Operator complexity: a journey to the edge of Krylov space
Operator complexity: a journey to the edge of Krylov space
Heisenberg time evolution under a chaotic many-body Hamiltonian H transforms an initially simple operator into an increasingly complex one, as it spreads over Hilbert space. Krylov complexity, or ‘K-complexity’, quantifies this growth with respect to a special basis, generated by H by successive nested commutators with the operator. In this work we study the evolution of K-complexity in finite-entropy systems for time scales greater than the scrambling time ts > log(S). We prove rigorous bounds on K-complexity as well as the associated Lanczos sequence and, using refined parallelized algorithms, we undertake a detailed numerical study of these quantities in the SYK4 model, which is maximally chaotic, and compare the results with the SYK2 model, which is integrable. While the former saturates the bound, the latter stays exponentially below it. We discuss to what extent this is a generic feature distinguishing between chaotic vs. integrable systems.
Rabinovici, E.
f97c4550-c0ad-4837-a529-42387151ea9d
Sánchez-Garrido, A.
6add42c4-992e-455c-a098-f26e85168537
Shir, Ruth
80e05c9b-9440-4c4d-a5b8-95cebfaa32f3
Sonner, J.
1d2008de-dbc3-4231-95e6-a3d2ec93d3c1
9 June 2021
Rabinovici, E.
f97c4550-c0ad-4837-a529-42387151ea9d
Sánchez-Garrido, A.
6add42c4-992e-455c-a098-f26e85168537
Shir, Ruth
80e05c9b-9440-4c4d-a5b8-95cebfaa32f3
Sonner, J.
1d2008de-dbc3-4231-95e6-a3d2ec93d3c1
Rabinovici, E., Sánchez-Garrido, A., Shir, Ruth and Sonner, J.
(2021)
Operator complexity: a journey to the edge of Krylov space.
JHEP, 2021, [62].
(doi:10.1007/JHEP06(2021)062).
Abstract
Heisenberg time evolution under a chaotic many-body Hamiltonian H transforms an initially simple operator into an increasingly complex one, as it spreads over Hilbert space. Krylov complexity, or ‘K-complexity’, quantifies this growth with respect to a special basis, generated by H by successive nested commutators with the operator. In this work we study the evolution of K-complexity in finite-entropy systems for time scales greater than the scrambling time ts > log(S). We prove rigorous bounds on K-complexity as well as the associated Lanczos sequence and, using refined parallelized algorithms, we undertake a detailed numerical study of these quantities in the SYK4 model, which is maximally chaotic, and compare the results with the SYK2 model, which is integrable. While the former saturates the bound, the latter stays exponentially below it. We discuss to what extent this is a generic feature distinguishing between chaotic vs. integrable systems.
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JHEP06(2021)062
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Accepted/In Press date: 30 May 2021
Published date: 9 June 2021
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Local EPrints ID: 501039
URI: http://eprints.soton.ac.uk/id/eprint/501039
ISSN: 1126-6708
PURE UUID: 18ca8098-ea8f-4705-a290-ca7f9c603096
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Date deposited: 21 May 2025 16:31
Last modified: 22 Aug 2025 02:43
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Author:
E. Rabinovici
Author:
A. Sánchez-Garrido
Author:
Ruth Shir
Author:
J. Sonner
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