Krylov complexity in quantum field theory, and beyond

TL;DR

This work probes Krylov complexity in quantum field theory by computing Lanczos coefficients across free massive fields, CFTs on spheres, holographic models, and lattice systems with UV cutoffs. It uncovers rich, nonuniversal asymptotics for b_n, including two-slopes and persistent staggering, and demonstrates that exponential growth of Krylov complexity satisfies a generalized MSS bound, though chaos interpretation hinges on UV-cutoff-induced asymptotics. The results reveal scenarios where QFT Krylov complexity diverges qualitatively from holographic complexity, and they highlight the essential role of UV structure in diagnosing chaotic dynamics. Collectively, the paper clarifies the limitations of using b_n as a universal chaos probe in continuum QFT and points to finite-cutoff frameworks as necessary to faithfully capture chaotic behavior, while raising several mathematical questions about the b_n–C(t) connection.

Abstract

We study Krylov complexity in various models of quantum field theory: free massive bosons and fermions on flat space and on spheres, holographic models, and lattice models with the UV-cutoff. In certain cases we find asymptotic behavior of Lanczos coefficients, which goes beyond previously observed universality. We confirm that in all cases the exponential growth of Krylov complexity satisfies the conjectural inequality, which generalizes the Maldacena-Shenker-Stanford bound on chaos. We discuss temperature dependence of Lanczos coefficients and note that the relation between the growth of Lanczos coefficients and chaos may only hold for the sufficiently late, truly asymptotic regime governed by the physics at the UV cutoff. Contrary to previous suggestions, we show scenarios when Krylov complexity in quantum field theory behaves qualitatively differently from the holographic complexity.

Figures (8)

• Figure 1: Lanczos coefficients for free massive scalar (left) and fermion (right) in $d=4$ dimensions. Values for the conformal (massless) cases we discussed in Dymarsky:2021bjq. For the scalar, introduction of $\tilde{m}=\beta m$ results in "persistent staggering" of $b_n$ around the conformal values (larger $\tilde{m}$ causes larger staggering amplitude) but does not change the slope $\pi n/\beta$. Lanczos coefficients split into two branches, see eq. \ref{['asymptm']}. For fermions, asymptotically, $b_n$ grow linearly with the slope $\pi n /\beta$ and the $m$-dependent intercept. The main effect of mass is non-vanishing $a_n=\tilde{m}(-1)^n$.
• Figure 2: Krylov complexity for the free massive scalar (left) and fermion (right) in $d=4$ dimensions. For the massless scalar $K=1+2\sinh^2(\pi t/\beta)$ is known analytically. The main effect of mass is the decrease of Krylov exponent $\lambda_K$.
• Figure 3: Left: Lanczos coefficients for free massless scalar compactified on ${\mathbb S}^3$ of radius $R=1$. Right: Krylov complexity $K(t)$ for radii $R=1$, flat space $K=1+2\sinh^2(\pi t)$, and $R=3/2$.
• Figure 4: Left: Lanczos coefficients for thermal AdS background \ref{['TAdS']} with $\beta=2\pi$ and $\Delta=1$. Right: Krylov complexity for these parameters (blue) superimposed with $K(t)$ for flat space 2d CFT result $K=1+2\sinh^2(\pi t)$ (orange).
• Figure 5: Both panels: Lanczos coefficients for the isotropic XY model \ref{['XYi']}.