Studying Phase Transition And Critical Exponent By Machine Learning

In recent years,machine learning methods have been preferred by physicists.It has been used to solve many problems of statistical physics,such as identifying the phase of matter and phase transitions.Restricted Boltzmann Machine(RBM)and Convolutional Neural Network(CNN)are two mainstream algorithm models for machine learning.This thesis mainly discuss how to use RBM to identify the product order of the Ashkin-Teller(AT)model and how to extract critical exponent of quantum hall plateau transitions by convolutional neural network.Traditional methods use the concept of order parameter to identify the.phase transition.Instead,we use machine learning method to identify the phase transition of Ashkin-Teller model,but the operator,however,is not directly present in our machine learning algorithm.This prompted us to define a non-trivial order parameter for the Ashkin-Teller model based on the machine learning method.Since machine learning can be used to identify phase transitions,an alternative question is:can machine be used to extract the critical exponent of phase transitions?If we can design a suitable method to make the machine detect the critical behavior near the critical point and extract the relevant scales to perform finite size scaling,then the machine might be able to extract the critical exponent.This thesis first gives a brief introduction to the related methods of machine learning and the network structure and training methods of neural networks.Then the structure,related properties and training methods of the RBM are introduced and analyzed in detail.The network structure and training method of the CNN are discussed next.The core of this paper discusses how to use the RBM and CNN to solve physical problems.We found out that the RBM was able to identify the paramagnetic phase and the product phase of the AT model.We also defined a special order parameter to identify the product phase using the properties of RBM.For the disordered Hofstadter model,we used a trial-and-error labelling scheme of the wave function to train the CNN.Based on the output performance curve of the neural network,we were able to define a characteristic energy scale for a particular system size.The characteristic energy scale corresponds to the location of the minimum of the perfonnance curve for the system size under consideration.We performed a finite size scaling of the characteristic energy scale for varies system sizes.We obtained the value of exponent v=2.22±0.04 for the plateau transition that is in agreement with the values obtained by using the conventional methods.Our methods confirmed that the machine learning method can be used to extract the critical behaviour.The application of these methods to other problems remains a future endeavour.