Finite-size Scaling Research Articles

The activated scaling behavior of quantum Griffiths singularity in two-dimensional superconductors

Quantum Griffiths singularity (QGS) is characterized by the divergence of the dynamical critical exponent with the activated scaling law and has been widely observed in various two-dimensional superconductors. Recently, the direct activated scaling analysis with the irrelevant correction was proposed and successfully used to analyze the experimental data of crystalline PdTe2 and polycrystalline -W films, which provides new evidence of QGS. Here we show that the direct activated scaling analysis is applicable to the experimental data in different superconducting films, including tri-layer Ga films and LaAlO3/SrTiO3 interface superconductors. When taking the irrelevant correction into account, we calculate the corrected sheet resistance at ultralow temperatures. The scaling behavior of the corrected resistance in a comparably large temperature regime and the theoretical fitting of the phase boundary give unambiguous evidence of QGS. Compared to previous methods based on finite size scaling, the direct activated scaling analysis represents a more direct and precise way to analyze the experimental data of QGS in diverse two-dimensional superconductors.

Finite-Size Relaxational Dynamics of a Spike Random Matrix Spherical Model.

We present a thorough numerical analysis of the relaxational dynamics of the Sherrington-Kirkpatrick spherical model with an additive non-disordered perturbation for large but finite sizes N. In the thermodynamic limit and at low temperatures, the perturbation is responsible for a phase transition from a spin glass to a ferromagnetic phase. We show that finite-size effects induce the appearance of a distinctive slow regime in the relaxation dynamics, the extension of which depends on the size of the system and also on the strength of the non-disordered perturbation. The long time dynamics are characterized by the two largest eigenvalues of a spike random matrix which defines the model, and particularly by the statistics concerning the gap between them. We characterize the finite-size statistics of the two largest eigenvalues of the spike random matrices in the different regimes, sub-critical, critical, and super-critical, confirming some known results and anticipating others, even in the less studied critical regime. We also numerically characterize the finite-size statistics of the gap, which we hope may encourage analytical work which is lacking. Finally, we compute the finite-size scaling of the long time relaxation of the energy, showing the existence of power laws with exponents that depend on the strength of the non-disordered perturbation in a way that is governed by the finite-size statistics of the gap.

Average entanglement entropy of midspectrum eigenstates of quantum-chaotic interacting Hamiltonians.

To which degree the average entanglement entropy of midspectrum eigenstates of quantum-chaotic interacting Hamiltonians agrees with that of random pure states is a question that has attracted considerable attention in the recent years. While there is substantial evidence that the leading (volume-law) terms are identical, which and how subleading terms differ between them is less clear. Here we carry out state-of-the-art full exact diagonalization calculations of clean spin-1/2 XYZ and XXZ chains with integrability breaking terms to address this question in the absence and presence of U(1) symmetry, respectively. We first introduce the notion of maximally chaotic regime, for the chain sizes amenable to full exact diagonalization calculations, as the regime in Hamiltonian parameters in which the level spacing ratio, the distribution of eigenstate coefficients, and the entanglement entropy are closest to the random matrix theory predictions. In this regime, we carry out a finite-size scaling analysis of the subleading terms of the average entanglement entropy of midspectrum eigenstates when different fractions ν of the spectrum are included in the average. We find indications that, for ν→0, the magnitude of the negative O(1) correction is only slightly greater than the one predicted for random pure states. For finite ν, following a phenomenological approach, we derive a simple expression that describes the numerically observed ν dependence of the O(1) deviation from the prediction for random pure states.

Formula omitted]-core decomposition of hypergraphs

In complex networks, many elements interact with each other in different ways. A hypergraph is a network in which group interactions occur among more than two elements. In this study, first, we propose a method to identify influential subgroups in hypergraphs, named (k,q)-core decomposition. The (k,q)-core is defined as the maximal subgraph in which each vertex has at least k hypergraph degrees and each hyperedge contains at least q vertices. The method contains a repeated pruning process until reaching the (k,q)-core, which shares similarities with a widely used k-core decomposition technique in a graph. Second, we analyze the pruning dynamics and the percolation transition with theoretical and numerical methods in random hypergraphs. We set up evolution equations for the pruning process, and self-consistency equations for the percolation properties. Based on our theory, we find that the pruning process generates a hybrid percolation transition for either k≥3orq≥3. The critical exponents obtained theoretically are confirmed with finite-size scaling analysis. Next, when k=q=2, we obtain a unconventional degree-dependent critical relaxation dynamics analytically and numerically. Finally, we apply the (k,q)-core decomposition to a real coauthorship dataset and recognize the leading groups at an early stage.

