Finite-size Scaling Approach Research Articles

Walking with the Atoms in a Chemical Bond: A Perspective Using Quantum Phase Transition.

Phase transitions happen at critical values of the controlling parameters, such as the critical temperature in classical phase transitions, and system critical parameters in the quantum case. However, true criticality happens only at the thermodynamic limit, when the number of particles goes to infinity with constant density. To perform the calculations for the critical parameters, a finite-size scaling approach was developed to extrapolate information from a finite system to the thermodynamic limit. With the advancement in the experimental and theoretical work in the field of ultra-cold systems, particularly trapping and controlling single atomic and molecular systems, one can ask: do finite systems exhibit quantum phase transition? To address this question, finite-size scaling for finite systems was developed to calculate the quantum critical parameters. The recent observation of a quantum phase transition in a single trapped 171 Yb+ ion indicates the possibility of quantum phase transitions in finite systems. This perspective focuses on examining chemical processes at ultra-cold temperatures, as quantum phase transitions-particularly the formation and dissociation of chemical bonds-are the basic processes for understanding the whole of chemistry.

Universal properties of dissipative Tomonaga-Luttinger liquids: Case study of a non-Hermitian XXZ spin chain

We demonstrate the universal properties of dissipative Tomonaga-Luttinger (TL) liquids by calculating correlation functions and performing finite-size scaling analysis of a non-Hermitian XXZ spin chain as a prototypical model in one-dimensional open quantum many-body systems. Our analytic calculation is based on effective field theory with bosonization, finite-size scaling approach in conformal field theory, and the Bethe-ansatz solution. Our numerical analysis is based on the density-matrix renormalization group generalized to non-Hermitian systems (NH-DMRG). We uncover that the model in the massless regime with weak dissipation belongs to the universality class characterized by the complex-valued TL parameter, which is related to the complex generalization of the $c=1$ conformal field theory. As the dissipation strength increases, the values of the TL parameter obtained by the NH-DMRG begin to deviate from those obtained by the Bethe-ansatz analysis, indicating that the model becomes massive for strong dissipation. Our results can be tested with the two-component Bose-Hubbard system of ultracold atoms subject to two-body loss.

Конечно-размерный скейлинг в ферромагнитных спиновых системах на решетке пирохлора

In this paper we present the results of the high-performance computations for the Ising model, the XY-model and the classical Heisenberg model for the pyrochlore lattice. We used Wolff and Swendsen-Wang cluster algorithms with GPU parallelization for the calculations. We obtained critical exponents and critical temperatures using finite-size scaling approach.

Geometry-induced entanglement in a mass-imbalanced few-fermion system

Many-body systems undergoing quantum phase transitions reveal substantial growth of non-classical correlations between different parties of the system. This behavior is manifested by characteristic divergences of the von Neumann entropy. Here we show, that very similar features may be observed in one-dimensional systems of a few strongly interacting atoms when the structural transitions between different spatial orderings are driven by a varying shape of an external potential. When the appropriate adaptation of the finite-size scaling approach is performed in the vicinity of the transition point, few-fermion systems display a characteristic power-law invariance of divergent quantities.

What does the instantaneous normal mode spectrum tell us about dynamical heterogeneity in glass-forming fluids?

We examine the instantaneous normal mode spectrum of model metallic and polymeric glass-forming liquids. We focus on the localized modes in the unstable part of the spectrum [unstable localized (UL) modes] and find that the particles making the dominant contribution to the participation ratio form clusters that grow upon cooling in a fashion similar to the dynamical heterogeneity in glass-forming fluids, i.e., highly mobile (or immobile) particles form clusters that grow upon cooling; however, a comparison of the UL mode clusters to the mobile and immobile particle clusters indicates that they are distinct entities. We also show that the cluster size provides an alternate method to distinguish localized and delocalized modes, offering a significant practical advantage over the finite-size scaling approach. We examine the trajectories of particles contributing most to the UL modes and find that they have a slightly enhanced mobility compared to the average, and we determine a characteristic time quantifying the persistence time of this excess mobility. This time scale is proportional to the structural relaxation time τα of the fluid, consistent with a prediction by Zwanzig [Phys. Rev. 156, 190 (1967)] for the lifetime of collective excitations in cooled liquids. Evidently, these collective excitations serve to facilitate relaxation but do not actually participate in the motion associated with barrier crossing events governing activated transport. They also serve as a possible concrete realization of the "facilitation" clusters postulated in previous modeling of glass-forming liquids.

