Finite-size Scaling Method Research Articles

Tricritical wetting in the two-dimensional Ising magnet due to the presence of localized non-magnetic impurities

Fixed vacancies (non-magnetic impurities) are placed along the centre of Ising strips in order to study the wetting behaviour in this confined system, by means of numerical simulations analysed with the aid of finite size scaling and thermodynamic integration methods. By considering strips of size () where short-range competitive surface fields () act along the M-direction, we observe localization–delocalization transitions of the interface between magnetic domains of different orientation (driven by the corresponding surface fields), which are the precursors of the wetting transitions that occur in the thermodynamic limit. By placing vacancies or equivalently non-magnetic impurities along the centre of the sample, we found that for low vacancy densities the wetting transitions are of second order, while by increasing the concentration of vacancies the transitions become of first order. Second- and first-order lines meet in tricritical wetting points (), where and are the magnitude of the surface field and the temperature, respectively. In the phase diagram, tricritical points shift from the high temperature and weak surface field regime at large vacancy densities to the , limit for low vacancy densities. By comparing the locations of the tricritical points with those corresponding to the case of mobile impurities, we conclude that in order to observe similar effects, in the latter the required density of impurities is much smaller (e.g. by a factor 3–5). Furthermore, a proper density of non magnetic impurities placed along the centre of a strip can effectively pin rather flat magnetic interfaces for suitable values of the competing surface fields and temperature.

The Yang-Mills gradient flow and SU(3) gauge theory with 12 massless fundamental fermions in a colour-twisted box

We perform the step-scaling investigation of the running coupling constant, using the gradient-flow scheme, in SU(3) gauge theory with twelve massless fermions in the fundamental representation. The Wilson plaquette gauge action and massless unimproved staggered fermions are used in the simulations. Our lattice data are prepared at high accuracy, such that the statistical error for the renormalised coupling, g_GF, is at the subpercentage level. To investigate the reliability of the continuum extrapolation, we employ two different lattice discretisations to obtain g_GF. For our simulation setting, the corresponding gauge-field averaging radius in the gradient flow has to be almost half of the lattice size, in order to have this extrapolation under control. We can determine the renormalisation group evolution of the coupling up to g^2_GF ~ 6, before the onset of the bulk phase structure. In this infrared regime, the running of the coupling is significantly slower than the two-loop perturbative prediction, although we cannot draw definite conclusion regarding possible infrared conformality of this theory. Furthermore, we comment on the issue regarding the continuum extrapolation near an infrared fixed point. In addition to adopting the fit ansatz a'la Symanzik for performing this task, we discuss a possible alternative procedure inspired by properties derived from low-energy scale invariance at strong coupling. Based on this procedure, we propose a finite-size scaling method for the renormalised coupling as a means to search for infrared fixed point. Using this method, it can be shown that the behaviour of the theory around g^2_GF ~ 6 is still not governed by possible infrared conformality.

Stochastic approximation of dynamical exponent at quantum critical point

We have developed a unified finite-size scaling method for quantum phase transitions that requires no prior knowledge of the dynamical exponent $z$. During a quantum Monte Carlo simulation, the temperature is automatically tuned by the Robbins-Monro stochastic approximation method, being proportional to the lowest gap of the finite-size system. The dynamical exponent is estimated in a straightforward way from the system-size dependence of the temperature. As a demonstration of our novel method, the two-dimensional $S=1/2$ quantum $XY$ model in uniform and staggered magnetic fields is investigated in the combination of the world-line quantum Monte Carlo worm algorithm. In the absence of a uniform magnetic field, we obtain the fully consistent result with the Lorentz invariance at the quantum critical point, $z=1$, i.e., the three-dimensional classical $XY$ universality class. Under a finite uniform magnetic field, on the other hand, the dynamical exponent becomes two, and the mean-field universality with effective dimension $(2+2)$ governs the quantum phase transition.

