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arXiv:2109.00254 · Physical Review Research

Machine learning of the XY model on a spherical Fibonacci lattice

Chen-Hui Song, Qu-Cheng Gao, Xu-Yang Hou, Xin Wang, Zheng Zhou, Yan He, Hao Guo, Chih-Chun Chien

Published Apr 1, 2022• 10 citations
arXivHTMLPDF

Abstract

The XY model on a spherical surface, inspired by recently realized atomic gases trapped in a spherical shell, is analyzed here. Instead of a traditional latitude-longitude lattice, we introduce the much more homogeneous Fibonacci lattice and use classical Monte Carlo (MC) simulations to generate spin configurations. The results clearly show that topological defects, in the form of vortices, must exist in the stable configuration on a sphere but vanish in a plane due to a mathematical theorem. Using these spin configurations as training samples, a graph-convolutional network-based method is implemented to recognize different phases and successfully predict the phase-transition temperature. We also apply the density-based spatial clustering of applications with noise, a powerful machine-learning algorithm, to monitor the path of two vortices with topological charges on the sphere during MC simulations. Our results provide reliable predictions for quantum simulators using polaritons or future space-based experiments on ultracold atoms in microgravity.

DOI
10.1103/physrevresearch.4.023005

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The XY model on a spherical surface, inspired by recently realized atomic gases trapped in a spherical shell, is analyzed here. Instead of a traditional latitude-longitude lattice, we introduce the much more homogeneous Fibonacci lattice and use classical Monte Carlo (MC) simulations to generate spin configurations. The results clearly show that topological defects, in the form of vortices, must exist in the stable configuration on a sphere but vanish in a plane due to a mathematical theorem. Using these spin configurations as training samples, a graph-convolutional network-based method is implemented to recognize different phases and successfully predict the phase-transition temperature. We also apply the density-based spatial clustering of applications with noise, a powerful machine-learning algorithm, to monitor the path of two vortices with topological charges on the sphere during MC simulations. Our results provide reliable predictions for quantum simulators using polaritons or future space-based experiments on ultracold atoms in microgravity.
Abstract We investigate the Ising model on a spherical surface, utilizing a Fibonacci lattice to approximate uniform coverage. This setup poses challenges in achieving consistent lattice distribution across the sphere for comparison with planar models. We employ Monte Carlo simulations, principal component analysis, graph convolutional networks (GCNs) to study spin configurations across a range of temperatures and to determine phase transition temperatures. The Fibonacci lattice, despite its uniformity, contains irregular sites that influence spin behavior. In the ferromagnetic case, sites with fewer neighbors exhibit a higher tendency for spin flips at low temperatures, though this effect weakens as temperature increases, leading to a higher phase transition temperature than in the planar Ising model. In the antiferromagnetic case, lattice irregularities induce geometric frustration, resulting in highly degenerate ground states and the phase transition temperature lower than the planar square lattice. Phase transition temperatures are derived through specific heat, magnetic susceptibility analysis and GCNs predictions, yielding T c values for both ferromagnetic and antiferromagnetic scenarios. This work emphasizes the impact of the Fibonacci lattice’s geometric properties-namely curvature and connectivity-on spin interactions in non-planar systems, with relevance to microgravity environments.
Abstract We investigate the ferromagnetic q -state Potts model on spherical Fibonacci graphs. These graphs are constructed by embedding quasi-uniform sites on a sphere and defining interactions via a chord-distance cutoff chosen so as to yield a network approximating four-neighbor connectivity. By combining Swendsen–Wang cluster Monte Carlo simulations with graph convolutional networks (GCNs), which operate directly on the adjacency structure and node spins, we develop a unified phase-classification framework applicable to both regular planar lattices and curved, irregular spherical graphs. Benchmarks on planar lattices demonstrate an efficient transfer strategy: after a fixed binarization of Potts spins into an effective Ising variable, a single GCN pre-trained on the Ising model can localize the transition region for different q values without retraining. Applying this strategy to spherical graphs, we find that curvature- and defect-induced connectivity irregularities induce only modest shifts in the inferred transition temperatures relative to their planar counterparts. Further analysis shows that the curvature-induced shift of the critical temperature is most pronounced at small q and diminishes rapidly as q increases. This trend is consistent with the physical picture that, in two dimensions, the Potts model undergoes a transition from a continuous phase transition to a weakly first-order one for q > 4, accompanied by a pronounced reduction in the correlation length.
In this paper, we apply machine learning methods to study phase transitions in certain statistical mechanical models on the two-dimensional lattices, whose transitions involve nonlocal or topological properties, including site and bond percolations, the XY model, and the generalized XY model. We find that using just one hidden layer in a fully connected neural network, the percolation transition can be learned and the data collapse by using the average output layer gives correct estimate of the critical exponent ν. We also study the Berezinskii-Kosterlitz-Thouless transition, which involves binding and unbinding of topological defects, vortices and antivortices, in the classical XY model. The generalized XY model contains richer phases, such as the nematic phase, the paramagnetic and the quasi-long-range ferromagnetic phases, and we also apply machine learning method to it. We obtain a consistent phase diagram from the network trained with only data along the temperature axis at two particular parameter Δ values, where Δ is the relative weight of pure XY coupling. Aside from using the spin configurations (either angles or spin components) as the input information in a convolutional neural network, we devise a feature engineering approach using the histograms of the spin orientations in order to train the network to learn the three phases in the generalized XY model and demonstrate that it indeed works. The trained network by using system size L×L can be used to the phase diagram for other sizes (L^{'}×L^{'}, where L^{'}≠L) without any further training.
This work aims at the goal whether the artificial intelligence can recognize phase transition without the prior human knowledge. If this becomes successful, it can be applied to, for instance, analyze data from quantum simulation of unsolved physical models. Toward this goal, we first need to apply the machine learning algorithm to well-understood models and see whether the outputs are consistent with our prior knowledge, which serves as the benchmark of this approach. In this work, we feed the compute with data generated by the classical Monte Carlo simulation for the XY model in frustrated triangular and union jack lattices, which has two order parameters and exhibits two phase transitions. We show that the outputs of the principle component analysis agree very well with our understanding of different orders in different phases, and the temperature dependences of the major components detect the nature and the locations of the phase transitions. Our work offers promise for using machine learning techniques to study sophisticated statistical models, and our results can be further improved by using principle component analysis with kernel tricks and the neural network method.
We use the Fortuin-Kasteleyn representation-based improved estimator of the correlation configuration as an alternative to the ordinary correlation configuration in the machine-learning study of the phase classification of spin models. The phases of classical spin models are classified using the improved estimators, and the method is also applied to the quantum Monte Carlo simulation using the loop algorithm. We analyze the Berezinskii-Kosterlitz-Thouless (BKT) transition of the spin-1/2 quantum XY model on the square lattice. We classify the BKT phase and the paramagnetic phase of the quantum XY model using the machine-learning approach. We show that the classification of the quantum XY model can be performed by using the training data of the classical XY model.
The authors extend the recent successful application of neural networks fordistinguishing phases of matter to the case of topologically driven phase transitions. They demonstrate that supervised networks correctly identify the Kosterlitz-Thouless transition in the two-dimensional classical XY model. However, their analysis suggests that a naive classification of phases is based on bulk features, rather than on detecting local topological defects. While they show that it is possible to fine-tune a network to recognize vortices, it generally remains a difficult task to detect topological defects without $a\phantom{\rule{0.333em}{0ex}}p\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}i$ knowledge.