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Recent advances in machine learning have become increasingly popular in the applications of phase transitions and critical phenomena. By machine learning approaches, we try to identify the physical characteristics in the two-dimensional percolation model. To achieve this, we adopt Monte Carlo simulation to generate dataset at first, and then we employ several approaches to analyze the dataset. Four kinds of convolutional neural networks (CNNs), one variational autoencoder (VAE), one convolutional VAE (cVAE), one principal component analysis (PCA), and one $k$-means are used for identifying order parameter, the permeability, and the critical transition point. The former three kinds of CNNs can simulate the two order parameters and the permeability with high accuracy, and good extrapolating performance. The former two kinds of CNNs have high anti-noise ability. To validate the robustness of the former three kinds of CNNs, we also use the VAE and the cVAE to generate new percolating configurations to add perturbations into the raw configurations. We find that there is no difference by using the raw or the perturbed configurations to identify the physical characteristics, under the prerequisite of corresponding labels. In the case of lacking labels, we use unsupervised learning to detect the physical characteristics. The PCA, a classical unsupervised learning, performs well when identifying the permeability but fails to deduce order parameter. Hence, we apply the fourth kinds of CNNs with different preset thresholds, and identify a new order parameter and the critical transition point. Our findings indicate that the effectiveness of machine learning still needs to be evaluated in the applications of phase transitions and critical phenomena.
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Topological materials discovery has emerged as an important frontier in condensed matter physics. Recent theoretical approaches based on symmetry indicators and topological quantum chemistry have been used to identify thousands of candidate topological materials, yet experimental determination of materials topology often poses significant technical challenges. X-ray absorption spectroscopy (XAS) is a widely-used materials characterization technique sensitive to atoms local symmetry and chemical environment; thus, it may encode signatures of materials topology, though indirectly. In this work, we show that XAS can potentially uncover materials topology when augmented by machine learning. By labelling computed X-ray absorption near-edge structure (XANES) spectra of over 16,000 inorganic materials with their topological class, we establish a machine learning-based classifier of topology with XANES spectral inputs. Our classifier correctly predicts 81% of topological and 80% of trivial cases, and can achieve 90% and higher accuracy for materials containing certain elements. Given the simplicity of the XAS setup and its compatibility with multimodal sample environments, the proposed machine learning-empowered XAS topological indicator has the potential to discover broader categories of topological materials, such as non-cleavable compounds and amorphous materials. It can also inform a variety of field-driven phenomena in situ, such as magnetic field-driven topological phase transitions.
We apply generalisations of the Swendson-Wang and Wolff cluster algorithms, which are based on the construction of Fortuin-Kasteleyn clusters, to the three-dimensional $pm 1$ random-bond Ising model. The behaviour of the model is determined by the temperature $T$ and the concentration $p$ of negative (anti-ferromagnetic) bonds. The ground state is ferromagnetic for $0 le p<p_c$, and a spin glass for $p_c < p le 0.5$ where $p_c simeq 0.222$. We investigate the percolation transition of the Fortuin-Kasteleyn clusters as function of temperature. Except for $p=0$ the Fortuin-Kasteleyn percolation transition occurs at a higher temperature than the magnetic ordering temperature. This was known before for $p=1/2$ but here we provide evidence for a difference in transition temperatures even for $p$ arbitrarily small. Furthermore, for all values of $p>0$, our data suggest that the percolation transition is universal, irrespective of whether the ground state exhibits ferromagnetic or spin-glass order, and is in the universality class of standard percolation. This shows that correlations in the bond occupancy of the Fortuin-Kasteleyn clusters are irrelevant, except for $p=0$ where the clusters are tied to Ising correlations so the percolation transition is in the Ising universality class.
We present an exhaustive mathematical analysis of the recently proposed Non-Poissonian Ac- tivity Driven (NoPAD) model [Moinet et al. Phys. Rev. Lett., 114 (2015)], a temporal network model incorporating the empirically observed bursty nature of social interactions. We focus on the aging effects emerging from the Non-Poissonian dynamics of link activation, and on their effects on the topological properties of time-integrated networks, such as the degree distribution. Analytic expressions for the degree distribution of integrated networks as a function of time are derived, ex- ploring both limits of vanishing and strong aging. We also address the percolation process occurring on these temporal networks, by computing the threshold for the emergence of a giant connected component, highlighting the aging dependence. Our analytic predictions are checked by means of extensive numerical simulations of the NoPAD model.
This work studies the jamming and percolation of parallel squares in a single-cluster growth model. The Leath-Alexandrowicz method was used to grow a cluster from an active seed site. The sites of a square lattice were occupied by addition of the equal size $k times k$ squares (E-problem) or a mixture of $k times k$ and $m times m$ ($m leqslant k$) squares (M-problem). The larger $k times k$ squares were assumed to be active (conductive) and the smaller $m times m$ squares were assumed to be blocked (non-conductive). For equal size $k times k$ squares (E-problem) the value of $p_j = 0.638 pm 0.001$ was obtained for the jamming concentration in the limit of $krightarrowinfty$. This value was noticeably larger than that previously reported for a random sequential adsorption model, $p_j = 0.564 pm 0.002$. It was observed that the value of percolation threshold $p_{mathrm{c}}$ (i.e., the ratio of the area of active $k times k$ squares and the total area of $k times k$ squares in the percolation point) increased with an increase of $k$. For mixture of $k times k$ and $m times m$ squares (M-problem), the value of $p_{mathrm{c}}$ noticeably increased with an increase of $k$ at a fixed value of $m$ and approached 1 at $kgeqslant 10m$. This reflects that percolation of larger active squares in M-problem can be effectively suppressed in the presence of smaller blocked squares.
Motivated by the importance of geometric information in real systems, a new model for long-range correlated percolation in link-adding networks is proposed with the connecting probability decaying with a power-law of the distance on the two-dimensional(2D) plane. By overlapping it with Achlioptas process, it serves as a gravity model which can be tuned to facilitate or inhibit the network percolation in a generic view, cover a broad range of thresholds. Moreover, it yields a set of new scaling relations. In the present work, we develop an approach to determine critical points for them by simulating the temporal evolutions of type-I, type-II and type-III links(chosen from both inter-cluster links, an intra-cluster link compared with an inter-cluster one, and both intra-cluster ones, respectively) and corresponding average lengths. Numerical results have revealed objective competition between fractions, average lengths of three types of links, verified the balance happened at critical points. The variation of decay exponents $a$ or transmission radius $R$ always shifts the temporal pace of the evolution, while the steady average lengths and the fractions of links always keep unchanged just as the values in Achlioptas process. Strategy with maximum gravity can keep steady average length, while that with minimum one can surpass it. Without the confinement of transmission range, $bar{l} to infty$ in thermodynamic limit, while $bar{l}$ does not when with it. However, both mechanisms support critical points. In two-dimensional free space, the relevance of correlated percolation in link-adding process is verified by validation of new scaling relations with various exponent $a$, which violates the scaling law of Weinribs.
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