Krylov complexity as an order parameter for deconfinement phase transitions at large $N$

Krylov complexity has been proposed as a diagnostic of chaos in non-integrable lattice and quantum mechanical systems, and if the system is chaotic, Krylov complexity grows exponentially with time. However, when Krylov complexity is applied to quantum field theories, even in free theory, it grows exponentially with time. This exponential growth in free theory is simply due to continuous momentum in non-compact space and has nothing to do with the mass spectrum of theories. Thus by compactifying space sufficiently, exponential growth of Krylov complexity due to continuous momentum can be avoided. In this paper, we propose that the Krylov complexity of operators such as $\mathcal{O}=\textrm{Tr}[F_{\mu\nu}F^{\mu\nu}]$ can be an order parameter of confinement/deconfinement transitions in large $N$ quantum field theories on such a compactified space. We explicitly give a prescription of the compactification at finite temperature to distinguish the continuity of spectrum due to momentum and mass spectrum. We then calculate the Krylov complexity of $\mathcal{N}=4, 0$ $SU(N)$ Yang-Mills theories in the large $N$ limit by using holographic analysis of the spectrum and show that the behavior of Krylov complexity reflects the confinement/deconfinement phase transitions through the continuity of mass spectrum.