Transfer-matrix study of a hard-square lattice gas with two kinds of particles and density anomaly

Using transfer matrix and finite-size scaling methods, we study the thermodynamic behavior of a lattice gas with two kinds of particles on the square lattice. Only excluded volume interactions are considered, so that the model is athermal. Large particles exclude the site they occupy and its four first neighbors, while small particles exclude only their site. Two thermodynamic phases are found: a disordered phase where large particles occupy both sublattices with the same probability and an ordered phase where one of the two sublattices is preferentially occupied by them. The transition between these phases is continuous at small concentrations of the small particles and discontinuous at larger concentrations, both transitions are separated by a tricritical point. Estimates of the central charge suggest that the critical line is in the Ising universality class, while the tricritical point has tricritical Ising (Blume-Emery-Griffiths) exponents. The isobaric curves of the total density as functions of the fugacity of small or large particles display a minimum in the disordered phase.

The Investigation of Critical Properties of the Transition from Modulated Phase into Paramagnetic Phase

The anisotropic Ising model with competing interactions in the region of transition from a modulated phase into paramagnetic state is investigated by the Monte-Carlo methods. By means of histogram analysis and the finite-size scaling method, the modulated – paramagnetic phase transition is shown to be a second order phase transition. Critical parameters and temperatures of phase transitions in this region are calculated.

Evaluation of Kirkwood-Buff integrals via finite size scaling: a large scale molecular dynamics study

Solvation of bio-molecules in water is severely affected by the presence of co-solvent within the hydration shell of the solute structure. Furthermore, since solute molecules can range from small molecules, such as methane, to very large protein structures, it is imperative to understand the detailed structure-function relationship on the microscopic level. For example, it is useful know the conformational transitions that occur in protein structures. Although such an understanding can be obtained through large-scale molecular dynamic simulations, it is often the case that such simulations would require excessively large simulation times. In this context, Kirkwood-Buff theory, which connects the microscopic pair-wise molecular distributions to global thermodynamic properties, together with the recently developed technique, called finite size scaling, may provide a better method to reduce system sizes, and hence also the computational times. In this paper, we present molecular dynamics trial simulations of biologically relevant low-concentration solvents, solvated by aqueous co-solvent solutions. In particular we compare two different methods of calculating the relevant Kirkwood-Buff integrals. The first (traditional) method computes running integrals over the radial distribution functions, which must be obtained from large system-size NVT or NpT simulations. The second, newer method, employs finite size scaling to obtain the Kirkwood-Buff integrals directly by counting the particle number fluctuations in small, open sub-volumes embedded within a larger reservoir that can be well approximated by a much smaller simulation cell. In agreement with previous studies, which made a similar comparison for aqueous co-solvent solutions, without the additional solvent, we conclude that the finite size scaling method is also applicable to the present case, since it can produce computationally more efficient results which are equivalent to the more costly radial distribution function method.

Quantum phase transitions in a chain with two- and four-spin interactions in a transverse field.

We use entanglement entropy and finite-size scaling methods to investigate the ground-state properties of a spin-1/2 Ising chain with two-spin (J(2)) and four-spin (J(4)) interactions in a transverse magnetic field (B). We concentrate our study on the unexplored critical region B=1 and obtain the phase diagram of the model in the (J(4)-J(2)) plane. The phases found include ferromagnetic (F), antiferromagnetic (AF), as well as more complex phases involving spin configurations with multiple periodicity. The system presents both first- and second-order transitions separated by tricritical points. We find an unusual phase boundary on the semi-infinite segment (J(4)<-1,J(2)=0) separating the F and AF phases.

Critical and Tricritical Wetting in the Two-Dimensional Blume–Capel model

The two-dimensional Blume–Capel model with free surfaces where a surface field $$H_1$$ acts and the “crystal field” (controlling the density of the vacancies) takes a value $$D _s$$ different from the value $$D$$ in the bulk, is studied by Monte Carlo methods. Using a recently developed finite size scaling method that studies thin films in a $$L \times M$$ geometry with antisymmetric surface fields $$(H_L=-H_1)$$ and keeps a generalized aspect ratio $$c = L^2/M$$ constant, surface phase diagrams are computed for several typical choices of the parameters. It is shown that both second order and first order wetting transitions occur, separated by tricritical wetting behavior. The special role of vacancies near the surface is investigated in detail.

