Dual applications of Chebyshev polynomials method: Efficiently finding thousands of central eigenvalues for many-spin systems

Abstract

Computation of a large group of interior eigenvalues at the middle spectrum is an important problem for quantum many-body systems, where the level statistics provides characteristic signatures of quantum chaos. We propose an exact numerical method, dual applications of Chebyshev polynomials (DACP), to simultaneously find thousands of central eigenvalues, where the level space decreases exponentially with the system size. To disentangle the near-degenerate problem, we employ twice the Chebyshev polynomials, to construct an exponential semicircle filter as a preconditioning step and to generate a large set of proper basis states in the desired subspace. Numerical calculations on Ising spin chain and spin glass shards confirm the correctness and efficiency of DACP. As numerical results demonstrate, DACP is 30 times faster than the state-of-the-art shift-invert method for the Ising spin chain while 8 times faster for the spin glass shards. In contrast to the shift-invert method, the computation time of DACP is only weakly influenced by the required number of eigenvalues, which renders it a powerful tool for large scale eigenvalues computations. Moreover, the consumed memory also remains a small constant (5.6 GB) for spin-1/2 systems consisting of up to 20 spins, making it desirable for parallel computing.

Stability of many-body localization in Kicked Ising model

“…To find eigenvectors |ψ n and the corresponding eigenphases e iφn of the unitary operator U KIM , we employ the POLFED algorithm [62]. The algorithm is based on a block Lanczos iteration [155][156][157] performed for a polynomial g K (U KIM ) of order K of the matrix U KIM (see [158][159][160][161] for similar techniques). The matrix g K (U KIM ) has the same eigenvectors |ψ n as U KIM , but its eigenvalues are equal to g K (e iφn ).…”