Determinant Quantum Monte Carlo Study Of Lattice Models Of Strongly Correlated Electron Systems

The physical phenomena of strongly correlated electron systems in condensed mat-ter physics have always been an important and active direction.For many years,the joint research work combining theories and experiments greatly expands our understanding of condensed matter physics.One important research object in strongly correlated electron systems is the high T_c cuprate superconductor.In its phase diagram,the antiferromagnetic Mott insulator near half filling is beyond the description of traditional band theory and is the result of electron-electron interaction.The pseudogap phase at the underdoped region has discontinuous Fermi surface namely Fermi arc in the Brillouin zone.The strange metal phase above the d-wave superconductivity dome has novel be-havior of electrical conductivity that defies the Landau Fermi liquid theory.It was also discovered that there exists fractional quasiparticle excitation namely anyon in fractional quantum Hall effect(FQHE).FQHEs are strongly correlated electron system that is beyond Landau's phase classification based on symmetry.Apart from these phenomena,there are strongly correlated systems worthy research such as the non-Fermi liquid in heavy fermion materials,charge-density-wave(CDW)phase in strong electron-phonon systems,phase transition in Dirac fermion systems in condensed matter physics,etc.It is very important to have theoretical methods,numerical methods and experimental approaches working together to study the physical systems in condensed matter physics.Especially in strongly correlated systems,pure theoretical analysis cannot lead to the complete solution to the problems due to their complexity and difficulty.The devel-opment and progress of numerical methods are thus very meaningful for the research of strongly correlated systems.Determinant quantum Monte Carlo(DQMC)is an effective method to study the strongly correlated electron problems.In recent years,re-searchers not only try to ease the notorious sign problem,but also spend quite some efforts to increase the efficiency of DQMC's performance on certain models.In this thesis,I study the Holstein model of electron-phonon coupling using DQMC.Firstly with the help of self-learning Monte Carlo(SLMC),we considerably increase the efficiency of numerical simulation of two dimensional square lattice Holstein model at half filling.Along this route,I study the Holstein model on honeycomb lattice and find that there are two phases,semimetal and CDW.The quantum phase transition seems to belong to chiral Ising universality class.I study the square lattice Hubbard model at half filling and finite temperatures and analyze the temperature evolution of local density of states.Combining with the results from slave-fermion approach,we show that the pseudogap phenomenon is the result of charge excitation renormalized by spin fluctuations.Together with the exponential tensor renormalization group(XTRG)method,we also study the doped Hubbard model,obtain the real space spin correlation function and compare it with that from cold atom experiments.Our results show that upon doping,the diagonal and third-nearest neighbor spin correlation functions show signs of sign change.At last,I study the model of orthogonal fermion coupled with Z₂ gauge field and Ising matter field.By tuning parameters in the model,we realize orthogonal metal phase,Fermi arc phase,deconfined Fermi liquid phase,s-wave superconducting phase,etc.Our numerical findings indicate that the numerical study of lattice model of strongly correlated electron systems can realize abundant and novel strongly correlating physical phenomena such as non-Fermi liquid and the analogous pseudogap phase that violates Luttinger's theorem.And unbiased large scale numerical simulations will certainly further bridge the gap between the experimental observations of real materials and various analytical theories.