Delta-Davidson method for interior eigenproblem in many-spin systems*

Abstract

Many numerical methods, such as tensor network approaches including density matrix renormalization group calculations, have been developed to calculate the extreme/ground states of quantum many-body systems. However, little attention has been paid to the central states, which are exponentially close to each other in terms of system size. We propose a delta-Davidson (DELDAV) method to efficiently find such interior (including the central) states in many-spin systems. The DELDAV method utilizes a delta filter in Chebyshev polynomial expansion combined with subspace diagonalization to overcome the nearly degenerate problem. Numerical experiments on Ising spin chain and spin glass shards show the correctness, efficiency, and robustness of the proposed method in finding the interior states as well as the ground states. The sought interior states may be employed to identify many-body localization phase, quantum chaos, and extremely long-time dynamical structure.

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

“…We remark that the plateau at the middle region in Figure 5 distinguishes the DACP method from those iterative filtering ones [10,20,23,24,28,29], although all of them make use of the Chebyshev polynomials. The iterative methods usually require a larger amount of filtrations and reorthogonalizations, thus a longer convergence time, in finding more eigenvalues [8,20,22].…”

“…The first one is the Dirac delta function δ(0), which was employed to efficiently find interior eigenvalues in Refs. [21][22][23]32]. The second is the rectangular function, which was also used for interior eigenvalue computations [24].…”

“…However, for large systems this method suffers rapid increases in computation time and memory consumption, due to the factorization of (H −λI) −1 . Other spectral filters have also been proposed, including the Dirac delta function filter δ(H − λI) [19][20][21][22][23] and the rectangular window function filter [24].…”