Krylov complexity of deformed conformal field theories

We consider a perturbative expansion of the Lanczos coefficients and the Krylov complexity for two-dimensional conformal field theories under integrable deformations. Specifically, we explore the consequences of $T{\bar{T}}$, $J{\bar{T}}$, and $J{\bar{J}}$ deformations, focusing on first-order corrections in the deformation parameter. Under $T\bar{T}$ deformation, we demonstrate that the Lanczos coefficients $b_n$ exhibit unexpected behavior, deviating from linear growth within the valid perturbative regime. Notably, the Krylov exponent characterizing the rate of exponential growth of complexity surpasses that of the undeformed theory for positive value of deformation parameter, suggesting a potential violation of the conjectured operator growth bound within the realm of perturbative analysis. One may attribute this to the existence of logarithmic branch points along with higher order poles in the autocorrelation function compared to the undeformed case. In contrast to this, both $J{\bar{J}}$ and $J{\bar{T}}$ deformations induce no first order correction to either the linear growth of Lanczos coefficients at large-$n$ or the Krylov exponent and hence the results for these two deformations align with those of the undeformed theory.