Krylov complexity in saddle-dominated scrambling

TL;DR

This paper shows that in integrable systems with saddle points, Krylov complexity exhibits exponential growth, indicating that such growth is not exclusive to chaotic systems but can also occur due to saddle-dominated dynamics.

Contribution

It demonstrates that exponential Krylov complexity growth can occur in integrable systems because of saddle points, challenging the notion that it solely indicates chaos.

Findings

Krylov complexity grows exponentially in saddle-dominated integrable systems.

Lanczos coefficients exhibit linear growth due to saddle points.

Exponential complexity growth is not exclusive to chaotic systems.

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.