Solving large-scale eigen problem in quantum few-body system on massive parallel computer

We present a benchmark study for solving large-scale eigenvalue problems arising in the quantum few-body systems composed of 4He atoms. Applying the variational method to the quantum mechanical equation leads to a generalized eigenvalue problem for a dense matrix whose rank is of the order of 104–105. After determining the matrix elements with our efficient parallel code, the diagonalization is performed using the EigenExa library, which is optimized for massively parallel computing. We employ Supercomputer Fugaku for these computations. A large-scale expansion with non-orthogonal basis functions often induces a numerical instability in the result, due to numerically near-zero or slightly negative eigenvalues in the overlap matrices. To avoid such an instability, in the construction of the transformation matrix composed of the eigenvectors of the overlap matrix, we need to introduce a truncation of small eigenvalues. We apply these procedures to a five-body 4He system with matrices of various ranks up to ∼ 300, 000, as well as changing the features of the basis functions, and find that the stable lowest eigenvalues can be successfully determined. We also measure the practical efficiency of the EigenExa library and examine the effect of the accuracy booster, which is introduced to reduce the numerical errors. These analyses provide us with knowledge of numerical criteria for solving large-scale eigenvalue problems as preparation for tackling systems with larger numbers of particles, whose matrix should have a million-scale rank.