Operator complexity: a journey to the edge of Krylov space - INSPIRE

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Operator complexity: a journey to the edge of Krylov space

E. Rabinovici
(
- Hebrew U.
)
,
A. Sánchez-Garrido
(
- Geneva U., Dept. Theor. Phys.
)
,
R. Shir
(
- Hebrew U.
)
,
J. Sonner
(
- Geneva U., Dept. Theor. Phys.
)

Sep 3, 2020

Published in:

JHEP 06 (2021) 062

Published: Jun 9, 2021

e-Print:

2009.01862 [hep-th]

DOI:

10.1007/JHEP06(2021)062

View in:

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Abstract: (Springer)

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.

Note:

v2: published version

AdS-CFT Correspondence
Field Theories in Lower Dimensions
Nonperturbative Effects
Random Systems
chaos
integrability
commutation relations
many-body problem
Hilbert space
Hamiltonian

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