Finite-size Scaling Research Articles

Orientational phase transition in monolayers of multipolar straight rigid rods: The case of 2-thiophene molecule adsorption on the Au(111) surface.

Monte Carlo simulations and finite-size scaling theory have been carried out to study the critical behavior and universality for the isotropic-nematic (IN) phase transition in a system of straight rigid pentamers adsorbed on a triangular lattice with polarized nonhomogeneous intermolecular interactions. The model was inspired by the deposition of 2-thiophene molecules over the Au(111) surface, which was previously characterized by experimental techniques and density functional theory. A nematic phase, observed experimentally by the formation of a self-assembled monolayer of parallel molecules, is separated from the isotropic state by a continuous transition occurring at a finite density. The precise determination of the critical exponents indicates that the transition belongs to the three-state Potts universality class. The finite-size scaling analysis includes the study of mutability and diversity. These two quantities are derived from information theory and they have not previously been considered as part of the conventional treatment of critical phenomena for the determination of critical exponents. The results obtained here contribute to the understanding of formation processes of self-assembled monolayers, phase transitions, and critical phenomena from novel compression algorithms for studying mutual information in sequences of data.

The BKT transition and its dynamics in a spin fluid.

We study the effect of particle mobility on phase transitions in a spin fluid in two dimensions. The presence of a phase transition of the BKT universality class is shown in an off-lattice model of particles with purely repulsive interaction employing computer simulations. A critical spin wave region 0 < T < TBKT is found with a nonuniversal exponent η(T) that follows the shape suggested by BKT theory, including a critical value consistent with ηBKT = 1/4. One can observe a transition from power-law decay to exponential decay in the static correlation functions at the transition temperature TBKT, which is supported by finite-size scaling analysis. A critical temperature TBKT = 0.17(1) is suggested. Investigations into the dynamic aspects of the phase transition are carried out. The short-time behavior of the incoherent spin autocorrelation function agrees with the Nelson-Fisher prediction, whereas the long-time behavior differs from the finite-size scaling known for the static XY model. Analysis of coherent spin wave dynamics shows that the spin wave peak is a propagating mode that can be reasonably well fitted by hydrodynamic theory. The mobility of the particles strongly enhances damping of the spin waves, but the model still lies within the dynamic universality class of the standard XY model.

Anderson localization transitions in disordered non-Hermitian systems with exceptional points

The critical exponents of continuous phase transitions of a Hermitian system depend on and only on its dimensionality and symmetries. This is the celebrated notion of the universality of continuous phase transitions. Here we numerically study the Anderson localization transitions in non-Hermitian two-dimensional (2D) systems with exceptional points by using the finite-size scaling analysis of the participation ratios. At the exceptional points of either second order or fourth order, two non-Hermitian systems with different symmetries have the same critical exponent $\ensuremath{\nu}\ensuremath{\simeq}2$ of correlation lengths, which differs from all known 2D disordered Hermitian and non-Hermitian systems. These feature is reminiscent of the superuniversality notion of Anderson localization transitions. In the symmetry-preserved and symmetry-broken phases, the non-Hermitian models with time-reversal symmetry and without spin-rotational symmetry, and without both time-reversal and spin-rotational symmetries, are in the same universality class of 2D Hermitian electron systems of Gaussian symplectic and unitary ensembles, where $\ensuremath{\nu}\ensuremath{\simeq}2.7$ and $\ensuremath{\nu}\ensuremath{\simeq}2.3$, respectively. The universality of the transition is further confirmed by showing that the critical exponent $\ensuremath{\nu}$ does not depend on the form of disorders and boundary conditions.

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.

Cubic fixed point in three dimensions: Monte Carlo simulations of the ϕ4 model on the simple cubic lattice

We study the cubic fixed point for $N=3$ and $4$ by using finite size scaling applied to data obtained from Monte Carlo simulations of the $N$-component $\phi^4$ model on the simple cubic lattice. We generalize the idea of improved models to a two-parameter family of models. The two-parameter space is scanned for the point, where the amplitudes of the two leading corrections to scaling vanish. To this end, a dimensionless quantity is introduced that monitors the breaking of the $O(N)$-invariance. For $N=4$, we determine the correction exponents $\omega_1=0.763(24)$ and $\omega_2=0.082(5)$. In the case of $N=3$, we obtain $Y_4=0.0142(6)$ for the RG-exponent of the cubic perturbation at the $O(3)$-invariant fixed point, while the correction exponent $\omega_2=0.0133(8)$ at the cubic fixed point. Simulations close to the improved point result in the estimates $\nu=0.7202(7)$ and $\eta=0.0371(2)$ of the critical exponents of the cubic fixed point for $N=4$. For $N=3$, at the cubic fixed point, the $O(3)$-symmetry is only mildly broken and the critical exponents differ only by little from those of the $O(3)$-invariant fixed point. We find $-0.00001 \lessapprox \eta_{cubic}- \eta_{O(3)} \lessapprox 0.00007$ and $\nu_{cubic}-\nu_{O(3)} =-0.00061(10)$.

