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.
Highlights
We propose an exact numerical method, dual applications of Chebyshev polynomials (DACP), to calculate thousands of eigenvalues at the middle of the energy band, which is enough to reveal the level statistics [26, 27]
We propose the DACP method to efficiently calculate a large scale of central eigenvalues with high precision, by employing twice the Chebyshev polynomials
Two key features distinguish DACP from the polynomial filtering methods: its computation time depends weakly on the number of required eigenvalues, while its memory overhead is independent of the system size
Summary
Energy level statistics provides an essential characterization of quantum chaos [1, 2]. We propose an exact numerical method, DACP, to calculate thousands of eigenvalues at the middle of the energy band, which is enough to reveal the level statistics [26, 27]. For spin systems, such a middle region usually indicates a peak of the density of states where the energy levels are nearly degenerate. The second application of the Chebyshev polynomial is to fast search a set of states to span the specific subspace, which consists of all the desired eigenstates Combining these two steps, the DACP method essentially transforms the original high-dimension eigenvalue problem to a low-dimension one.
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