Euclidean operator growth and quantum chaos
We consider growth of local operators under Euclidean time evolution in lattice systems with local interactions. We derive rigorous bounds on the operator norm growth and then proceed to establish an analog of the Lieb-Robinson bound for the spatial growth. In contrast to the Minkowski case when bal...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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American Physical Society
2020-11-01
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| Series: | Physical Review Research |
| Online Access: | http://doi.org/10.1103/PhysRevResearch.2.043234 |
| _version_ | 1827285906901434368 |
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| author | Alexander Avdoshkin Anatoly Dymarsky |
| author_facet | Alexander Avdoshkin Anatoly Dymarsky |
| author_sort | Alexander Avdoshkin |
| collection | DOAJ |
| description | We consider growth of local operators under Euclidean time evolution in lattice systems with local interactions. We derive rigorous bounds on the operator norm growth and then proceed to establish an analog of the Lieb-Robinson bound for the spatial growth. In contrast to the Minkowski case when ballistic spreading of operators is universal, in the Euclidean case spatial growth is system-dependent and indicates if the system is integrable or chaotic. In the integrable case, the Euclidean spatial growth is at most polynomial. In the chaotic case, it is the fastest possible: exponential in 1D, while in higher dimensions and on Bethe lattices local operators can reach spatial infinity in finite Euclidean time. We use bounds on the Euclidean growth to establish constraints on individual matrix elements and operator power spectrum. We show that one-dimensional systems are special with the power spectrum always being superexponentially suppressed at large frequencies. Finally, we relate the bound on the Euclidean growth to the bound on the growth of Lanczos coefficients. To that end, we develop a path integral formalism for the weighted Dyck paths and evaluate it using saddle point approximation. Using a conjectural connection between the growth of the Lanczos coefficients and the Lyapunov exponent controlling the growth of the out-of-time-ordered correlators (OTOCs), we propose an improved bound on chaos valid at all temperatures. |
| first_indexed | 2024-04-24T10:22:35Z |
| format | Article |
| id | doaj.art-381af61ff6084bb18a454d0c601d4e2b |
| institution | Directory Open Access Journal |
| issn | 2643-1564 |
| language | English |
| last_indexed | 2024-04-24T10:22:35Z |
| publishDate | 2020-11-01 |
| publisher | American Physical Society |
| record_format | Article |
| series | Physical Review Research |
| spelling | doaj.art-381af61ff6084bb18a454d0c601d4e2b2024-04-12T17:03:54ZengAmerican Physical SocietyPhysical Review Research2643-15642020-11-012404323410.1103/PhysRevResearch.2.043234Euclidean operator growth and quantum chaosAlexander AvdoshkinAnatoly DymarskyWe consider growth of local operators under Euclidean time evolution in lattice systems with local interactions. We derive rigorous bounds on the operator norm growth and then proceed to establish an analog of the Lieb-Robinson bound for the spatial growth. In contrast to the Minkowski case when ballistic spreading of operators is universal, in the Euclidean case spatial growth is system-dependent and indicates if the system is integrable or chaotic. In the integrable case, the Euclidean spatial growth is at most polynomial. In the chaotic case, it is the fastest possible: exponential in 1D, while in higher dimensions and on Bethe lattices local operators can reach spatial infinity in finite Euclidean time. We use bounds on the Euclidean growth to establish constraints on individual matrix elements and operator power spectrum. We show that one-dimensional systems are special with the power spectrum always being superexponentially suppressed at large frequencies. Finally, we relate the bound on the Euclidean growth to the bound on the growth of Lanczos coefficients. To that end, we develop a path integral formalism for the weighted Dyck paths and evaluate it using saddle point approximation. Using a conjectural connection between the growth of the Lanczos coefficients and the Lyapunov exponent controlling the growth of the out-of-time-ordered correlators (OTOCs), we propose an improved bound on chaos valid at all temperatures.http://doi.org/10.1103/PhysRevResearch.2.043234 |
| spellingShingle | Alexander Avdoshkin Anatoly Dymarsky Euclidean operator growth and quantum chaos Physical Review Research |
| title | Euclidean operator growth and quantum chaos |
| title_full | Euclidean operator growth and quantum chaos |
| title_fullStr | Euclidean operator growth and quantum chaos |
| title_full_unstemmed | Euclidean operator growth and quantum chaos |
| title_short | Euclidean operator growth and quantum chaos |
| title_sort | euclidean operator growth and quantum chaos |
| url | http://doi.org/10.1103/PhysRevResearch.2.043234 |
| work_keys_str_mv | AT alexanderavdoshkin euclideanoperatorgrowthandquantumchaos AT anatolydymarsky euclideanoperatorgrowthandquantumchaos |