Numerical analysis of percolation cluster size distribution in two-dimensional and three-dimensional lattices

To investigate the statistical behavior in the sizes of finite clusters for percolation, cluster size distribution n s (p) for site and bond percolations at different lattices and dimensions was simulated using a modified algorithm. An equation to approximate the finite cluster size distribution n s (p) was obtained and expressed as: log (n s (p)) = as − b log s + c. Based on the analysis of simulation data, we found that the equation is valid for p from 0 to 1 on site and for the bond percolation of two-dimensional (2D) and three-dimensional (3D) lattices. Furthermore, the relationship between the coefficients of the equation and the occupied ratio p was studied using the finite-size scaling method. When $$x = D(p - p_c )L^{y_t }$$ , p < p c , and D was a nonuniversal metric factor. a was found to be related only to p, and the a-x curves of different lattices were nearly overlapped; b was related to the dimensions and p, and the scaled data of the b of all lattices with the same dimension tended to fall on the same curves. Unlike a and b, c apparently had a quadratic relation with x in 2D lattices and linear relation with x in 3D lattices. The results of this paper could significantly reduce the amount of tasks required to obtain numerical data of on the cluster size distribution for p from 0 to p c .

Combined Scaling of Fluid Flow and Seismic Stiffness in Single Fractures

The connection between fluid flow and seismic stiffness in single fractures is governed by the geometry of the fracture through the size and spatial distributions of the void and contact areas. Flow and stiffness each exhibit scaling behavior as the scale of observation shifts from local to global sample sizes. The purpose of this study was to explore the joint scaling of both properties using numerical models. Finite-size scaling methods are used to extract critical thresholds and power laws for fluid flow through weakly correlated fractures under increasing load. An important element in the numerical fracture deformation is the use of extended boundary conditions that simulate differences between laboratory cores relative to in situ field studies. The simulated field conditions enable joint scaling of flow and stiffness to emerge with the potential to extrapolate from small laboratory samples to behavior on the field scale.

Critical behavior of a quantum chain with four-spin interactions in the presence of longitudinal and transverse magnetic fields.

We study the ground-state properties of a spin-1/2 model on a chain containing four-spin Ising-like interactions in the presence of both transverse and longitudinal magnetic fields. We use entanglement entropy and finite-size scaling methods to obtain the phase diagrams of the model. Our numerical calculations reveal a rich variety of phases and the existence of multicritical points in the system. We identify phases with both ferromagnetic and antiferromagnetic orderings. We also find periodically modulated orderings formed by a cluster of like spins followed by another cluster of opposite like spins. The quantum phases in the model are found to be separated by either first- or second-order transition lines.

Finite-size scaling method for the Berezinskii–Kosterlitz–Thouless transition

We test an improved finite-size scaling method for reliably extracting the critical temperature TBKT of a Berezinskii–Kosterlitz–Thouless (BKT) transition. Using known single-parameter logarithmic corrections to the spin stiffness ρs at TBKT in combination with the Kosterlitz–Nelson relation between the transition temperature and the stiffness, ρs(TBKT) = 2TBKT/π, we define a size-dependent transition temperature TBKT(L1,L2) based on a pair of system sizes L1,L2, e.g., L2 = 2L1. We use Monte Carlo data for the standard two-dimensional classical XY model to demonstrate that this quantity is well behaved and can be reliably extrapolated to the thermodynamic limit using the next expected logarithmic correction beyond the ones included in defining TBKT(L1,L2). For the Monte Carlo calculations we use GPU (graphical processing unit) computing to obtain high-precision data for L up to 512. We find that the sub-leading logarithmic corrections have significant effects on the extrapolation. Our result TBKT = 0.8935(1) is several error bars above the previously best estimates of the transition temperature, TBKT ≈ 0.8929. If only the leading log-correction is used, the result is, however, consistent with the lower value, suggesting that previous works have underestimated TBKT because of the neglect of sub-leading logarithms. Our method is easy to implement in practice and should be applicable to generic BKT transitions.

