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The Ising model [70] is regarded as one of the most fundamental magnetic models used to study the nature of phase transitions from a microscopic viewpoint in statistical physics. The energy of any particular state is given by the Ising Hamiltonian
$ \begin{array}{*{20}{l}} E=-H\sum\limits_{i}\sigma_i-J\sum\limits_{\langle ij\rangle}\sigma_i\sigma_j. \end{array} $
(1) Here,
$ \sigma_i $ is the spin of the ith site. It equals either 1 (spin up) or –1 (spin down). We consider the model on a cubic lattice with periodic boundary conditions and set$ J=1 $ as the energy unit. The first sum describes the interaction of the spins with an external magnetic field H. The energy is minimum when a spin points parallel to the external magnetic field. The second sum is taken only over pairs$ (i,j) $ that are nearest neighbors in the grid, and it describes the interaction of the spins with each other. The interaction energy of a pair of adjacent spins is minimum when they point in the same direction. We usually normalize quantities of interest by the number of degrees of freedom and then work with the average energy per spin,$ \epsilon=\langle E\rangle/N $ , and the average magnetization per spin,$ m=\dfrac{1}{N}\langle\sum_{i}\sigma_i\rangle $ . The average magnetization can be regarded as the order parameter of this model.The phase diagram of the 3D Ising model [34, 37] is shown in Fig. 1. The horizontal and vertical axes in the figure are the reduced temperature
$ t=(T-T_c)/T_c $ and external magnetic field H, respectively. When decreasing the temperature under zero magnetic field, i.e., along the direction of line (i) shown in the plot, a second-order phase transition with spontaneous twofold symmetry breaking occurs at the CP$ T_c $ . A discrete Z(2) spin inversion symmetry is broken in the ferromagnetic (order) phase below$ T_c $ and is restored in the paramagnetic (disordered) phase at temperatures above$ T_c $ . On the other hand, when we change the external magnetic field with a fixed temperature below$ T_c $ , i.e., along the direction of line (ii), a first-order phase transition occurs between two phases corresponding to$ m<0 $ and$ m>0 $ . The stable phase is the one for which m has the same sign as H, but the magnetization remains nonzero even in the limit$ H\rightarrow 0 $ . The system therefore undergoes a first-order phase transition in which m changes discontinuously as it crosses the coexistence curve at$ H=0 $ . The size of the discontinuity decreases with increasing temperature and reaches zero at the critical temperature$ T_c $ , i.e., the coexistence curve ends at a CP. When we change the external magnetic field with a fixed temperature beyond$ T_c $ , i.e., along the direction of line (iii), the transition is a smooth crossover. Based on non-universal mapping between Ising variables$ (t,H) $ and QCD coordinates$ (T,\mu_B) $ in the phase diagrams, we can construct an equation of state matching the first principle lattice QCD calculations, which can be employed in hydrodynamic simulations of relativistic heavy-ion collisions [32–35].A Monte Carlo (MC) simulation of the 3D Ising model on a size of
$ L\times L\times L $ is implemented via the Metropolis algorithm. This algorithm is a simple and widely used approach to generate the canonical ensemble. It allows us to efficiently obtain a large number of uncorrelated sample configurations over a wide temperature range. We generate$ 20000 $ independent spin configurations at each selected temperature and external magnetic field. Approximately$ 90 $ % of event samples are randomly chosen as inputs to train the ML model, and the rest are used to test its performance. To reduce correlations of samples, we take 50 sweeps between every two independent configurations [71].ML as a tool for identifying phase transitions has recently garnered significant attention in this field. Supervised learning, a commonly used method, is performed with a training set to teach models to yield the desired output. The algorithm measures its accuracy through the loss function, which is adjusted until the error is sufficiently minimized. After training, the neural network can recognize unseen samples and predict the correct label, illustrating that it has learned important features that can be used for classification tasks.
In deep learning, a convolutional neural network (CNN) [53] is a class of deep neural networks, mostly applied to analyze visual imagery. It is inspired by biological processes in which the connectivity pattern between neurons resembles the organization of the animal visual cortex. The CNN is known as the shift or space invariant of multilayer perceptrons for certain architectures [72]. It has the distinguishing features of local connectivity, shared weights, pooling, etc. Motivated by previous studies [54, 62, 63, 66], we apply ML techniques to the 3D Ising model. Supervised learning with a deep CNN architecture is used to explore the phase transitions and structure of the system.
Machine learning phase transitions of the three-dimensional Ising universality class
- Received Date: 2022-07-20
- Available Online: 2023-03-15
Abstract: Exploration of the QCD phase diagram and critical point is one of the main goals in current relativistic heavy-ion collisions. The QCD critical point is expected to belong to a three-dimensional (3D) Ising universality class. Machine learning techniques are found to be powerful in distinguishing different phases of matter and provide a new way to study the phase diagram. We investigate phase transitions in the 3D cubic Ising model using supervised learning methods. It is found that a 3D convolutional neural network can be trained to effectively predict physical quantities in different spin configurations. With a uniform neural network architecture, it can encode phases of matter and identify both second- and first-order phase transitions. The important features that discriminate different phases in the classification processes are investigated. These findings can help study and understand QCD phase transitions in relativistic heavy-ion collisions.