Research On Parallel Exact Diagonalization Algorithm For Quantum Many-body Systems

Exact Diagonalization is a reliable numerical method for studying various problems in quantum many-body physics.Its research scope includes the chemical properties of strongly correlated lattice models,nuclear structure and quantum field in condensed mat-ter physics.The main limitation of the Exact Diagonalization algorithm is that the compu-tational load and memory demand increase exponentially with the increase of the system size.The storage methods of the matrix and vector in the algorithm need to be optimized to realize the ability to calculate the extreme eigenvalues of large-scale systems.As the amount of calculation increases,the computation time must be controlled within a rea-sonable time by massive parallelism.In order to solve the above problems,the main research contents of this paper are as follows:1.Aiming at the storage bottleneck of Exact Diagonalization,the CSR storage format was used to storage the Hamiltonian matrix to reduce the memory consumption,and the Hubbard model with 12 sites was solved on the premise of maintaining the computational efficiency.Through the strategy of dynamic calculation of Hamiltonian matrix elements,which greatly reduced the memory comsumption of Exact Diagonalization while increas-ing the computational amount of the program,we realized the exact diagonalization of the Hubbard model with 24,32 and 36 sites.2.A parallel QR algorithm is designed by transforming QR decomposition into the calculation of column vector q₁?q_n,which is used to solve all eigenvalues and eigen-vectors of the Hamiltonian matrix.When solving the Hubbard model with 8 sites,the maximum speedup is 1.42.3.In view of the shortcoming that the algorithm generating all basis state costs a lot of time for the large system,an new algorithm calculating all basis state by permutation and combination law of electrons is proposed and the time complexity of the algorithm is reduced from O(2^M×M)to O(M!/(N_?!(M-N_?)!)).When calculating all basis state of the system with 36 sites and 8 electrons(N_?=N_?=4),the time taken was reduced from6837.4s to 1.3s,achieving a speedup of 5251.4.In order to accelerate the process of solving the quantum many-body system,a parallel MF-Lanczos algorithm was designed.In the calculation of matrix vector multi-plication in a single Lanczos iteration,the calculation process of the matrix H is evenly divided into different processes according to rows of the matrix to accelerate the pro-cess.When solving the Hubbard model with 36 sites and 10 electrons,the program was extended to 31416 processes at most and maintained a parallel efficiency of 53.17%.