论文标题
Chebyshev多项式的双重应用有效地计算了许多旋转系统中的数千个中央特征值
Dual application of Chebyshev polynomial for efficiently computing thousands of central eigenvalues in many-spin systems
论文作者
论文摘要
众所周知，光谱的统计特性提供了量子混乱的基本特征。因此，对于量子多体系统，在中间频谱上计算大量内部特征值是一个重要的问题。我们提出了Chebyshev多项式（DACP）方法的双重应用，以有效地找到成千上万的中央特征值，这些中央值在系统大小方面彼此相近。为了应对近定位问题，我们使用Chebyshev多项式来构建半圆形滤波器的指数为预处理步骤，并生成一大批适当的状态作为所需子空间的基础。此外，DACP欠其计算时间不受所需特征值数量的影响。关于Ising自旋链和自旋玻璃碎片的数值实验显示了所提出方法的正确性和效率。正如我们的结果所证明的那样，DACP比Ising自旋链的最新移位旋转方法快30倍，而自旋玻璃碎片的速度快8倍。记忆需求量表的尺寸更好，并且可能比转移方法少100倍。
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.
