Krylov complexity in inverted harmonic oscillator

Abstract

Recently, the out-of-time-ordered correlator(OTOC) and Krylov complexity have been studied actively as a measure of operator growth. OTOC is known to exhibit exponential growth in chaotic systems, which was confirmed in many previous works. However, in some non-chaotic systems, it was observed that OTOC shows chaotic behavior and cannot distinguish saddle-dominated scrambling from chaotic systems. For K-complexity, in the universal operator growth hypothesis, it was stated that Lanczos coefficients show linear growth in chaotic systems, which is the fastest. But recently, it appeared that Lanczos coefficients and K-complexity show chaotic behavior in the LMG model and cannot distinguish saddle-dominated scrambling from chaos. In this paper, we compute Lanczos coefficients and K-complexity in an inverted harmonic oscillator. We find that they exhibit chaotic behavior, which agrees with the case of the LMG model. We also analyze bounds on the quantum Lyapunov coefficient and the growth rate of Lanczos coefficients and find that there is a difference with the chaotic system. Microcanonical K-complexity is also analyzed and compared with the OTOC case.