Quantum criticality analysis by finite-size scaling and exponential basis sets

We combine the finite-size scaling method with the mesh-free spectral method to calculate quantum critical parameters for a given Hamiltonian. The basic idea is to expand the exact wave function in a finite exponential basis set and extrapolate the information about system criticality from a finite basis to the infinite basis set limit. The used exponential basis set, though chosen intuitively, allows handling a very wide range of exponential decay rates and calculating multiple eigenvalues simultaneously. As a benchmark system to illustrate the combined approach, we choose the Hulthen potential. The results show that the method is very accurate and converges faster when compared with other basis functions. The approach is general and can be extended to examine near-threshold phenomena for atomic and molecular systems based on even-tempered exponential and Gaussian basis functions.

Ground-state stability and criticality of two-electron atoms with screened Coulomb potentials using the B-splines basis set

We applied the finite-size scaling method using the B-splines basis set to construct the stability diagram for two-electron atoms with a screened Coulomb potential. The results of this method for two-electron atoms are very accurate in comparison with previous calculations based on Gaussian, Hylleraas and finite-element basis sets. The stability diagram for the screened two-electron atoms shows three distinct regions, i.e. a two-electron region, a one-electron region and a zero-electron region, which correspond to stable, ionized and double ionized atoms, respectively. In previous studies, it was difficult to extend the finite-size scaling calculations to large molecules and extended systems because of the computational cost and the lack of a simple way to increase the number of Gaussian basis elements in a systematic way. Motivated by recent studies showing how one can use B-splines to solve Hartree–Fock and Kohn–Sham equations, this combined finite-size scaling using the B-splines basis set might provide an effective systematic way to treat criticality of large molecules and extended systems. As benchmark calculations, the two-electron systems show the feasibility of this combined approach and provide an accurate reference for comparison.

Thermodynamic glass transition in a spin glass without time-reversal symmetry.

Spin glasses are a longstanding model for the sluggish dynamics that appear at the glass transition. However, spin glasses differ from structural glasses in a crucial feature: they enjoy a time reversal symmetry. This symmetry can be broken by applying an external magnetic field, but embarrassingly little is known about the critical behavior of a spin glass in a field. In this context, the space dimension is crucial. Simulations are easier to interpret in a large number of dimensions, but one must work below the upper critical dimension (i.e., in d<6) in order for results to have relevance for experiments. Here we show conclusive evidence for the presence of a phase transition in a four-dimensional spin glass in a field. Two ingredients were crucial for this achievement: massive numerical simulations were carried out on the Janus special-purpose computer, and a new and powerful finite-size scaling method.

Monte Carlo simulations of an unconventional phase transition for a two-dimensional dimerized quantum Heisenberg model

Motivated by the indication of a new critical theory for the spin-1/2 Heisenberg model with a spatially staggered anisotropy on the square lattice, we reinvestigate the phase transition of this model induced by dimerization using first-principle Monte Carlo simulations. We focus on studying the finite-size scaling of ${\ensuremath{\rho}}_{s1}2L$ and ${\ensuremath{\rho}}_{s2}2L$, where $L$ stands for the spatial box size used in the simulations and ${\ensuremath{\rho}}_{si}$, with $i\ensuremath{\in}{1,2}$, is the spin-stiffness in the $i$-direction. Remarkably, while we observe a large correction to scaling for the observable ${\ensuremath{\rho}}_{s1}2L$, the data for ${\ensuremath{\rho}}_{s2}2L$ exhibit a good scaling behavior without any indication of a large correction. As a consequence, we are able to obtain a numerical value for the critical exponent $\ensuremath{\nu}$, which is consistent with the known $O(3)$ result with moderate computational effort. Further, we additionally carry out an unconventional finite-size scaling analysis with which we assume that the ratio of the spatial winding numbers squared is fixed through all simulations. The theoretical correctness of our idea is argued and its validity is confirmed. Using this unconventional finite-size scaling method, even from ${\ensuremath{\rho}}_{s1}L$, which receives the most serious correction among the observables considered in this study, we are able to arrive at a value for $\ensuremath{\nu}$ consistent with the expected $O(3)$ value. A detailed investigation to compare these two finite-size scaling methods should be performed.

