Krylov complexity in conformal field theory

Krylov complexity, or K-complexity for short, has recently emerged as a new probe of chaos in quantum systems. It is a measure of operator growth in Krylov space, which conjecturally bounds the operator growth measured by the out of time ordered correlator (OTOC). We study Krylov complexity in conformal field theories by considering arbitrary 2d CFTs, free field, and holographic models. We find that the bound on OTOC provided by Krylov complexity reduces to bound on chaos of Maldacena, Shenker, and Stanford. In all considered examples including free and rational CFTs Krylov complexity grows exponentially, in stark violation of the expectation that exponential growth signifies chaos.

Figure 1

Lanczos coefficients $b_n$ for free massless scalar $\varphi$ in $d=4$ ($\mathrm{\Delta }=1$, blue), $d=5$ ($\mathrm{\Delta }=3/2$, orange), $d=6$ ($\mathrm{\Delta }=2$, green) dimensions, and for the composite operator ${\varphi }^2$ in $d=5$ dimensions ($\mathrm{\Delta }=3$, red); dashed lines of the appropriate color show asymptotic behavior of $b_n$ as given by (16).

Figure 2

Krylov complexity $K_{\mathcal{O}}$ shown in logarithmic scale for free scalar in $d=4$ (blue), $d=5$ (orange), $d=6$ (green) dimensions and for generalized free field with $\mathrm{\Delta }=10$ (brown). Blue curve is known analytically, $\ln (1+2{\sinh }^2(\pi t/\beta ))$. All four curves exhibit an apparent linear growth of $\ln K_{\mathcal{O}}\propto 2\pi t/\beta$ at late times.

Figure 3

Left panel. Lanczos coefficients $b_n$ for generalized free field (13) with $\mathrm{\Delta }=10$ (blue) vs approximation for small $n$ (15) (orange) and asymptotic behavior for large $n$ (16) (red line). Right panel. Lanczos coefficients $b_n$ for generalized free field (13) of dimension $\mathrm{\Delta }=8.5$ (blue) and for holographic operator $O=\int d^{3}x\mathcal{O}$ of effective dimension $\mathrm{\Delta }=8.5$, while $\mathcal{O}$ has dimension $\mathrm{\Delta }=10$ (orange). The same effective dimension means both sequences have the same asymptotic behavior $b_n\approx \pi (n+9)$.