Static universality of the Ising and Blume–Capel models on two-dimensional Penrose tiles

We study the phase transitions and the critical behavior of the Ising and Blume–Capel models on two kinds of two-dimensional Penrose tiles employing Monte Carlo methods and detailed finite-size scaling analysis. The Wolff algorithm is used for the Ising model whereas the Wang–Landau and Metropolis algorithms are used for the Blume–Capel model to obtain critical temperatures and exponents. The phase diagram of the Blume–Capel model reveals the presence of the double transition, the reentrant behavior, and the tricriticality in the vicinity of the critical single-ion anisotropy. Our results indicate that continuous phase transitions of the Ising and Blume–Capel models on the Penrose tiles belong to the two-dimensional Ising universality class. However, the scaling functions depend on the spin magnitude, anisotropy, and graph shape. Lastly, we verify that the critical Binder cumulant is also universal but depends on the shape of the graph.

Mean-field theory of chaotic insect swarms.

The harmonically confined Vicsek model displays qualitative and quantitative features observed in natural insect swarms. It exhibits a scale-free transition between single and multicluster chaotic phases. Finite-size scaling indicates that this unusual phase transition occurs at zero confinement [Phys. Rev. E 107, 014209 (2023)2470-004510.1103/PhysRevE.107.014209]. While the evidence of the scale-free-chaos phase transition comes from numerical simulations, here we present its mean-field theory. Analytically determined critical exponents are those of the Landau theory of equilibrium phase transitions plus dynamical critical exponent z=1 and a new critical exponent φ=0.5 for the largest Lyapunov exponent. The phase transition occurs at zero confinement and noise in the mean-field theory. The noise line of zero largest Lyapunov exponents informs observed behavior: (i) the qualitative shape of the swarm (on average, the center of mass rotates slowly at the rate marked by the winding number and its trajectory fills compactly the space, similarly to the observed condensed nucleus surrounded by vapor) and (ii) the critical exponents resemble those observed in natural swarms. Our predictions include power laws for the frequency of the maximal spectral amplitude and the winding number.

Laplacian roughening model on kagome and honeycomb lattices

We investigate the Laplacian roughening model on kagome and honeycomb lattices. By using the Wang–Landau algorithm, we obtain the density of states, and calculate both the specific heat and the partition function zeros. Critical exponents and transition temperature are measured by means of the finite-size scaling analysis. The results indicate strong evidences that the system undergoes a single first-order phase transition on both lattices. On triangular and square lattices, a single second-order transition was found in our previous studies. It seems that the type of transition in the Laplacian roughening model depends on the substrate lattice structures.

Identifying many-body localization transition through global quantum discord

We investigate the many-body localization (MBL) transitions in a spin-1/2 Heisenberg chain with an on-site random magnetic field by employing global quantum discord (GQD). We use the disorder-averaged GQD to estimate the MBL critical point, which is found to be around at by making a finite-size scaling analysis. We further compare our results of GQD with those of half-chain entanglement entropy (EE) that is promising in the study of MBL. We show that GQD can exclude the finite-size interference under the same condition, which implies that GQD is more robust than the half-chain EE in characterizing MBL.

Universality class of the mode-locked glassy random laser

By means of enhanced Monte Carlo numerical simulations parallelized on GPUs we study the critical properties of the spin-glass-like model for the mode-locked glassy random laser, a 44-spin model with complex spins with a global spherical constraint and quenched random interactions. Implementing two different boundary conditions for the mode frequencies we identify the critical points and the critical indices of the random lasing phase transition with finite size scaling techniques. The outcome of the scaling analysis is that the mode-locked random laser universality class is compatible with a mean-field one, though different from the mean-field class of the Random Energy Model and of the glassy random laser in the narrow band approximation, that is, the fully connected version of the present model. The low temperature (high pumping) phase is finally characterized by means of the overlap distribution and evidence for the onset of replica symmetry breaking in the lasing regime is provided.

