Use Numerical Methods To Study Quantum Phase Transitions

Tight binding lattice models describe low energy physics for particles in periodic potentials.Lattice models are fundamental for condensed matter physics and are most popular for quantum simulation experiments.The study of lattice models is important for studies of crystalline materials and quantum computing.For long time the major ways to study lattice models are analytical computations.The band theory for no interaction systems and the bosonization method for 1D interacting systems are the examples.However,the models that can be solved analytically are rare and special.Analytical methods are not enough to study the quick emerging lattice models.With the development of classical computers,numerical methods are able to solve this problem.A computer can quickly handle vast amount of data which is impossible for a human's work even in a life time.Many restrictions of parameters for analytical approximations are absent for numerical methods.Some famous numerical methods for studying lattice models are exact diagonalization(ED),density matrix renormalization group(DMRG)and Monte Carlo methods,which apply for different kinds of problems.This paper describes some of these numerical methods,including ED and DMRG,and use them to study several quantum many body systems.Using the basic ED method as the first part of the paper,we studied the simple problem of basis truncations.By simplifying the ED method for no interaction systems,we could handle quite large systems.And we used it to study quantum phases of Boson Hofstadter ladder with staggered hopping.A complete phase diagram was obtained.By modifying the code to study the quench dynamics of fermion systems,we showed numerically the overlaps and correlation functions in quench dynamics can reveal the difference of topological number for ground state phases.As the second part of the paper,we summarized the use of DMRG,including its characters and restrictions.Then we showed how this method is used to find current phases for strong interaction Hofstadter ladder with staggered hopping.By calculating the average chiral currents and overlaps,we obtained the whole phase diagram.Finally we summarized the influence of staggered hopping on Hofstadter ladder,including the ability of driving the Mott-superfluid transition and the different behavior of current phases in Mott and superfluid regimes.