Studying Phase Transition In The Percolation And XY Models By Machine Learning Method

In this paper,we apply machine learning methods to study phase transitions in certain statistical mechanical models,whose transitions involve non-local or topological properties,including site percolation,bond percolation,the XY model and the generalized XY model.We use a kind of unsupervised method t-distributed stochastic neighbor embedding,which could classify the configurations of percolation models and distinguish percolation and non percolation as well as critical region.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 ?.We also study the Berezinskii-Kosterlitz-Thouless transition,which involves topological defects—vortices and antivortices,in the classical XY model.As pointed out by M.Beach,A.Golubeva and R.Melko [Phys.Rev.B 97,045207(2018)],using spin orientations rather than vortex configuration gives better results in learning the transition,we thus use the spin components and angles as the input information in a convolutional neural network and verify that this indeed works for learning the transition in the XY model on both the square and honeycomb latticesTo go beyond the XY model,we apply also the machine learning methods to the generalized XY model,which consists of a nematic phase,in addition to the paramagnetic and the quasi-long-range ferromagentic phase.We find that using the histograms of the spin orientations actually works for both the XY model and the generalized XY model in learning the transition.We also train the network to learn the three phases(quasi-long-range ferromagnetic,nematic and paramanetic phases)using such a feature engineering approach.in the appendix,other work introduced.The title is supersolid phase on a one dimensional Anyon-hubbard model.The main content of this thesis is arranged as follows:In chapter I,we introduce the background knowledge,application,classfication of machine learning and our motivation about this paper.We also describe the site percolation model,the bond percolation model and the partition functions.Finally we introduce the classical XY model,classical generalized XY model and show the corresponding Hamiltonians.In chapter II,we describe machine learning algorithm and Monte Carlo algorithm used in this paper.Machine learning methods contain: t-distributed stochastic neighbor embedding,fully-connected neural network and convolutional neural network.Monte Carlo methods includes: Metropolis algorithm,Wolff algorithm,and Swendsen-Wang algorithm.In chapter III,we apply the machine learning algorithm to simulate the sitepercolation model and bond percolation model on the square lattice and triangular lattice respectively.In chapter IV,summary and prospect are presented.In appendix,we briefly introduce other work,Supersolid phase on a one dimensional Anyon-Hubbard model.