Dimerization of Polyglutamine within the PRIME20 Model using Stochastic Approximation Monte Carlo

AbstractThis study presents a numerical investigation of the dimerization of polyglutamine homo‐peptides of varying length. It employs the PRIME20 intermediate resolution protein model and studies it with a flat‐histogram type Monte Carlo simulation that gives access to the thermodynamic equilibrium of this model over the complete control parameter range (for the simulations this is temperature). For densities comparable to typical in vitro experimental conditions, this study finds that the aggregation and folding of the polyglutamine chains occur concurrently. However, as a function of chain length the sequence of establishment of intra‐ and intermolecular hydrogen bonding contacts changes. Chains longer than about N = 24 polyglutamine repeat units fold first and then aggregate. This agrees well with the experimental finding that, beyond N = 24 the single polyglutamine chain is the critical nucleus for the aggregation of amyloid fibrils. A finite size scaling of the ordering temperatures reveals that for this chain length (and longer chains) folding occurs at physiological (respectively larger) temperatures, whereas shorter chains are disordered at physiological conditions.

Multiple localization transitions and novel quantum phases induced by a staggered on-site potential

We propose a one-dimensional generalized Aubry-Andr\'e-Harper (AAH) model with off-diagonal hopping and staggered on-site potential. We find that the localization transitions could be multiple reentrant with the increasing of staggered on-site potential. The multiple localization transitions are verified by the quantum static and dynamic measurements such as the inverse or normalized participation ratios, fractal dimension, and survival probability. Based on the finite-size scaling analysis, we also obtain an interesting intermediate phase where the extended, localized, and critical states are coexistent in certain regimes of model parameters. These results are quite different from those in the generalized AAH model with off-diagonal hopping, and can help us to find novel quantum phases and new localization phenomena in the disordered systems.

Anomalous finite-size effect on the magnetostructural transition in CrN

We report a systematic investigation of the finite-size effect on magnetostructural transition in CrN based on a series of nanoparticle samples with various sizes prepared by a high-pressure method, leading to the discovery of anomalous critical behaviors near transition, especially for spin correlation length. Such critical anomalies are presumably attributed to the strong spin-lattice coupling in this nitride, because the spin correlation of fluctuation can be greatly affected by the concurrent structural distortion. Using the conventional Gibbs-Thomson theorem and size-dependent effect, the scaling relationship between transition temperature and spin correlation length is deduced by involving a term of the transition-induced volume change that rationalizes the role that the lattice degree of freedom played during magnetic ordering. A phenomenological model is also established for understanding the kinetics and dynamics of magnetic fluctuation at transition in a finite-size system. In addition, our results also show that CrN nanoparticles possess a compressed surface shell, which can be described by Laplace pressure, obeying a unique $\frac{1}{{d}^{1.8}}$ finite-size scaling law.

Finite-size excess-entropy scaling for simple liquids.

Explicit and implicit size effects in computer simulations result from considering systems with a fixed number of particles and periodic boundary conditions, respectively. We investigate these effects in the relation D*(L) = A(L) exp(α(L)s2(L)) between reduced self-diffusion coefficient D*(L) and two-body excess entropy s2(L) for prototypical simple-liquid systems of linear size L. To this aim, we introduce and validate a finite-size two-body excess entropy integral equation. Our analytical arguments and simulation results show that s2(L) exhibits a linear scaling with 1/L. Since D*(L) displays a similar behavior, we show that the parameters A(L) and α(L) are also linearly proportional to 1/L. By extrapolating to the thermodynamic limit, we report the coefficients A∞ = 0.048 ± 0.001 and α∞ = 1.000 ± 0.013 that agree well with the universal values available in the literature [M. Dzugutov, Nature 381, 137-139 (1996)]. Finally, we find a power law relation between the scaling coefficients for D*(L) and s2(L), suggesting a constant viscosity-to-entropy ratio.

