Krylov complexity in large-$q$ and double-scaled SYK model

Abstract

Considering the large-q expansion of the Sachdev-Ye-Kitaev (SYK) model in the two-stage limit, we compute the Lanczos coefficients, Krylov complexity, and the higher Krylov cumulants in subleading order, along with the t/q effects. The Krylov complexity naturally describes the "size" of the distribution while the higher cumulants encode richer information. We further consider the double-scaled limit of SYK q at infinite temperature, where q ∼ √ N . In such a limit, we find that the scrambling time shrinks to zero, and the Lanczos coefficients diverge. The growth of Krylov complexity appears to be "hyperfast", which is previously conjectured to be associated with scrambling in de Sitter space.

Operator growth in open quantum systems: lessons from the dissipative SYK

“…A distinct approach is to generalize the "moment method" [52], which was previously studied in closed SYK [1], and its higher-order corrections [33]. Rather than constructing a basis, this method focuses on the auto-correlation function, or its derivatives around t = 0, known as the moments:…”

“…In this paper, we aim to study the operator growth in the dissipative Sachdev-Ye-Kitaev (SYK) model from the point of view of the K-complexity [1] (for an incomplete list of studies, see [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38] and the references therein). The SYK model is a (0 + 1)-dimensional quantum mechanical model which consists of N fermions where q of them are interacting at a time [39].…”

“…The chaotic nature of this model is well captured by OTOC, satisfying the chaos bound [45]. Several recent studies have also studied Kcomplexity in SYK analytically for closed system [1,33] and numerically for closed [21,22] and open/dissipative systems [20]. The numerical results are performed for finite N and finite q (especially q = 4) [20][21][22], where the Krylov bases have been constructed using the Lanczos algorithm [46].…”

“…Then we give an analytical check by generalizing the method of moments [52]. Previously, a specific version of this method was applied to closed systems [1,33]. We extract the analytical form of two sets of Lanczos coefficients, which correspond to the primary diagonal and primary off-diagonal elements of the Lindbladian matrix written in Krylov basis [19].…”

Assessing the saturation of Krylov complexity as a measure of chaos

“…The K complexity is a quantitative measure of this dynamic and it has recently caused a great deal of activity within the community. Many works study this metric for a wide variety of systems [11][12][13][14][15][16][17][18][19].…”