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

### Submission summary

 As Contributors: Haoyu Guan Preprint link: scipost_202106_00048v3 Code repository: https://doi.org/10.5281/zenodo.5513873 Data repository: https://doi.org/10.5281/zenodo.5513873 Date accepted: 2021-11-01 Date submitted: 2021-10-24 17:19 Submitted by: Guan, Haoyu Submitted to: SciPost Physics Academic field: Physics Specialties: Quantum Physics Approach: Computational

### 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.

###### Current status:
Publication decision taken: accept

Editorial decision: For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)

### List of changes

1. We revise the first paragraph of the Conclusion Section to obtain a clearer expression.
2. We add the relationship between x and i, and illustrate the process of block filtering by a 2x2 matrix example in Appendix C.
3. We revise the last sentence of Appendix E to make it clearer.
4. We have proofread the manuscript again.