Krylov complexity in the IP matrix model II

TL;DR

This paper investigates how Krylov complexity transitions from oscillatory behavior at zero temperature to exponential growth at high temperatures in the IP matrix model, revealing temperature-dependent dynamics and corrections.

Contribution

It extends previous work by analyzing the temperature dependence of Krylov complexity and includes $1/N^2$ corrections in the IOP model, showing different growth behaviors.

Findings

Krylov complexity oscillates at zero temperature

Exponential growth occurs at high temperature

$1/N^2$ corrections suppress exponential growth in the IOP model

Abstract

We continue the analysis of the Krylov complexity in the IP matrix model. In a previous paper, for a fundamental operator, it was shown that at zero temperature, the Krylov complexity oscillates and does not grow, but in the infinite temperature limit, the Krylov complexity grows exponentially in time as $\sim \exp\left( {\mathcal{O}\left( {\sqrt{t}}\right)} \right)$. We study how the Krylov complexity changes from a zero-temperature oscillation to an infinite-temperature exponential growth. At low temperatures, the spectral density is approximated as collections of infinite Wigner semicircles. We showed that this infinite collection of branch cuts yields linear growth to the Lanczos coefficients and gives exponential growth of the Krylov complexity. Thus the IP model for any nonzero temperature shows exponential growth for the Krylov complexity even though the Green function decays by a power law in time. We also study the Lanczos coefficients and the Krylov complexity in the IOP matrix model taking into account the $1/N^2$ corrections. There, the Lanczos coefficients are constants and the Krylov complexity does not grow exponentially as expected.