Development And Application Of Quantum Monte Carlo Method

Strongly correlated quantum many-body systems possess rich physical phenomena,making them an essential topic in condensed matter physics.Physicists have systematically explored the strong correlated systems and developed various theories and numerical methods.Strong correlated systems not only possess rich physical phenomena,but also have great potential applications.However,due to the complexity of strong correlated system,especially its intrinsic strong correlation nature,many conventional theoretical methods fail to work.Therefore,it is of great scientific significance to develop efficient and accurate theoretical or numerical methods for strongly correlated systems.Among various theoretical methods,quantum Monte Carlo method is considered to be a very effective numerical method for strongly correlated systems.However,the current quantum Monte Carlo method still has two formidable obstacles.First,the computational cost is very heavy,which usually has the cubic scaling with the size of the system,or even higher.For example,among the known calculation methods for fermions,the calculation of determinant is essential.The complexity of determinant calculation has a cubic scaling with the size of the system.Thus,with the increase of the system,the amount of calculation will increase sharply,which undoubtedly limits its application.Second,when quantum Monte Carlo method is used to simulate the fermion systems,the sign problem is another huge obstacle.Now,it is urgent to develop new numerical methods.In view of the difficulties and current situation in the study of strong correlated systems,this thesis has made progress in four aspects:1)A new expression of path integral formula.By combining the off-diagonal operator in Hamiltonian with its Hermite conjugate into a pair,we present a new formula for path integral.The new formula has several obvious advantages:(1)there is no need to introduce Hubbard-Stratonovich transformation,so it does not involve the calculation of auxiliary field;(2)The determinant is absent,which simplifies the computational complexity of the system;(3)The accuracy of Suzuki-Trotter decomposition is improved;(4)It provides a new perspective and scheme for thinking and solving sign problem.2)New Monte Carlo algorithm.In order to obtain the path with non-zero weight,we design a new algorithm by combining multiple-time-slice threading and local heat bath technique.The preliminary test shows that the computational cost of our method has square scaling with the size of the system,which is comparable to the usual first principles calculation.As an application,we calculate the results of one-dimensional Hubbard model at finite temperature,which are consistent with the accurate values.3)A generic solution of fermion sign problem.Based on the new path integral formula,we find a universal strategy to solve the sign problem.In this strategy,paths with negative weight are completely offset or replaced by some paths with positive weight.Through rigorous mathematical proof,we find that physical quantities can be exactly calculated in a specific path space.In these special spaces,each path has a positive weight.According to this finding,we have provided a scheme to deal with the fermion sign problem.As an example,the current method is applied to the twodimensional Hubbard model,and the results do manifest the correctness.4)Studies on one-dimensional and two-dimensional impurity systems.Based on our quantum Monte Carlo method,we have studied the one-and two-dimensional Hubbard model with one impurity.The results show that,due to the influence of the impurity the correlation function gradually appears periodic oscillation with the increase of system size and the density correlation is promoted.