Dual application of Chebyshev polynomial for efficiently computing thousands of central eigenvalues in many-spin systems

Abstract

It is known that the statistical properties of the spectrum provide an essential characterization of quantum chaos. The computation of a large group of interior eigenvalues at the middle spectrum is thus an important problem for quantum many-body systems. We propose a dual application of Chebyshev polynomial (DACP) method to effciently find thousands of central eigenvalues, which are exponentially close to each other in terms of the system size. To cope with the near-degenerate problem, we use the Chebyshev polynomial to both construct an exponential of semicircle filter as the preconditioning step and generate a large set of proper states as the basis of the desired subspace. Besides, DACP owes an excellent property that its computation time is not influenced by the required number of eigenvalues. Numerical experiments on Ising spin chain and spin glass shards show the correctness and effciency of the proposed method. As our results demonstrate, DACP is a factor of 30 faster than the state-of-the-art shift-invert method for the Ising spin chain while 8 times faster for the spin glass shards. The memory requirements scale better with system size and could be a factor of 100 less than in the shift-invert approach.