Operator complexity: a journey to the edge of Krylov space

Sep 3, 2020
Published in:
  • JHEP 06 (2021) 062
  • Published: Jun 9, 2021
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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 ts_{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 SYK4_{4} model, which is maximally chaotic, and compare the results with the SYK2_{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