Extracting critical exponents by finite-size scaling with convolutional neural networks

Machine learning has been successfully applied to identify phases and phase transitions in condensed matter systems. However, quantitative characterization of the critical fluctuations near phase transitions is lacking. In this study we propose a finite-size scaling approach based on a convolutional neural network and analyze the critical behavior of a quantum Hall plateau transition. The localization length critical exponent learned by the neural network is consistent with the value obtained by conventional approaches. We show that the general-purposed method can be used to extract critical exponents in models with drastically different physics and input data, such as the two-dimensional Ising model and 4-state Potts model.

The droplet formation-dissolution transition in different ensembles: Finite-size scaling from two perspectives

The formation and dissolution of a droplet is an important mechanism related to various nucleation phenomena. Here, we address the droplet formation-dissolution transition in a two-dimensional Lennard-Jones gas to demonstrate a consistent finite-size scaling approach from two perspectives using orthogonal control parameters. For the canonical ensemble, this means that we fix the temperature while varying the density and vice versa. Using specialised parallel multicanonical methods for both cases, we confirm analytical predictions at fixed temperature (rigorously only proven for lattice systems) and corresponding scaling predictions from expansions at fixed density. Importantly, our methodological approach provides us with reference quantities from the grand canonical ensemble that enter the analytical predictions. Our orthogonal finite-size scaling setup can be exploited for theoretical and experimental investigations of general nucleation phenomena – if one identifies the corresponding reference ensemble and adapts the theory accordingly. In this case, our numerical approach can be readily translated to the corresponding ensembles and thereby proves very useful for numerical studies of equilibrium cluster formation, in general.

Casimir amplitudes in topological quantum phase transitions

Topological phase transitions constitute a new class of quantum critical phenomena. They cannot be described within the usual framework of the Landau theory since, in general, the different phases cannot be distinguished by an order parameter, neither can they be related to different symmetries. In most cases, however, one can identify a diverging length at these topological transitions. This allows us to describe them using a scaling approach and to introduce a set of critical exponents that characterize their universality class. Here we consider some relevant models of quantum topological transitions associated with well-defined critical exponents that are related by a quantum hyperscaling relation. We extend to these models a finite-size scaling approach based on techniques for calculating the Casimir force in electromagnetism. This procedure allows us to obtain universal Casimir amplitudes at their quantum critical points. Our results verify the validity of finite-size scaling in these systems and confirm the values of the critical exponents obtained previously.

Finite-Size Scaling Analysis of the Conductivity of Dirac Electrons on a Surface of Disordered Topological Insulators

Two-dimensional (2D) massless Dirac electrons appear on a surface of three-dimensional topological insulators. The conductivity of such a 2D Dirac electron system is studied for strong topological insulators in the case of the Fermi level being located at the Dirac point. The average conductivity $\langle\sigma\rangle$ is numerically calculated for a system of length $L$ and width $W$ under the periodic or antiperiodic boundary condition in the transverse direction, and its behavior is analyzed by applying a finite-size scaling approach. It is shown that $\langle\sigma\rangle$ is minimized at the clean limit, where it becomes scale-invariant and depends only on $L/W$ and the boundary condition. It is also shown that once disorder is introduced, $\langle\sigma\rangle$ monotonically increases with increasing $L$. Hence, the system becomes a perfect metal in the limit of $L \to \infty$ except at the clean limit, which should be identified as an unstable fixed point. Although the scaling curve of $\langle\sigma\rangle$ strongly depends on $L/W$ and the boundary condition near the unstable fixed point, it becomes almost independent of them with increasing $\langle\sigma\rangle$, implying that it asymptotically obeys a universal law.

Analysis of localization-delocalization transitions in corner-sharing tetrahedral lattices

We study the critical behavior of the Anderson localization-delocalization transition in corner-sharing tetrahedral lattices. We compare our results obtained by three different numerical methods namely the multifractal analysis, the Green resolvent method, and the energy-level statistics which yield the singularity strength, the decay length of the wave functions, and the (integrated) energy-level distribution, respectively. From these measures a finite-size scaling approach allows us to determine the critical parameters simultaneously. With particular emphasis we calculate the propagation of the statistical errors by a Monte-Carlo method. We find a high agreement between the results of all methods and we can estimate the highest critical disorder $W_\mathrm{c}=14.474(8)$ at energy $E_\mathrm{c}=-4.0$ and the critical exponent $\nu=1.565(11)$. Our results agree with a previous study by Fazileh et al. but improve accuracy significantly.

Critical properties of short-range Ising spin glasses on a Wheatstone-bridge hierarchical lattice.