Effect of Substrate Thickness on Responsivity of Free-Membrane Bolometric Detectors

The effect of substrate thickness and its Debye temperature on the bolometric response of the freestanding-membrane type of superconductor transition-edge detectors is investigated. The investigation is based on the calculation of the variation of the specific heat per unit volume and the heat conductivity using the finite-size scaling method for different Debye temperatures and micron size thickness of substrate. We also calculated the heat capacity and time constant for various materials with Debye temperatures in the range of 600-1400 K. The calculated results are compared with previously reported response values obtained theoretically and experimentally for the thermal-based infrared (IR) detectors. The IR response versus substrate thickness of these types of devices for different modulation frequencies is also presented. This inspected response at low thicknesses of substrate shows anomalistic behavior with respect to the previously reported results of response analyses. Here, we also obtained the optimized substrate thickness and Debye temperatures for maximum responsivity of these types of devices.

Far-from-equilibrium growth of thin films in a temperature gradient

The irreversible growth of thin films under far-from-equilibrium conditions is studied in (2+1)-dimensional strip geometries. Across one of the transverse directions, a temperature gradient is applied by thermal baths at fixed temperatures between T(1) and T(2), where T(1)<T(c)(hom)<T(2) and T(c})(hom)=0.69(1) is the critical temperature of the system in contact with an homogeneous thermal bath. By using standard finite-size scaling methods, we characterized a continuous order-disorder phase transition driven by the thermal bath gradient with critical temperature T(c)=0.84(2) and critical exponents ν=1.53(6), γ=2.54(11), and β=0.26(8), which belong to a different universality class from that of films grown in an homogeneous bath. Furthermore, the effects of the temperature gradient are analyzed by means of a bond model that captures the growth dynamics. The interplay of geometry and thermal bath asymmetries leads to growth bond flux asymmetries and the onset of transverse ordering effects that explain qualitatively the shift in the critical temperature.

Diffusive domain coarsening: Early time dynamics and finite-size effects

We study the diffusive dynamics of phase separation in a symmetric binary (A + B) mixture with a 50:50 composition of A and B particles, following a quench below the demixing critical temperature, both in spatial dimensions d=2 and d=3. The particular focus of this work is to obtain information about the effects of system size and correction to the growth law via the appropriate application of the finite-size scaling method to the results obtained from the Kawasaki exchange Monte Carlo simulation of the Ising model. Observations of only weak size effects and a very small correction to scaling in the growth law are significant. The methods used in this work and information thus gathered will be useful in the study of the kinetics of phase separation in fluids and other problems of growing length scale. We also provide a detailed discussion of the standard methods of understanding simulation results which may lead to inappropriate conclusions.

Continuity of the explosive percolation transition

The explosive percolation problem on the complete graph is investigated via extensive numerical simulations. We obtain the cluster-size distribution at the moment when the cluster size heterogeneity becomes maximum. The distribution is found to be well described by the power-law form with the decay exponent τ=2.06(2), followed by a hump. We then use the finite-size scaling method to make all the distributions at various system sizes up to N=2(37) collapse perfectly onto a scaling curve characterized solely by the single exponent τ. We also observe that the instant of that collapse converges to a well-defined percolation threshold from below as N→∞. Based on these observations, we show that the explosive percolation transition in the model should be continuous, contrary to the widely spread belief of its discontinuity.