Local integrals of motion in quasiperiodic many-body localized systems

Local integrals of motion play a central role in the understanding of many-body localization in many-body quantum systems in one dimension subject to a random external potential, but the question of how these local integrals of motion change in a deterministic quasiperiodic potential is one that has received significantly less attention. Here we develop a powerful new implementation of the continuous unitary transform formalism and use this method to directly compute both the effective Hamiltonian and the local integrals of motion for many-body quantum systems subject to a quasiperiodic potential. We show that the effective interactions between local integrals of motion retain a strong fingerprint of the underlying quasiperiodic potential, exhibiting sharp features at distances associated with the incommensurate wavelength used to generate the potential. Furthermore, the local integrals of motion themselves may be expressed in terms of an operator expansion which allows us to estimate the critical strength of quasiperiodic potential required to lead to a localization/delocalization transition, by means of a finite size scaling analysis.

Entanglement gap in 1D long-range quantum spherical models

We investigate the finite-size scaling of the entanglement gap in the one-dimensional long-range quantum spherical model (QSM). We focus on the weak long-range QSM, for which the thermodynamic limit is well-defined. This model exhibits a continuous phase transition, separating a paramagnetic from a ferromagnet phase. The universality class of the transition depends on the long-range exponent α. We show that in the thermodynamic limit the entanglement gap is finite in the paramagnetic phase, and it vanishes in the ferromagnetic phase. In the ferromagnetic phase the entanglement gap is understood in terms of standard magnetic correlation functions. The half-system entanglement gap decays as , where the constant C α depends on the low-energy properties of the model and L is the system size. This reflects that the lower part of the dispersion is affected by the long range physics. Finally, multiplicative logarithmic corrections are absent in the scaling of the entanglement gap, in contrast with the higher-dimensional case.

Tensor network based finite-size scaling for two-dimensional Ising model

We propose a scheme to perform tensor network based finite-size scaling analysis for two-dimensional classical models. In the tensor network representation of the partition function, we use higher-order tensor renormalization group (HOTRG) method to coarse grain the weight tensor. The renormalized tensor is then used to construct the approximated transfer matrix of an infinite strip of finite width. By diagonalizing the transfer matrix we obtain the correlation length, the magnetization, and the energy density which are used in finite-size scaling analysis to determine the critical temperature and the critical exponents. As a benchmark we study the two-dimensional classical Ising model. We show that the critical temperature and the critical exponents can be accurately determined. With HOTRG bond dimension $D=70$, the absolute errors of the critical temperature $T_c$ and the critical exponent $\nu$, $\beta$ are at the order of $10^{-7}, 10^{-5}$, $10^{-4}$ respectively. Furthermore, the results can be systematically improved by increasing the bond dimension of the HOTRG method. Finally, we study the length scale induced by the finite cut-off in bond dimension and elucidate its physical meaning in this context.

The Influence of Dimensionality on the Charge‐Density‐Wave Transition and its Application on Mid‐Infrared Photodetection

Abstract2D charge density wave (CDW) materials receive much attention for high responsivity and broadband photodetection in recent years due to their collective electron transport and narrow bandgap. However, the high dark current density problem hinders their real application. Here, a sharp CDW transition in quasi‐1D (TaSe4)2I is reported and applied for broadband photodetection. Especially at mid‐infrared region, the device shows both high photo responsivity of 1.18 ×103 A W−1 and large light on/off ratio of 80, which is superior to 2D CDW TaS2 and most reported low‐dimensional materials. Such high performance relies on two aspects. One is the much lower dark current density resulting from the pseudo gap associated with 1D Luttinger liquid state, which is supported by finite size scaling of nonlinear I–V at variable temperatures and occurrence of 1D structural phase transition consolidated by in situ Raman spectroscopy. The other is the high photocurrent associated with the “Fröhlich superconductivity” state, manifested by an ultrasensitive switching, which can be only accessible in 1D CDW materials, in agreement with the authors’ density functional theory calculation. This work thus reveals the pivotal role of dimensionality in CDW phase transition and paves a way for implementing highly sensitive broadband photodetector.

