Besides the Berezinskii-Kosterlitz-Thouless phase transition, the two-dimensional generalized XY model, identified by a generalization parameter $q$ (as proposed by Romano and Zagrebnov), can also support a first-order phase transition, starting from a critical value ${q}_{c}$ . However, the value of ${q}_{c}$ at which this transition takes place is unknown. In this paper, we take two approaches to accurately determine the critical parameter ${q}_{c}$ . Furthermore, we show that the model is characterized by three distinct regions concerning both first-order and Berezinskii-Kosterlitz-Thouless phase transitions. Finally, the underlying mechanism governing such transitions is presented, along with an estimation of the critical temperatures.Besides the Berezinskii-Kosterlitz-Thouless phase transition, the two-dimensional generalized XY model, identified by a generalization parameter $q$ (as proposed by Romano and Zagrebnov), can also support a first-order phase transition, starting from a critical value ${q}_{c}$. However, the value of ${q}_{c}$ at which this transition takes place is unknown. In this paper, we take two approaches to accurately determine the critical parameter ${q}_{c}$. Furthermore, we show that the model is characterized by three distinct regions concerning both first-order and Berezinskii-Kosterlitz-Thouless phase transitions. Finally, the underlying mechanism governing such transitions is presented, along with an estimation of the critical temperatures.

arXiv:2406.04059Physical review. B./Physical review. BSep 24, 20241 citation

Machine learning of phase transitions in the percolation and XY models

Wanzhou Zhang, Jiayu Liu, Tzu-Chieh Wei

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.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.

arXiv:1804.02709Physical review. EMar 29, 2019114 citations

Besides the Berezinskii-Kosterlitz-Thouless phase transition, the two-dimensional generalized XY model, identified by a generalization parameter

q

(as proposed by Romano and Zagrebnov), can also support a first-order phase transition, starting from a critical value

{q}_{c}

. However, the value of

{q}_{c}

at which this transition takes place is unknown. In this paper, we take two approaches to accurately determine the critical parameter

{q}_{c}

. Furthermore, we show that the model is characterized by three distinct regions concerning both first-order and Berezinskii-Kosterlitz-Thouless phase transitions. Finally, the underlying mechanism governing such transitions is presented, along with an estimation of the critical temperatures.