Exploring The Phase Transition In The Three Dimensional Ising Model With Deep Learning Method

Quantum Chromodynamics(QCD),as a gauge theory describing strong interactions,predicts that when energies and energy densities are large enough,quarks and gluons can be deconfined from hadrons to form a new state of matter called quark-gluon plasma(QGP).Various theoretical models predict that the transition from the hadronic phase to the QGP phase belongs to the first-order phase transition in the region of low temperature and high baryon chemical potential.The end point of the first-order phase transition line is the so-called QCD critical point,and the phase transition at this point belongs to a second-order transition.In current relativistic heavy ion collision experiment,experimentalists plant to use heavy ion accelerators to produce a high-temperature and high-density environment in a very short time,which is hopped to produce the deconfined quark gluon plasma from hadronic matter.A lot of experimental results have suggested that quark gluon plasma has been generated during these collisions.However,the QGP seems to exists only for a very short time and thus it is impossible to observe it directly.Since we know little about the properties of the high-temperature and high-density nuclear matter,it is of great interests to study the phase structure of the new matter in heavy-ion collisions.Exploring the structure of QCD phase diagram and locating the critical point of QCD is one of the main goals in the current high energy collisions.This is a big challenge from both theoretical and experimental aspects.In this aim,the second beam energy scan program(BES?)at the Relativistic Heavy Ion Collider(RHIC)and a series of collision experiments on the Large Hadron Collider(LHC)are dedicated to locating the QCD critical point.However,there is no solid conclusion in experiments because the problems of the choice of critical related observable,the influence of non-critical effects and other effects.In theoretical side,lattice QCD simulations cannot be performed in finite baryon chemical potentials.So the calculation of the existence and position of the critical point from the first principle is still a big challenge.In this work,we try to start from a phenomenological point of view.A three-dimensional Ising model,which belongs to the same universal class as the QCD critical point is studied.We can explore the QCD phase diagram through a parametrization of the scaling equation of state in Ising model together with a parametrized map from Ising variables to QCD coordinates.For the phase classification and phase transition recognition,the common used tra-ditional thermodynamic method is to choose an order parameter through consideration of symmetry of the system,and then use the order parameter to distinguish different phases.However,there is no clear order parameter in the QCD phase transition at present.To solve this problem,we plan to use modern data analysis methods,such as deep learning method,to directly extract phase and signals of phase transition only from the spin configuration of Ising model.Current studies have shown that both fully connected neural networks and two-dimensional convolutional neural networks(CNN)can be successfully used to recog-nize the phase transition in the two-dimensional Ising model.It proves the feasibility and accuracy of deep learning methods to study phase transitions.In this paper,we use the deep learning method in the machine learning technology to study the classification of first-order and second-order phase transitions in the three-dimensional Ising model.A series of spin configurations of the three-dimensional cubic Ising model are generated by Metropolis algorithm,which are used as the input of the neural network.We choose supervised learning method and use the one-hot encoding.A three-dimensional convolutional neural network is trained through the input spin configu-ration and its corresponding label.The weight parameters in the network architecture are continuously updated during the training process.After all the parameters are optimized,the network can be tested to recognize and classify the unknown in the test dataset.In order to compare the accuracy and efficiency of the deep learning algorithms,we use two different networks,i.e.,fully connected neural network and convolutional neural network,in the study on phase transitions in three-dimensional Ising model.The results show that both the convolutional neural network and the fully connected neural network can well identify different phases in Ising model.And the convolutional neural network has been found to be with higher accuracy.It is found that the accuarcy increases with the increase of system size.Finally,we use the convolutional neural networks to predict the the energy of the system in different temperatures.The predictions of the CNN are in good agreement with the results obtained by the Monte Carlo simulation method.The deep learning method can identify different phases with high accuracy in the three-dimensional Ising model without knowing the order parameters in advance.There-fore it is suggested to be an effective tool for phase classification.It can be achieved by building a network framework,selecting appropriate input and output and doing an iter-ative training.In the next,we will study how the neural network weights as a important feature to access the information of phase transitions.We hope to use the deep learning method to find characteristic quantities in the unknown phase transition and then can be directly used in real experiments to explore the QCD phase transition.