Operator complexity: a journey to the edge of Krylov space

A bstract 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 succ...

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Vydáno v:The journal of high energy physics Ročník 2021; číslo 6; s. 1 - 24
Hlavní autoři: Rabinovici, E., Sánchez-Garrido, A., Shir, R., Sonner, J.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2021
Springer Nature B.V
SpringerOpen
Témata:
AdS-CFT Correspondence
Algorithms
Black holes
Classical and Quantum Gravitation
Commutators
Complexity
Elementary Particles
Evolution
Field Theories in Lower Dimensions
High energy physics
Hilbert space
Nonperturbative Effects
Numerical analysis
Parallel processing
Physical properties
Physics
Physics and Astronomy
Quantum Field Theories
Quantum Field Theory
Quantum Physics
Random Systems
Regular Article - Theoretical Physics
Relativity Theory
String Theory
AdS-CFT Correspondence
Field Theories in Lower Dimensions
Random Systems
Nonperturbative Effects
ISSN:1029-8479, 1029-8479
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Popis
Shrnutí:A bstract 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.
Bibliografie:ObjectType-Article-1
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ISSN:1029-8479
1029-8479
DOI:10.1007/JHEP06(2021)062