An Ising spin-glass model with nearest-neighbor interactions, following a symmetric probability distribution, is investigated on a hierarchical lattice of the Wheatstone-bridge family characterized by a fractal dimension D≈3.58. The interaction distribution considered is a stretched exponential, which has been shown recently to be very close to the fixed-point coupling distribution, and such a model has been considered lately as a good approach for Ising spin glasses on a cubic lattice. An exact recursion procedure is implemented for calculating site magnetizations, mi=〈Si〉T, as well as correlations between pairs of nearest-neighbor spins, 〈SiSj〉T (〈〉T denote thermal averages), for a given set of interaction couplings on this lattice. From these local magnetizations and correlations, one can compute important physical quantities, such as the Edwards-Anderson order parameter, the internal energy, and the specific heat. Considering extrapolations to the thermodynamic limit for the order parameter, such as a finite-size scaling approach, it is possible to obtain directly the critical temperature and critical exponents. The transition between the spin-glass and paramagnetic phases is analyzed, and the associated critical exponents β and ν are estimated as β=0.82(5) and ν=2.50(4), which are in good agreement with the most recent results from extensive numerical simulations on a cubic lattice. Since these critical exponents were obtained from a fixed-point distribution, they are universal, i.e., valid for any coupling distribution considered.

Beyond the percolation universality class: the vertex split model for tetravalent lattices

We propose a statistical model defined on tetravalent three-dimensional lattices in general and the three-dimensional diamond network in particular where the splitting of randomly selected nodes leads to a spatially disordered network, with decreasing degree of connectivity. The terminal state, that is reached when all nodes have been split, is a dense configuration of self-avoiding walks on the diamond network. Starting from the crystallographic diamond network, each of the four-coordinated nodes is replaced with probability p by a pair of two edges, each connecting a pair of the adjacent vertices. For all values the network percolates, yet the fraction fp of the system that belongs to a percolating cluster drops sharply at pc = 1 to a finite value . This transition is reminiscent of a percolation transition yet with distinct differences to standard percolation behaviour, including a finite mass of the percolating clusters at the critical point. Application of finite size scaling approach for standard percolation yields scaling exponents for that are different from the critical exponents of the second-order phase transition of standard percolation models. This transition significantly affects the mechanical properties of linear-elastic realizations (e.g. as custom-fabricated models for artificial bone scaffolds), obtained by replacing edges with solid circular struts to give an effective density ϕ. Finite element methods demonstrate that, as a low-density cellular structure, the bulk modulus K shows a cross-over from a compression-dominated behaviour, with , at p = 0 to a bending-dominated behaviour with at p = 1.

Percolation Theory Generates a Physically Based Description of Tortuosity in Saturated and Unsaturated Porous Media

Tortuosity is a property of porous media that is invoked and used in the literature on hydrology, soil science, physics, and engineering. It has been defined in a variety of ways, one of which is a purely geometrical concept. In this study, we focused on the geometrical tortuosity and developed a model based on percolation theory and the finite-size scaling approach. Our result, developed for porous media of any saturation, expresses the tortuosity as a power-law function of the water content, the critical water content, and the system size. The model parameters include the fractal dimension of either the backbone (the flow-carrying part of porous media) or the optimal path (for crossing the system between two opposite faces), which have clear physical meaning. The results may be combined with power laws, which percolation theory provides for the hydraulic or electrical conductivity, to develop the appropriate form that expresses the connectivity as a function of the water content of a pore space. Comparison with numerical simulations and experiments reported in the literature indicated that the model estimates the tortuosity accurately. Our model and results leads us to conclude that pore connectivity and tortuosity should be treated as two distinct properties and that significant uncertainty in results for the saturation dependence of the hydraulic conductivity is traceable to the confusion that mixes the two as a single concept.

Non-mean-field behavior of critical wetting transition for short-range forces

Critical wetting transition for short-range forces in three dimensions (d=3) is reinvestigated by means of Monte Carlo simulation. Using an anisotropic finite size scaling approach, as well as approaches that do not rely on finite size scaling, we show that the critical wetting transition shows clear deviation from mean-field behavior. We estimate that the effective critical exponent ν_{∥}^{eff}=1.76 ± 0.08 for J/kT=0.35 and ν_{∥}^{eff}=1.85 ± 0.07 for J/kT=0.25. These values are in accord with predictions of Parry et al. [Phys. Rev. Lett. 100, 136105 (2008)]. We also point out that the anisotropic finite size scaling approach in d=3 requires additional modification in order to reach full consistency of simulational results.

