Krylov complexity in saddle-dominated scrambling

Abstract

In semi-classical systems, the exponential growth of the out-of-timeorder correlator (OTOC) is believed to be the hallmark of quantum chaos. However, on several occasions, it has been argued that, even in integrable systems, OTOC can grow exponentially due to the presence of unstable saddle points in the phase space. In this work, we probe such an integrable system exhibiting saddle-dominated scrambling through Krylov complexity and the associated Lanczos coefficients. In the realm of the universal operator growth hypothesis, we demonstrate that the Lanczos coefficients follow the linear growth, which ensures the exponential behavior of Krylov complexity at early times. The linear growth arises entirely due to the saddle, which dominates other phase-space points even away from itself. Our results reveal that the exponential growth of Krylov complexity can be observed in integrable systems with saddle-dominated scrambling and thus need not be associated with the presence of chaos.

Classical and quantum butterfly effect in nonlinear vector mechanics

“…Second, it is promising to study other diagnostics of quantum chaos in our system, e.g., the Lanczos coefficients and Krylov complexity [22][23][24][25][26][27][28]. Due to simplicity of the model (1.3), these quantities should also be amenable to analytical calculations.…”

“…The oldest and most famous example of such a diagnostic is the statistics of energy level spacings [4][5][6][7][8]. There are also definitions of quantum chaos that rely on the calculation of dynamical entropy [9,10], decoherence [11], entanglement [12,13], out-of-time ordered correlation functions [14][15][16][17], spectral form factor [18][19][20][21], Krylov complexity [22][23][24][25][26][27][28], and Hilbert-space geometry [29,30]. Furthermore, various diagnostics of quantum chaos are believed to be related to each other and form the "web of diagnostics" [31,32].…”

Krylov complexity in quantum field theory, and beyond

“…There is non-trivial evidence supporting the connection between the behavior of b n and integrability/chaos, yet it does not seem to be universal. In particular a possible stronger formulation, relating the linear growth of b n specifically to chaotic behavior of the underlying systems is apparently wrong [4,5]. We observe that for continuous systems and local operator A, Lanczos coefficients always exhibit linear growth.…”

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

“…In the last few years, many indirect probes of scrambling and quantum chaos have been proposed. These include operator distribution [4][5][6][7][8], out-of-time-ordered-correlators (OTOCs) [9][10][11][12][13][14], and Krylov complexity [15][16][17][18]. These probes have been tested against quantum mechanical realizations of known classical chaotic systems, purely quantum systems, and black holes in holography.…”

“…As a probe of quantum-chaotic dynamics, it has been studied extensively over the past few years. Applications of Krylov complexity extend from a few body quantum systems to field theories [16][17][18].…”