Operator complexity: a journey to the edge of Krylov space
- URL: http://arxiv.org/abs/2009.01862v2
- Date: Tue, 22 Jun 2021 19:47:44 GMT
- Title: Operator complexity: a journey to the edge of Krylov space
- Authors: E. Rabinovici, A. S\'anchez-Garrido, R. Shir and J. Sonner
- Abstract summary: Krylov complexity, or K-complexity', quantifies this growth with respect to a special basis.
We study the evolution of K-complexity in finite-entropy systems for time scales greater than the scrambling time.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- 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 $t_s>\log (S)$. 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
SYK$_4$ model, which is maximally chaotic, and compare the results with the
SYK$_2$ 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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