Diagrammatic Monte Carlo Method And The Application On Many-fermion System

This thesis is a summary of the author's recent three year's works,which is focused on the diagrammatic Monte Carlo(Diag-MC)methods and the application on the Fermi liquid.Diag-MC is a powerful method for studying the interacting fermionic models.For the Feynman diagram expansions of the interacting system,The high-order ampli-tudes are easy to obtain with the help of the Monte Carlo numerical method,while the analytic methods could do nothing.Furthermore,the Diag-MC method works directly in the thermodynamic limit and overcomes the conventional sign problem exponentially dependent on the system volume.Firstly,we study the Fermionic sign structure of high-order Feynman diagrams in a many-fermion system.The sign cancellation between scattering amplitudes makes fermions different from bosons.We systematically investigate Feynman diagrams'fermionic sign structure in a representative many-fermion system—a uniform Fermi gas with Yukawa interaction.We analyze the role of the crossing symmetry and the global gauge symmetry in the fermionic sign cancellation.The symmetry arguments are then used to identify the sign-canceled groups of diagrams.Sign-structure analysis has two applications.Numerically,it leads to a cluster diagrammatic Monte Carlo al-gorithm for fast diagram evaluations.This algorithm is about 105 times faster than the conventional approaches in the sixth order.Analytically,our analysis systematically reveals the relevant diagrams that dominate the dynamics.Then we apply the Diag-MC method to the three-dimensional uniform electron gas.We establish the quasiparticle interaction in the three-dimensional uniform electron gas by using this controlled effective field theory approach.We accurately determine the angle-resolved Landau interaction function as well as the Landau parameters.Several emergent features of the forward-scattering electron interaction on the Fermi surface are identified.In particular,we find that the different scattering channels of two electrons with parallel momenta and opposite spins almost perfectly cancel out,indicating the electrons are nearly asymptotically free in this limit.The thesis is organized as follows:Chapter One is a simple introduction to the basic Monte Carlo method and the sign problem.We introduce the Feynman diagram technique and counter term renormalization method in Chapter Two.Chapter Three provides some details of our diagrammatic Monte Carlo methods,which is an integra-tion of the Feynman diagram technique,the Monte Carlo method,and the counter term renormalization method.Chapter Four presents our work on the Fermionic sign struc-ture of high-order Feynman diagrams in a many-fermion system.The knowledge of sign cancellation is used to group the Feynman diagrams and improve the efficiency of the Diag-MC method.Chapter Five mainly focuses on the application of the Diag-MC to the three-dimensional uniform electron gas.We establish the angle-resolved Landau interaction function and physical scattering amplitude.