaling exponent). Our analysis involved 10 Fagus sylvatica L. trees sampled at two distinct sites in the Italian Apennines. Our results strongly emphasize that vessel diameter follows a predictable pattern with the distance from the stem apex and β~0.20 remains consistent across cambial age and climates. This finding supports the hypothesis that trees do not alter their axial configuration represented by scaling of vessel diameter to compensate for hydraulic limitations imposed by tree height during growth. The study further indicates that within-tree variability significantly contributes to the overall variance of the vessel diameter-stem length exponent. Understanding the factors that contribute to the intraindividual variability in the widening exponent is essential, particularly in relation to interspecific responses and adaptations to drought stress.

Behavior of quantum coherence in quantum phase transitions of two-dimensional XY and ising models

We investigate the quantum coherence behavior of the ground states of 2D Heisenberg XY model and 2D Ising model with a transverse field on square lattices using the Quantum Renormalization Group (QRG) method. Our analysis focused on the ground state density matrix and its marginal states, revealing non-analytic behavior of quantum coherence (especially two-site coherence) near the critical point. This behavior allowed us to detect quantum phase transitions (QPT) in these models. By examining the scaling behavior of the maximum derivative of quantum coherence with system size, we determined the critical exponent of coherence for both models and the length exponent of the Ising model. Additionally, we investigated the time evolution of coherence in both models. Our results closely align with those obtained from entanglement analysis, that is while quantum coherence requires fewer computational calculations compared to discord and entanglement approaches.

Finding critical points and correlation length exponents using finite size scaling of Gini index.

The order parameter for a continuous transition shows diverging fluctuation near the critical point. Here we show, through numerical simulations and scaling arguments, that the inequality (or variability) between the values of an order parameter, measured near a critical point, is independent of the system size. Quantification of such variability through the Gini index (g) therefore leads to a scaling form g=G[|F-F_{c}|N^{1/dν}], where F denotes the driving parameter for the transition (e.g., temperature T for ferromagnetic to paramagnetic transition, or lattice occupation probability p in percolation), N is the system size, d is the spatial dimension and ν is the correlation length exponent. We demonstrate the scaling for the Ising model in two and three dimensions, site percolation on square lattice, and the fiber bundle model of fracture.

Schramm-Loewner Evolution in 2D Rigidity Percolation.

Amorphous solids may resist external deformation such as shear or compression, while they do not present any long-range translational order or symmetry at the microscopic scale. Yet, it was recently discovered that, when they become rigid, such materials acquire a high degree of symmetry hidden in the disorder fluctuations: their microstructure becomes statistically conformally invariant. In this Letter, we exploit this finding to characterize the universality class of central-force rigidity percolation (RP), using Schramm-Loewner evolution (SLE) theory. We provide numerical evidence that the interfaces of the mechanically stable structures (rigid clusters), at the rigidification transition, are consistently described by SLE_{κ}, showing that this powerful framework can be applied to a mechanical percolation transition. Using well-known relations between different SLE observables and the universal diffusion constant κ, we obtain the estimation κ∼2.9 for central-force RP. This value is consistent, through relations coming from conformal field theory, with previously measured values for the clusters' fractal dimension D_{f} and correlation length exponent ν, providing new, nontrivial relations between critical exponents for RP. These findings open the way to a fine understanding of the microstructure in other important classes of rigidity and jamming transitions.

Possible origin for the similar phase transitions in k-core and interdependent networks

The models of k-core percolation and interdependent networks (IN) have been extensively studied in their respective fields. A recent study has revealed that they share several common critical exponents. However, several newly discovered exponents in IN have not been explored in k-core percolation, and the origin of the similarity still remains unclear. Thus, in this paper, by considering k-core percolation on random networks, we first verify that the two newly discovered exponents (fractal fluctuation dimension, , and correlation length exponent, ) observed in d-dimensional IN spatial networks also exist with the same values in k-core percolation. That is, the fractality of the k-core giant component fluctuations is manifested by a fractal fluctuation dimension, , within a correlation size Nʹ that scales as , with . Here we define, and . This implies that both models, IN and k-core, feature the same scaling behaviors with the same critical exponents, further reinforcing the similarity between the two models. Furthermore, we suggest that these two models are similar since both have two types of interactions: short-range (SR) connectivity links and long-range (LR) influences. In IN the LR are the influences of dependency links while in k-core we find here that for k = 1 and k = 2 the influences are SR and in contrast for the influence is LR. In addition, analytical arguments for a universal hyper-scaling relation for the fractal fluctuation dimension of the k-core giant component and for IN as well as for any mixed-order transition are established. Our analysis enhances the comprehension of k-core percolation and supports the generalization of the concept of fractal fluctuations in mixed-order phase transitions.

