Krylov Complexity in Free and Interacting Scalar Field Theories with Bounded Power Spectrum

TL;DR

This work analyzes Krylov complexity and Lanczos coefficients for free and interacting scalar quantum field theories at finite temperature in $d$ dimensions. By introducing mass and ultraviolet cutoffs (hard and smooth) in continuous momentum space, it shows mass-induced staggering of the Lanczos coefficients into two families and a reduced exponential growth rate of Krylov complexity, with oscillations and altered asymptotics. A hard UV cutoff saturates the Lanczos spectrum, triggering a transition from exponential to linear growth of Krylov complexity, while a smooth cutoff modulates the high-frequency tail and staggering. In 4D, one-loop interactions (marginally irrelevant and relevant deformations) generate further modifications to the spectral power and Lanczos slopes, revealing nuanced interplay between IR mass scales, UV regularization, and operator growth. The paper also identifies conditions under which staggering is absent and clarifies connections to lattice regularization and RG dynamics, suggesting avenues for future exploration of asymptotic growth rates and other operators.

Abstract

We study a notion of operator growth known as Krylov complexity in free and interacting massive scalar quantum field theories in $d$-dimensions at finite temperature. We consider the effects of mass, one-loop self-energy due to perturbative interactions, and finite ultraviolet cutoffs in continuous momentum space. These deformations change the behavior of Lanczos coefficients and Krylov complexity and induce effects such as the "staggering" of the former into two families, a decrease in the exponential growth rate of the latter, and transitions in their asymptotic behavior. We also discuss the relation between the existence of a mass gap and the property of staggering, and the relation between our ultraviolet cutoffs in continuous theories and lattice theories.

Figures (23)

• Figure 1: Lanczos coefficients $b_{n}$ for $m \beta=80$ in $d=5$. The squares represent the results obtained numerically from \ref{['eq:Moments5dLargeBetaMass']}, while the circles represent the approximate results Eq. \ref{['eq-bn-largemassv2']}, which are valid for $m \beta \gg n$. In both cases, the top family of $b_{n}$ corresponds to odd $n$, while the lower one corresponds to even $n$.
• Figure 2: Mass-dependence of the constants in the linear fitting (\ref{['linearfittingbnd5']}) for $d=5$ and $\beta=1$. We also plot a straight line $\gamma_{\textrm{odd}}-\gamma_{\textrm{even}}=m$ in the right figure to compare the mass-dependence.
• Figure 3: Krylov complexity of free scalar theories with $\beta=1$. The vertical axis is in a logarithmic scale. Linear growth in the log plot implies an exponential growth of the Krylov complexity.
• Figure 4: Mass-dependence of $\tilde{\lambda}_{K}$ for $\beta=1$. The linear fitting to determine $\tilde{\lambda}_{K}$ is done in the range $1.5\le \pi t/\beta \le 2.0$. We also plot the fitting curves of $\tilde{\lambda}^{(d)}_{K}$ (\ref{['eq:LambdaKMass']}).
• Figure 5: Krylov entropy of free scalar theories with $\beta=1$ and $d=5$.