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 TT̄, JT, and JJ deformations, focusing on first-order corrections in the deformation parameter. Under TT deformation, we demonstrate that the Lanczos coefficients bₙ 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, suggesting a potential violation of the conjectured operator growth bound within the realm of perturbative analysis. In contrast to this, both JJ and JT 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.