Study On The Second-order Phase Transition In 3D Ising Model By Machine Learning And Finite-size Scaling Method

Quantum Chromodynamics(QCD)is a gauge theory that describes strong interactions between quarks and gluons.QCD predicts that quarks trapped inside the hadron will be deconfined to form a so-called quark-gluon plasma(QGP)when temperature and density of the system is high enough.QCD predicts that the transition from hadron to QGP is of the first-order in low temperature and high baryon chemical potential region.The end point of the first order phase transition is the critical point.The region with high temperature and low baryon chemical potential is a crossover region.The exploration of the QCD phase diagram and location of the critical point is one of the most important tasks in current heavy-ion collisions.Theoretical studies have shown that the phase transition of the QCD critical point may belong to a three-dimensional(3D)Ising universal class.In ours work,we choose the 3D Ising model to study how to determine the critical temperature and critical exponent with the help of finite-size scaling theory and machine learning method.The unique feature of our method comparing to traditional statistical one is that we do not need to know order parameters or Hamiltonian of the system in advance,but only to input spin configuration information of each grid in 3D Ising model at different temperatures and sizes.Then we use machine learning method to get the critical temperature and critical exponent of 3D Ising.The analytical method used in this paper is the deep learning method in machine learning techniques.Machine learning is a kind of artificial intelligence,which uses predesigned algorithm to learn the generative or discriminant model on a big data set.The techniques of machine learning include supervised machine learning(such as deep learning),unsupervised machine learning and reinforcement machine learning.Supervised machine learning refers to use a given sample(sample consists of both the data and label corresponding to the data)for training and update the parameters in the model.By using this trained model and mapping all the input into the corresponding output,we can complete the classification or regression tasks.The theory of finite-size scaling points out that if a second-order or first-order phase transition occurs at a limited system size,the chosen observable at different sizes can be scaled to the universal form under finite-size scaling.Therefore,an appropriate order parameter should be selected first in the traditional thermodynamic method.The order parameter indicates the degree of order of the system.Then we can use this order parameter by a finite-size scaling method to get the critical temperature and critical exponent.However,the order parameters of the QCD phase transition cannot be given theoretically until now.And several proposed observables,such as higher-order cumulant of the conserved charges(net-baryon,net-charge,net-strangeness)or baryon density,that may be related to the order parameters are suggested to be measured in experiments.In our work,we do not need to know the order parameters,but only the spin configuration information at different temperatures in the Ising model.Then we can extract the critical temperature and critical exponent.In 3D Ising model,when the external field is zero,the system will undergo a transition from an ordered phase to a disordered phase with increasing temperature of the system.We call the temperature of transition between the ordered and disordered phases the critical temperature.We firstly do a regression of the magnetization(m)in the 3D Ising model through a supervised learning method.The m obtained by machine learning is compared with the one given by Monte Carlo.It is found that the magnetizations obtained by the two methods are in good agreement.Then,we classify the ordered phase and disordered phase of the 3D Ising model with six different sizes by supervised machine learning.It is found that the supervised machine learning can be used to effectively classify ordered phase and disordered one.The accuracy of classification of ordered and disordered phases increased with increasing sizes of the system.And the accuracy reached above 98\%when the size is large.We plot the classification curve of two phases in six different sizes on the same figure.It is found that the classification curves of all sizes intersect at a point,which corresponds to the theoretical critical temperature of the 3D Ising model.We try different values of the parameters in the finite-size scaling method to scale the curve of different sizes.When a selected exponent value equals to a specific value,the curves of different sizes coincide with each other.It is found that the selected value equals the theoretic value of the critical exponent in the 3D Ising model.Finally,we classify two different phases(M<0 and M>0)in the first-order phase transition by the same network architecture of supervised learning as used in the second-order transition.We find that machine learning technique can also classify the two different phases in the first-order phase transition with very high accuracy.