Abstract
Recently, Krylov complexity was proposed as a measure of complexity and chaoticity of quantum systems. We consider the stadium billiard as a typical example of the quantum mechanical system obtained by quantizing a classically chaotic system, and numerically evaluate Krylov complexity for operators and states. Despite no exponential growth of the Krylov complexity, we find a clear correlation between variances of Lanczos coefficients and classical Lyapunov exponents, and also a correlation with the statistical distribution of adjacent spacings of the quantum energy levels. This shows that the variances of Lanczos coefficients can be a measure of quantum chaos. The universality of the result is supported by our similar analysis of Sinai billiards. Our work provides a firm bridge between Krylov complexity and classical/quantum chaos.
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Acknowledgments
The work of K. H. was supported in part by JSPS KAKENHI Grant No. JP22H01217, JP22H05111 and JP22H05115. The work of K. M. was supported in part by JSPS KAKENHI Grant No. JP20K03976, JP21H05186 and JP22H01217. The work of N. T. was supported in part by JSPS KAKENHI Grant No. JP18K03623 and JP21H05189. The work of R. W. was supported by Grant-in-Aid for JSPS Fellows No. JP22KJ1940.
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Hashimoto, K., Murata, K., Tanahashi, N. et al. Krylov complexity and chaos in quantum mechanics. J. High Energ. Phys. 2023, 40 (2023). https://doi.org/10.1007/JHEP11(2023)040
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DOI: https://doi.org/10.1007/JHEP11(2023)040