Quantum Monte Carlo Simulations Of Fermion-boson Lattice Systems

Fermion-boson lattice model is a model which describe the coupling between fermion and bosonic quantum fluctuations.As the model directly captures the quantum fluctuations of order parameter in correlated electron systems,it is meaningful for exploring the quantum critical behavior,thermodynamics and many possible new phases in quantum critical region.In this thesis,we will mainly discuss quantum Monte Carlo simulation of fermion-boson lattice model,including topological phase transition driven by the coupling between the Dirac fermion and Ising nematic quantum fluctuations,non-fermi-liquid at itinerant ferromagnetic quantum critical point as well as our attempt on the improvements on the algorithm of quantum Monte Carlo.At first,interaction-driven topological phase transitions in Dirac semimetals are investigated by means of large-scale quantum Monte Carlo simulations.The interaction among Dirac fermions is introduced by coupling them to Ising spins that realize the quantum dynamics of the two-dimensional transverse field Ising model.The ground state phase diagram,in which the tuning parameters are the transverse field and the coupling between fermion and Ising spins,is determined.At weak and intermediate coupling,a second-order Ising quantum phase transition and a first-order topological phase transition between two topologically distinct Dirac semimetals are observed.Interestingly,at the latter,the Dirac points smear out to form nodal lines in the Brillouin zone,and collective bosonic fluctuations with SO(4)symmetry are strongly enhanced.At strong coupling,these two phase boundaries merge into a first-order transition.Secondly,we studied the non-fermi-liquid behavior at the itinerant ferromagnetic quantum critical point.We construct a two-dimensional lattice model of fermions coupled to Ising ferromagnetic critical fluctuations.Using extensive sign-problem-free quantum Monte Carlo simulations,we show that the model realizes a continuous itinerant quantum phase transition.In comparison with other similar itinerant quantum critical points,our quantum critical point shows less strong superconductivity instability,making the system an ideal platform for the exploration of pristine itinerant quantum critical point.Remarkably,clear signatures of non-Fermi-liquid behavior in the fermion propagators are observed at the quantum critical point.The critical fluctuations at the quantum critical point partially resemble Hertz-Millis-Moriya behavior.However,careful scaling analysis reveals that the quantum critical point belongs to a different universality class,deviating from both(2+1)d Ising and Hertz-Millis-Moriya predictions.Finally,we introduce a new quantum Monte Carlo method,namely,selflearning quantum Monte Carlo method,for solving(critical)slow down in the quantum Monte Carlo simulations.We implement it in the framework of determinantal quantum Monte Carlo,and calculate a problem of the bosonic fluctuations coupled to a Fermi surface.Due to the high speed-up of the self-learning quantum Monte Carlo,we can calculate systems as large as 100 × 100 at critical point and obtain critical exponents with high precision.