Optimization On The Density Matrix Renormalization Group Method

Strongly correlated system is an important research field of condensed matter physics,which contains abundant physical phenomena,such as various quantum phases and quantum phase transitions.Many strongly correlated phenomena are difficult to deal with qualitatively by perturbation theory,then numerical meth-ods are very important in this field.In particular,for low-dimensional strongly correlated lattice models,the density matrix renormalization group(DMRG)is proposed as one of the most important numerical methods,which is mainly used to solve the ground state and some low-lying excited states.Lots of DMRG cal-culations of these models show extremely high accuracy.At present,the method has been extended to other problems and some other research areas,for exam-ples:time-dependent problems,finite-temperature studies,two-dimensional sys-tems,quantum chemistry,etc.In addition,this method can be described based on the language of matrix product states and matrix product operators,and this greatly promotes the generalization of quantum many-body numerical methods,for exam-ples:the multiscale entanglement renormalization ansatz methods,the projected entangled pair state methods,etc.Though the dimension of Hilbert space which DMRG focus on is greatly reduced with keeping a small number of states in each DMRG step,it is also very huge in calculations of many problems,and this re-sults that long time and large storage space is needed by DMRG calculations to converge to ground states with high accuracy.Currently,lots of optimizations and improvements of this method has been studied to improve the accuracy and reduce the computation time and memory cost in practical applications.This paper has made further optimization and improvement based on previous research works,and it mainly includes following two parts.The real-space parallel DMRG is proposed to accelerate the DMRG calcula-tions by multiple nodes,and it has been applied to study some quasi-two-dimensional lattice models.It may achieve almost ideal acceleration for the reasonable initial interval division.However,it is difficult to determine good initial partitionings before real-space parallel DMRG calculations for some lattice models.Here we present a new real-space parallel DMRG method with dynamical boundaries.This method greatly reduces the waiting time between two neighboring nodes which occurs in the method with fixed boundaries due to unreasonable initial interval division and greatly improves the parallel efficiency.In DMRG calculations on shared bonds,better initial wave functions are provided in this paper,which ac-celerates the convergence of diagonalization methods.The improved method is applied to calculations of ground state energy of the Heisenberg model on two-leg ladder and the water molecule in quantum chemistry.The maximum parallel effi-ciency achieved in 4 nodes are 91%and 76%for these two models,respectively.Meanwhile,numerical calculations show that the speedup of the parallel method with fixed boundaries seriously depends on the initial interval division,however the method with dynamical boundaries can always obtain a higher speedup.With the development of high-performance computing,the graphic process-ing unit(GPU)can be applied to general-purpose computing and provides sev-eral times higher performance than the central processing unit(CPU).Developing heterogeneous parallel algorithms based on CPU and GPU becomes an effective method to optimize many numerical methods.In this paper,a new CPU-GPU het-erogeneous parallel DMRG algorithm is implemented.Compared with the previ-ous heterogeneous parallel algorithms,a tensor contraction method with less com-putation cost is used in diagonalizing Hamiltonian.In addition,a heterogeneous parallel algorithm is also presented for the truncation procedure.Here we apply the optimized DMRG algorithm to the Fermi Hubbard model on a four-leg lad-der with next-nearest interaction.In the calculation of the ground state,the charge stripe consistent with the previous numerical work are obtained.Numerical results shows that the GPU memory cost is much less than the previous strategy,and the speedup increases with the number of kept states.In all our calculations,the maxi-mum GPU memory cost is less than 12 gigabytes,the number of kept states can be up to 10~4,and the maximum speedup is 2.7.All calculations indicate that this het-erogeneous parallel algorithm can be applied to study the ground state properties of some models.This thesis focuses on the improvement of the efficiency of the DMRG method and does some tests on some basic models.The research results obtained have yet to be further studied.It is hoped that it will make progress in studying more practical systems.