Anomalous finite-size scaling in higher-order processes with absorbing states.

Here we study standard and higher-order birth-death processes on fully connected networks, within the perspective of large-deviation theory [also referred to as the Wentzel-Kramers-Brillouin (WKB) method in some contexts]. We obtain a general expression for the leading and next-to-leading terms of the stationary probability distribution of the fraction of "active" sites as a function of parameters and network size N. We reproduce several results from the literature and, in particular, we derive all the moments of the stationary distribution for the q-susceptible-infected-susceptible (q-SIS) model, i.e., a high-order epidemic model requiring q active ("infected") sites to activate an additional one. We uncover a very rich scenario for the fluctuations of the fraction of active sites, with nontrivial finite-size-scaling properties. In particular, we show that the variance-to-mean ratio diverges at criticality for [1≤q≤3], with a maximal variability at q=2, confirming that complex-contagion processes can exhibit peculiar scaling features including wild variability. Moreover, the leading order in a large-deviation approach does not suffice to describe them: next-to-leading terms are essential to capture the intrinsic singularity at the origin of systems with absorbing states. Some possible extensions of this work are also discussed.

Magnetic properties of two dimensional V 2 O 3 monolayer

Two dimensional (2D) magnetic materials with unique properties offer an ideal platform to study magnetic phase transitions in the low-dimensional systems. In this study, we implement extensive Monte Carlo simulations to investigate the magnetic properties of the 2D V 2 O 3 monolayer. By using finite-size scaling tools, we obtain the critical temperature and critical exponents of the monolayer precisely. Our numerical results demonstrate that 2D V 2 O 3 monolayer belongs to the two-dimensional Ising universality class. Besides, we examine the hysteresis characteristics of the structure and show that V 2 O 3 monolayer is a hard ferromagnetic material with a high coercive field at room temperature.

The fifty-year quest for universality in percolation theory in high dimensions

Although well described by mean-field theory in the thermodynamic limit, scaling has long been puzzling for finite systems in high dimensions. This raised questions about the efficacy of the renormalization group and foundational concepts such as universality, finite-size scaling and hyperscaling, until recently believed not to be applicable above the upper critical dimension. Significant theoretical progress has been made resolving these issues, and tested in numerous simulational studies of spin models. This progress rests upon superlinearity of correlation length, a notion that for a long time encountered resistance but is now broadly accepted. Percolation theory brings added complications such as proliferation of interpenetrating clusters in apparent conflict with suggestions coming from random-graph asymptotics and a dearth of reliable simulational guidance. Here we report on recent theoretical progress in percolation theory in the renormalization group framework in high dimensions that accommodates superlinear correlation and renders most of the above concepts mutually compatible under different boundary conditions. Results from numerical simulations for free and periodic boundary conditions which differentiate between previously competing theories are also presented. Although still fragmentary, these Monte Carlo results support the new framework which restores the renormalization group and foundational concepts on which it rests.

Scaling properties of scale-free networks in degree-thresholding renormalization flows

We study the statistical properties of observables of scale-free networks in degree-thresholding renormalization (DTR) flows. For Barabási-Albert (BA) scale-free networks with different sizes, we find that their structural and dynamical observables have similar scaling behavior in the DTR flow. The finite-size scaling analysis confirms this view and reveals a scaling function with a single scaling exponent that collectively captures the changes of these observables. Furthermore, for the scale-free network with a single initial size, we use its DTR snapshots as the original networks in the DTR flows, then perform a similar finite-size scaling analysis. Interestingly, the initial network and its snapshots share the same scaling exponent as the BA synthetic network. Our findings have important guiding significance for analyzing the structure and dynamic behavior of large-scale networks. Such as, in large-scale simulation scenarios with high time complexity, the DTR snapshot could serve as a substitute or guide for the initial network and then quickly explore the scaling behavior of initial networks.

Non-equilibrium relaxations: ageing and finite-size effects

The long-time behaviour of spin-spin correlators in the slow relaxation of systems undergoing phase-ordering kinetics is studied in geometries of finite size. A phenomenological finite-size scaling ansatz is formulated and tested through the exact solution of the kinetic spherical model, quenched to below the critical temperature, in 2&lt; d &lt; 4 dimensions.

Percolation threshold and electrical conductivity of conductive polymer composites filled with curved fibers in two-dimensional space.