An allometry-based approach for understanding forest structure, predicting tree-size distribution and assessing the degree of disturbance

Tree-size distribution is one of the most investigated subjects in plant population biology. The forestry literature reports that tree-size distribution trajectories vary across different stands and/or species, whereas the metabolic scaling theory suggests that the tree number scales universally as −2 power of diameter. Here, we propose a simple functional scaling model in which these two opposing results are reconciled. Basic principles related to crown shape, energy optimization and the finite-size scaling approach were used to define a set of relationships based on a single parameter that allows us to predict the slope of the tree-size distributions in a steady-state condition. We tested the model predictions on four temperate mountain forests. Plots (4 ha each, fully mapped) were selected with different degrees of human disturbance (semi-natural stands versus formerly managed). Results showed that the size distribution range successfully fitted by the model is related to the degree of forest disturbance: in semi-natural forests the range is wide, whereas in formerly managed forests, the agreement with the model is confined to a very restricted range. We argue that simple allometric relationships, at an individual level, shape the structure of the whole forest community.

Spin glasses in a field: Three and four dimensions as seen from one space dimension

We study the existence of a line of transitions of an Ising spin glass in a magnetic field-known as the de Almeida-Thouless line-using one-dimensional power-law diluted Ising spin-glass models. We choose the power-law exponent to have values that approximately correspond to three- and four-dimensional nearest-neighbor systems and perform a detailed finite-size scaling analysis of the data for large linear system sizes, both using a new approach proposed recently [Phys. Rev. Lett. 103, 267201 (2009)], as well as traditional approaches. Our results for the model corresponding to a three-dimensional system are consistent with there being no de Almeida-Thouless line, although the new finite-size scaling approach does not rule one out. For the model corresponding to four space dimensions, the new and traditional finite-size scaling analyses give conflicting results, indicating the need for a better understanding of finite-size scaling of spin glasses in a magnetic field.

Stabilizing Topological Phases in Graphene via Random Adsorption

We study the possibility of realizing topological phases in graphene with randomly distributed adsorbates. When graphene is subjected to periodically distributed adatoms, the enhanced spin-orbit couplings can result in various topological phases. However, at certain adatom coverages, the intervalley scattering renders the system a trivial insulator. By employing a finite-size scaling approach and Landauer-Büttiker formula, we show that the randomization of adatom distribution greatly weakens the intervalley scattering, but plays a negligible role in spin-orbit couplings. Consequently, such a randomization turns graphene from a trivial insulator into a topological state.

Finite-Size Scaling Approach for Critical Wetting: Rationalization in Terms of a Bulk Transition with an Order Parameter Exponent Equal to Zero

Clarification of critical wetting with short-range forces by simulations has been hampered by the lack of accurate methods to locate where the transition occurs. We solve this problem by developing an anisotropic finite-size scaling approach and show that then the wetting transition is a "bulk" critical phenomenon with order parameter exponent equal to zero. For the Ising model in two dimensions, known exact results are straightforwardly reproduced. In three dimensions, it is shown that previous estimates for the location of the transition need revision, but the conclusions about a slow crossover away from mean-field behavior remain unaltered.

Quantum Critical Dynamics Simulation of Dirty Boson Systems

Recently, the scaling result z=d for the dynamic critical exponent at the Bose glass to superfluid quantum phase transition has been questioned both on theoretical and numerical grounds. This motivates a careful evaluation of the critical exponents in order to determine the actual value of z. We study a model of quantum bosons at T=0 with disorder in 2D using highly effective worm MonteCarlo simulations. Our data analysis is based on a finite-size scaling approach to determine the scaling of the quantum correlation time from simulation data for boson world lines. The resulting critical exponents are z=1.8±0.05, ν=1.15±0.03, and η=-0.3±0.1, hence suggesting that z=2 is not satisfied.

Finite-size scaling study of the vapor-liquid critical properties of confined fluids: Crossover from three dimensions to two dimensions

We perform histogram-reweighting grand canonical Monte Carlo simulations of the Lennard-Jones fluid confined between two parallel hard walls and determine the vapor-liquid critical and coexistence properties in the range of sigma<or=H<or=6sigma and 10sigma<or=L(x),L(y)<or=28sigma, where H is the wall separation, L(x)=L(y) is the system size and sigma is the characteristic length. By matching the probability distribution of the ordering operator, P(M), to the three-dimensional (3D) and two-dimensional (2D) Ising universality classes according to the mixed-field finite-size scaling approach, we establish a "phase diagram" in the (H,L) plane, showing the boundary between four types of behavior: 3D, quasi-3D, quasi-2D, and 2D. In order to facilitate 2D critical point calculation, we present a four-parameter analytical expression for the 2D Ising universal distribution. We show that the infinite-system-size critical points obtained by extrapolation from the apparent 3D and 2D critical points have only minor differences with each other. In agreement with recent reports in the literature [Jana et al., J. Chem. Phys. 130, 214707 (2009)], we find departure from linearity in the relationship between critical temperature and inverse wall separation, as well as nonmonotonic dependence of the critical density and the liquid density at coexistence upon wall separation. Additional studies of the ST2 model of water show similar behavior, which suggests that these are quite general properties of confined fluids.