Quantum Phases Of Bosons On A Two-leg Ladder In Artificial Gauge Field

The strong correlation system is one of the most important research fields in condensed matter physics due to tremendous novel physical effects,such as high temperature superconductivity and quantum phase transition.Since interactions among particles may lead to the failure of energy band theory,and it is often difficult to obtain a rigorous solution through analytical derivations,more numerical methods are applied to solve such problems.The tensor network method is one of them,especially the density matrix renormalization group(DMRG)algorithm,which can obtain very accurate results in onedimensional systems.In addition,the Harper-Hofstadter model has drawn widespread attention due to its relationship with quantum Hall effects and topological insulators.A limit is the ladder geometry where the system is extended in one direction but only contains two legs in the other direction which also harbors a large variety of quantum phases.In chapter 1,we briefly describe the numerical methods used in the strong correlation system and the development process of the Harper-Hofstadter model.In chapter 2,we introduce the principle and numerical tricks of the density matrix renormalization group.We use the matrix product state(MPS)language to describe the basic content and implementation of single-site DMRG,and introduce several important numerical tricks,including good quantum number and subspace singular value decomposition(SVD),compression of matrix product operator and subspace expansion.In chapter 3,we first introduce the history of the Harper-Hofstadter model,and then present the single-particle model of the Harper-Hofstadter ladder,in which the ground state of the Meissner-vortex phase transion occurs under the change of hopping or flux.Then we describe its implementation in cold atom experiments.Finally,based on the research of single-component hard-core and soft-core bosonic systems,the DMRG method is used to numerically study the quantum phases and quantum phase transitions of the two-component hard-core ladder model under the inter-species interactions.We find that when two kinds of bosons have the opposite flux,the system will undergo a phase transition from the vortex Mott insulator to the Meissner spin density wave state.If they have the same flux on the contrary,the spin density wave will vanish at larger interaction while the Meissner state still remains.