Quantifying the influence of fiber curvature on the percolation behavior of flexible conductive fiber and further on the electrical conductivity of conductive polymer composites (CPCs) is crucial for the design of CPCs. This study considers CPCs as a random packing of soft curved discorectangles (CDCRs) in a polymer matrix. The geometry of CDCR is developed, and an inter-CDCR contact detection algorithm is used to generate a random packing structure of CDCRs. The effects of aspect ratio α and bending central angles θ of CDCR on the percolation threshold ϕc of the overlapped CDCR system in a two-dimensional plane are then investigated using the finite-size scaling method. The result reveals that ϕc decreases monotonically as α grows and increases monotonically as θ rises. A shape-independent power law formula, denoted as ϕc = 2.2015 A-0.8172dex is developed to quantify the relationship between the Adex and ϕc. A comparison of our numerical simulations, published data, and predictions verifies the reliability and universality of the fitting model. Subsequently, a resistor network searching algorithm (RNSA) is proposed to construct the random resistor network model (RRNM). A power law model, denoted as is developed to evaluate the effects of the normalized reduced density (η - ηc)/ηc on the effective conductivity σeff of CPC. Comparing our predictions with data from the literature and our simulation verifies the reliability of our RNSA and the fitting model. This paper's methodology and findings may provide a theoretical hint for the CPC's design.

Scaling behaviour under the influence of a homogeneous size-dependent perturbation

We study the finite-size scaling behaviour at the critical point, resulting from the addition of a homogeneous size-dependent perturbation, decaying as an inverse power of the system size. The scaling theory is first formulated in a general framework and then illustrated using three concrete problems for which exact results are obtained.

Finite-size scaling of the majority-voter model above the upper critical dimension

The majority-voter model is studied by Monte Carlo simulations on hypercubic lattices of dimension d = 2 to 7 with periodic boundary conditions. The critical exponents associated to the finite-size scaling of the magnetic susceptibility are shown to be compatible with those of the Ising model. At dimension d = 4, the numerical data are compatible with the presence of multiplicative logarithmic corrections. For d ≥ 5, the estimates of the exponents are close to the prediction d/2 when taking into account the dangerous irrelevant variable at the Gaussian fixed point. Moreover, the universal values of the Binder cumulant are also compatible with those of the Ising model. This indicates that the upper critical dimension of the majority-voter model is not dc = 6 as claimed in the literature, but dc = 4 like the equilibrium Ising model.

The external field effect on the opinion formation based on the majority rule and the q-voter models on the complete graph

We investigate the external field effect on opinion formation based on the majority rule and [Formula: see text]-voter models on a complete graph. The external field can be considered as the mass media in the social system, with the probability [Formula: see text] agents following the mass media opinion. Based on our Monte Carlo simulation, the mass media effect is not strong enough to make the system reach a homogeneous state (complete consensus) with the magnetization [Formula: see text] for all values of [Formula: see text], indicating the existence of a usual phase transition for all values of [Formula: see text]. In the [Formula: see text]-voter model, the mass media eliminates the usual phase transition at [Formula: see text]. We obtain the model’s critical point and scaling parameters using the finite-size scaling analysis and obtain that both models have the same scaling parameters. The external field effect decreases both models’ relaxation time and the relaxation time following the power-law relation such as [Formula: see text], where [Formula: see text] is the population size and [Formula: see text] depends on the probability [Formula: see text]. In the majority rule model, [Formula: see text] follows a linear relation, and in the q-voter model, [Formula: see text] follows a power-law relation.

Diffusivity of porous media considering the coupling influence of pore shape- and size-polydispersities on the percolation: Theoretical and numerical studies

Aggressive media in the environment will penetrate into the porous media through the connected pore channels, causing damage to the microstructure and reducing the working performance. In statistical physics, the connectivity of porous network is usually described by the concept of percolation. A critical yet unresolved subject has been how to adequately capture the percolation thresholds of these complex porous network including polyshaped–polysized pores, and their quantitative implications on the transport property of porous media. This paper presents theoretical and numerical approaches for precisely determining the effects of pore shape- and size-polydispersities on percolation thresholds and diffusivity of porous media. Combining the Monte Carlo simulation with finite-size scaling analysis, the statistical values of percolation thresholds are obtained. Incorporating the proposed pore size-polydispersity degree with the excluded volume, the influences of pore shape- and size-polydispersities on the percolation thresholds are characterized. Substituting the percolation thresholds into generalized effective medium theory, the diffusivities of porous media are theoretically predicted. Moreover, the lattice Boltzmann method is employed to numerically calculate the diffusivity. Comparison with the theoretical, numerical and experimental results reveals the present model can accurately predict the percolation thresholds and diffusivities of porous media. The results indicate that the enhancing of the pore excluded volume and the decline of the pore size-polydispersity degree will decrease the percolation threshold and increase the diffusivity. This work can provide novel insights into understanding the complex interactions between the composition (pore shape- and size-polydispersities), microstructure (percolation threshold), and macro-property (diffusivity) of porous media.

