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.
Highlights
Complete numerical analysis of K-complexity for all time scales, including scrambling and well beyond, in concrete models demonstrating the above-described time-dependence
Since for chaotic systems K ∼ D2 and in general D ∼ eS, where S is the entropy of the system, we find that the saturation value of K-Entropy is essentially of order S, while the saturation value of K-complexity is of order e2S
We have found an algorithmic expression for the dimension of Krylov space K
Summary
We begin by providing the definition of Krylov space and stressing its relation to the structure of the spectrum of the system under consideration (and to its integrable or chaotic character). If the operator O has a non-vanishing projection over several eigenstates of the Liouvillian that share the same degenerate eigenvalue, this phase only contributes one dimension to the Krylov space. |K0) = Oaa |ωaa) = Oaa |Ea Ea| ≡ OaaPa. To summarize, K is determined by exploiting the advantages of the basis in operator space induced by the eigenbasis of the Hamiltonian; crucially, the determinant of a suitable matrix containing the representation of these nested commutators in the given basis reduces to a Vandermonde determinant of energy differences. The eigenvalues of the Liouvillian are precisely all possible energy differences, ωab = Ea − Eb. The zero phase is always at least D times degenerate, the Krylov dimension is bounded by (2.7), for any non-zero operator. We have confirmed both expectations by studying the chaotic SYK4 model (see numerics below) as well as RMT (see appendix A), and the integrable SYK2 model, see section 3 below
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