Effects of Topological Properties with Local Variable Apertures on Solute Transport through Three-Dimensional Discrete Fracture Networks

In this study, the effects of topological properties with local variable apertures on fluid flow and solute transport through three-dimensional (3D) discrete fracture networks (DFNs) were investigated. A series of 3D DFNs with different fracture density, length, and aperture distribution were generated. The fluid flow and solute transport through the models were simulated by combining the MATLAB code and COMSOL Multiphysics. The effects of network topology and aperture heterogeneity on fluid flow and transport process were analyzed. The results show that the fluid flow and solute transport exhibit a strong channeling effect even in the DFNs with identical aperture, in which the areas of fast and slow migration fit well with the high- and low-flow regions, respectively. More obvious preferential paths of flow and migration are observed in individual fractures for the DFN with heterogeneous aperture than the model with identical aperture. Increasing the fracture length exponent reduces the available flow and transport paths for sparse fracture networks but does not significantly change the flow and transport channels for dense fracture networks. The breakthrough curves (BTCs) shift towards the right and slightly lag behind as the fracture density decreases or the aperture heterogeneity increases. The advection–diffusion equation (ADE) model cannot properly capture the evolution of BTCs for 3D DFNs, especially the long tails of BTCs. Compared to the ADE model, the mobile-immobile model (MIM) model separating the liquid phase into flowing and stagnate regions is proven to better fit the BTCs of 3D DFNs with heterogeneous aperture.

Dimensional analysis of literature formulas to estimate the characteristic flood response time in ungauged basins: A velocity-based approach

When investigating the wide literature dealing with the assessment of the critical time scale for basin hydrological response, several aspects still to be clarified can be acknowledged. Despite the high sensitivity of design flood peaks to the estimated time parameter value, there is still no agreement on the conceptual and operational definitions of the basin response time, resulting in several different approaches and formulations available.In our work, we suggest a conceptual approach to either reject or recommend formulas of the characteristic basin response time for ungauged basins, with the aim of devising some practical steps in the choice of a robust formulation to be used in hydrological modelling and flood hydrograph design. To this end, 29 empirical and semi-empirical formulas, all containing a basin length and slope, have been carefully selected and their structure compared in dimensional terms, using a simple hydraulic reasoning (e.g., the Chezy formula) as indicator of hydraulic consistency. 13 hydraulically consistent formulas have been identified.Starting from wave celerities and using the river network morphology of 135 watersheds in north-western Italy, we have then investigated and compared the variability of the average flow velocities estimated using all formulas whose input data are available within the study area. By comparing the magnitude and basin scale dependence of the inferred velocities with the values observed in the literature, which generally increase with basin size, some formulas are considered not reasonable, while 5 of them are identified as more robust, i.e. consistent with the observations. These are the formulas of Chow (1962), NERC (1975), SCS (1954), McEnroe and Zhao (1999) and Watt and Chow (1985).Our findings lead to identify analytically the relationships between the exponents of each formula and those of the scaling law linking the length and slope of the basin. These relationships, driving the increase or decrease in the velocity values with basin size, allow us to identify the range of length and slope exponents in the characteristic time formulas for which velocity increases with basin area, as literature suggests. Based on the same relationships, one of the 5 formulas above can be adopted in practical applications and a guideline for calibrating new formulations can be followed.

Finite-size analysis in neural network classification of critical phenomena.

We analyze the problem of supervised learning of ferromagnetic phase transitions from the statistical physics perspective. We consider two systems in two universality classes, the two-dimensional Ising model and two-dimensional Baxter-Wu model, and perform careful finite-size analysis of the results of the supervised learning of the phases of each model. We find that the variance of the neural network (NN) output function (VOF) as a function of temperature has a peak in the critical region. Qualitatively, the VOF is related to the classification rate of the NN. We find that the width of the VOF peak displays the finite-size scaling governed by the correlation length exponent ν of the universality class of the model. We check this conclusion using several NN architectures-a fully connected NN, a convolutional NN, and several members of the ResNet family-and discuss the accuracy of the extracted critical exponents ν.

Effects of quantum quench on entanglement dynamics in antiferromagnetic Ising model

We study the relationship between quench dynamics of entanglement and quantum phase transition in the antiferromagnetic Ising model with the Dzyaloshinskii–Moriya (DM) interaction by using the quantum renormalization-group method and the definition of negativity. Two types of quench protocols (i) adding the DM interaction suddenly and (ii) rotating the spins around x axis are considered to drive the dynamics of the system, respectively. By comparing the behaviors of entanglement in both types of quench protocols, the effects of quench on dynamics of entanglement are studied. It is found that there is the same characteristic time at which the negativity firstly reaches its maximum although the system shows different dynamical behaviors. Especially, the characteristic time can accurately reflect the quantum phase transition from antiferromagnetic to saturated chiral phases in the system. In addition, the correlation length exponent can be obtained by exploring the nonanalytic and scaling behaviors of the derivative of the characteristic time.

Swarming Transition in Super-Diffusive Self-Propelled Particles.

A super-diffusive Vicsek model is introduced in this paper that incorporates Levy flights with exponent α. The inclusion of this feature leads to an increase in the fluctuations of the order parameter, ultimately resulting in the disorder phase becoming more dominant as α increases. The study finds that for α values close to two, the order-disorder transition is of the first order, while for small enough values of α, it shows degrees of similarities with the second-order phase transitions. The article formulates a mean field theory based on the growth of the swarmed clusters that accounts for the decrease in the transition point as α increases. The simulation results show that the order parameter exponent β, correlation length exponent ν, and susceptibility exponent γ remain constant when α is altered, satisfying a hyperscaling relation. The same happens for the mass fractal dimension, information dimension, and correlation dimension when α is far from two. The study reveals that the fractal dimension of the external perimeter of connected self-similar clusters conforms to the fractal dimension of Fortuin-Kasteleyn clusters of the two-dimensional Q=2 Potts (Ising) model. The critical exponents linked to the distribution function of global observables vary when α changes.