Finite-Size Scaling Theory at a Self-Dual Quantum Critical Point

The nondivergence of the generalized Grüneisen ratio (GR) at a quantum critical point (QCP) has been proposed to be a universal thermodynamic signature of self-duality. We study how the Kramers–Wannier-type self-duality manifests itself in the finite-size scaling behavior of thermodynamic quantities in the quantum critical regime. While the self-duality cannot be realized as a unitary transformation in the total Hilbert space for the Hamiltonian with the periodic boundary condition, it can be implemented in certain symmetry sectors with proper boundary conditions. Therefore, the GR and the transverse magnetization of the one-dimensional transverse-field Ising model exhibit different finite-size scaling behaviors in different sectors. This implies that the numerical diagnosis of self-dual QCP requires identification of the proper symmetry sectors.

Topological random fractals

The search for novel topological quantum states has recently moved beyond naturally occurring crystalline materials to complex and engineered systems. In this work we generalize the notion of topological electronic states to random lattices in non-integer dimensions. By considering a class D tight-binding model on critical clusters resulting from a two-dimensional site percolation process, we demonstrate that these topological random fractals exhibit the hallmarks of topological insulators. Specifically, our large-scale numerical studies reveal that topological random fractals display a robust mobility gap, support quantized conductance and represent a well-defined thermodynamic phase of matter. The finite-size scaling analysis further suggests that the critical properties are not consistent with the expectations of class D systems in two dimensions, hinting to the nontrivial relationship between fractal and integer-dimensional topological states. Our results establish topological random fractals as the most complex systems known to support nontrivial band topology with their distinct unique properties.

Ionic liquid gating-tuned superconductor–insulator transition in PrBa2Cu3O7/La2CuO4 bilayers

Recently, high-temperature superconductivity has been found at the interface between two Mott insulators, PrBa2Cu3O7 and La2CuO4. In this work, we demonstrate our tuning of this interface superconductor by ionic liquid gating. By varying gate voltage, VG, a superconductor–insulator phase transition was achieved. Hall-effect measurements suggest that the carrier density, which can be tuned continuously from 1.86 × 1014 to 3.78 × 1014 cm−2 for VG varying from −0.3 to −3 V, is the tuning parameter. A finite-size scaling analysis was used to analyze the superconductor–insulator phase transition, which gives a critical resistance of 14–17 kΩ and a critical exponent of zν ≈ 1.3–1.4. These values are comparable with those obtained in previous studies on ultrathin clean cuprate films gated by ionic liquid.

Taming pseudofermion functional renormalization for quantum spins: Finite temperatures and the Popov-Fedotov trick

The pseudo-fermion representation for $S=1/2$ quantum spins introduces unphysical states in the Hilbert space which can be projected out using the Popov-Fedotov trick. However, state-of-the-art implementation of the functional renormalization group method for pseudo-fermions have so far omitted the Popov-Fedotov projection. Instead, restrictions to zero temperature were made and absence of unphysical contributions to the ground-state was assumed. We question this belief by exact diagonalization of several small-system counterexamples where unphysical states do contribute to the ground state. We then introduce Popov-Fedotov projection to pseudo-fermion functional renormalization, enabling finite temperature computations with only minor technical modifications to the method. At large and intermediate temperatures, our results are perturbatively controlled and we confirm their accuracy in benchmark calculations. At lower temperatures, the accuracy degrades due to truncation errors in the hierarchy of flow equations. Interestingly, these problems cannot be alleviated by switching to the parquet approximation. We introduce the spin projection as a method-intrinsic quality check. We also show that finite temperature magnetic ordering transitions can be studied via finite-size scaling.

Quantum extraordinary-log universality of boundary critical behavior

The recent discovery of extraordinary-log universality has generated intense interest in classical and quantum boundary critical phenomena. Despite tremendous efforts, the existence of quantum extraordinary-log universality remains extremely controversial. Here, by utilizing quantum Monte Carlo simulations, we study the quantum edge criticality of a two-dimensional Bose-Hubbard model featuring emergent bulk criticality. On top of an insulating bulk, the open edges experience a Kosterlitz-Thouless-like transition into the superfluid phase when the hopping strength is sufficiently enhanced on edges. At the bulk critical point, the open edges exhibit the special, ordinary, and extraordinary critical phases. In the extraordinary phase, logarithms are involved in the finite-size scaling of two-point correlation and superfluid stiffness, which admit a classical-quantum correspondence for the extraordinary-log universality. Thanks to modern quantum emulators for interacting bosons in lattices, the edge critical phases might be realized in experiments.