Supersolid Of Extended Bose-hubbard Model Of Cavity

The optical lattice provides a very clean and controllable experimental platform for the study of cold atoms.The loading of cold atoms on the optical lattice in an optical cavity could introduce the coupling interaction between photons and atoms,which helps us to study the quantum phases such as superfluid and supersolid and the superradiant transition.Besides the supersolid drived by the nearest-neighbor interactions,the positive and negative signs of the atom-photon coupling alternate with the position and they could also lead to the supersolid.In this thesis,we use the mean-field method and the density-matrixrenormalization-group method based on matrix product states to study the extended Bose-Hubbard model in a cavity.The model studied in this thesis is based on the standard Bose-Hubbard model,and the coupling between photons and atoms and the repulsive interaction between atoms are introduced.By using the mean-field method,we show the global ground-state phase diagrams and detailed descriptions of the extended hard-core and soft-core Bose-Hubbard model in a cavity.It is found that a small range of supersolid phase exists between the solid phases and the superfluid phases in the extended hard-core Bose-Hubbard model.Besides,for the extended soft-core Bose-Hubbard modelon the one dimensional lattices in a cavity,we find that there is a big range of supersolid phase.To ensure that the supersolid phase of the mean-field method is stable,we verify the mean-field results by using the density-matrixrenormalization-group method.Moreover,we also use machine learning method to distinguish the ordered phase and disordered phase of one-dimensional quantum Ising model in a transverse field.The structure of this thesis is arranged as follows:In the first chapter,we describe the Ising model in a traverse field and the extended Bose-Hubbard model in a cavity.And then we introduce the research background and motivations of finding the supersolid from the extended Bose-Hubbard model in a cavity and using machine learning method to identify quantum phases of quantum Ising model in a traverse field.In the second chapter,the mean-field method and density-matrixrenormalization-group method are introduced to simulate the extended BoseHubbard model in a cavity.In the third chapter,the extended hard-core and soft-core Bose-Hubbard model on the one-dimensional lattices in a cavity is studied by using mean-field method,and the results are given.By using the exact density-matrixrenormalization-group method,the structural factor,momentum distribution and correlation function are given to verify the important research results of meanfield method.In the fourth chapter,the machine learning method used in this paper is introduced,such as t-distributed stochastic neighbor embedding,fully connected network and convolutional neural network.The extended configuration of onedimensional quantum Ising model is generated by zero temperature quantum Monte-Carlo method,and the magnetization and magnetic susceptibility are measured.By using the machine learning method,the distribution results of t-distributed stochastic neighbor embedding of the extended configuration,the average values of the output neuron and the effective magnetic susceptibility with the transverse field are given to determine the ordered and disordered phase of the quantum Ising model.Finally,chapter V shows the summaries and outlooks.