Finite-size scaling, phase coexistence, and algorithms for the random cluster model on random graphs

Pour Δ≥5 et q grand en fonction de Δ, nous donnons une description détaillée de la transition de phase du modèle de composantes connexes aléatoires (i.e., le modèle FK) sur des graphes Δ-réguliers aléatoires. En particulier, nous déterminons la distribution limite des poids des phases ordonnées et désordonnées au point critique et prouvons la décroissance exponentielle des corrélations et le comportement gaussien des fluctuations loin du point critique. Nos techniques sont basées sur l’utilisation de modèles de polymères et l’expansion en clusters pour contrôler les écarts par rapport aux états fondamentaux ordonnés et désordonnés. Ces techniques produisent également des algorithmes de comptage et d’échantillonnage efficaces pour les modèles de Potts et FK sur des graphes Δ-réguliers aléatoires à toutes les températures lorsque q est grand. Cela inclut la température critique à laquelle on sait que la dynamique de Glauber et de Swendsen–Wang pour le modèle de Potts mélangent lentement. Nous prouvons en outre de nouveaux résultats de mélange lent pour les chaînes de Markov, notamment que la dynamique de Swendsen–Wang mélange exponentiellement lentement tout au long d’un intervalle ouvert contenant la température critique. Ceci n’était auparavant connu qu’à la température critique. Beaucoup de nos résultats s’appliquent plus généralement aux graphes Δ-réguliers qui satisfont une borne inférieure sur le nombre d’arêtes quittant chaque « petit ensemble » de sommets dans le graphe.

Neural network approach to scaling analysis of critical phenomena.

Determining the universality class of a system exhibiting critical phenomena is one of the central problems in physics. There are several methods to determine this universality class from data. As methods to collapse plots onto scaling functions, polynomial regression, which is less accurate, and Gaussian process regression, which provides high accuracy and flexibility but is computationally expensive, have been proposed. In this paper, we propose a regression method using a neural network. The computational complexity is linear only in the number of data points. We demonstrate the proposed method for the finite-size scaling analysis of critical phenomena in the two-dimensional Ising model and bond percolation problem to confirm the performance. This method efficiently obtains the critical values with accuracy in both cases.

Wetting transition in the transverse-field spin- 12 XY model with boundary fields

The wetting transition in the transverse-field spin-$\frac{1}{2}$ XY model with opposite boundary fields ${h}_{L}^{x}{h}_{R}^{x}<0$ is studied analytically and numerically. We find that the phase diagram is complex and that the wetting transition is of three types: first, second, and fourth order. The energy gap is obtained analytically, and the magnetization profile, correlation functions, and wetting layer thickness are obtained numerically. For $|{h}_{L}^{x}|,|{h}_{R}^{x}|<{h}_{w}$, a first-order phase transition occurs at ${h}_{L}^{x}=\ensuremath{-}{h}_{R}^{x}$, where ${h}_{w}$ is the continuous wetting transition point. For $|{h}_{R}^{x}|$ larger than ${h}_{w}$, the continuous wetting transition occurs at ${h}_{L}^{x}={h}_{w}$, and vice versa. For $g\ensuremath{\ne}1\ensuremath{-}{\ensuremath{\gamma}}^{2}$, the wetting transition is second order, and commensurate and incommensurate phases occur for $g<1\ensuremath{-}{\ensuremath{\gamma}}^{2}$ and $g>1\ensuremath{-}{\ensuremath{\gamma}}^{2}$, respectively. For $g=1\ensuremath{-}{\ensuremath{\gamma}}^{2}$, the wetting transition is fourth order. For this fourth-order phase transition, the third derivative of the surface magnetization oscillates and diverges near the transition point. The correlation length exponent is $\ensuremath{\nu}=2$, and the dynamic exponent is $z=2$. Thus, this fourth-order transition belongs to a new universality class. The wetting behavior is induced by asymmetric boundary fields ${h}_{L}^{x}=\ensuremath{-}{h}_{R}^{x}$ and its finite-size scaling is discussed.