Operator dynamics in Lindbladian SYK: a Krylov complexity perspective

TL;DR

This paper investigates operator growth in dissipative SYK models using Krylov complexity, revealing linear growth, saturation behavior, and spectral properties, and connects these findings to quantum measurement processes.

Contribution

It provides an analytical and numerical study of operator growth in dissipative SYK models through Krylov complexity, including the effects of dissipation and spectral analysis.

Findings

Krylov complexity exhibits linear growth in dissipative SYK.

Dissipative timescale grows logarithmically with dissipation strength.

Operator growth behavior is linked to spectral properties and quantum measurement.

Abstract

We use Krylov complexity to study operator growth in the $q$-body dissipative SYK model, where the dissipation is modeled by linear and random $p$-body Lindblad operators. In the large $q$ limit, we analytically establish the linear growth of two sets of coefficients for any generic jump operators. We numerically verify this by implementing the bi-Lanczos algorithm, which transforms the Lindbladian into a pure tridiagonal form. We find that the Krylov complexity saturates inversely with the dissipation strength, while the dissipative timescale grows logarithmically. This is akin to the behavior of other $\mathfrak{q}$-complexity measures, namely out-of-time-order correlator (OTOC) and operator size, which we also demonstrate. We connect these observations to continuous quantum measurement processes. We further investigate the pole structure of a generic auto-correlation and the high-frequency behavior of the spectral function in the presence of dissipation, thereby revealing a general principle for operator growth in dissipative quantum chaotic systems.