Nonlocal Interactions in Quantum Systems
23 Oct 2015 NUS mathematicians have developed a computational method for fast and accurate evaluation of nonlocal interactions in quantum systems.
Nonlocal (long-range) interactions are encountered in modeling a variety of problems from quantum physics and chemistry to materials science and biology. Typical examples include the Coulomb interaction for electronic structure calculations in materials simulation and design via the density functional theory (DFT) and the dipole-dipole interaction in quantum chemistry, dipolar Bose-Einstein condensation and Rydberg molecules. Due to high dimensions and/or the singularity in the interaction kernel, it is a very challenging and demanding problem to efficiently and accurately evaluate nonlocal interaction and the related interaction energy.
A team led by Prof BAO Weizhu from the Department of Mathematics in NUS has proposed an efficient computational method for fast and accurate evaluation of the nonlocal (long-range) interaction and the related interaction energy via the nonuniform fast Fourier transform (NUFFT). The key ideas are to reformulate the nonlocal interaction in the Fourier (phase) space, to adopt spherical/polar coordinates in phase space in three/two dimensions so that the singularity in the origin is canceled naturally and to accelerate the computation via NUFFT. The computational method is easy to be implemented and very efficient with O(N ln N) operations with N the total number of discretization points and it can achieve spectral accuracy.
The computational method has been applied to simulate ground state and dynamics of the Schrodinger equation with nonlocal interactions for modeling many-body quantum system with binary Coulomb interaction and the Gross-Pitaevskii equation for modeling dipolar degenerate quantum gas with/without angular momentum rotation and/or spin-orbit coupling. It is expected that the method and its code for the evaluation of nonlocal interactions will be integrated into some DFT packages for electronic structure calculations in materials simulation and design.
Figure shows the dynamics of the Schrodinger equation with the Coulomb interaction and a honeycomb potential in 2D. [Image credit: BAO Weizhu]
1. Bao W, Jiang S, Tang Q, Zhang Y. "Computing the ground state and dynamics of the nonlinear Schrodinger equation with nonlocal interactions via the nonuniform FFT.’’ Journal of Computational Physics, 296 (2015) 72.
2. Jiang S, Greengard L, Bao W. “Fast and accurate evaluation of nonlocal Couolmb and dipole-dipole interactions via the nonuniform FFT.” SIAM Journal on Scientific Computing, 36 (2014) B777.