State dependence of Krylov complexity in 2d CFTs

We compute the Krylov Complexity of a light operator O-L in an eigenstate of a 2d CFT at large central charge c. The eigenstate corresponds to a primary operator O-H under the state-operator correspondence. We observe that the behaviour of K-complexity is different (either bounded or exponential) depending on whether the scaling dimension of OH is below or above the critical dimension h(H) = c/24, marked by the 1st order Hawking-Page phase transition point in the dual AdS(3) geometry. Based on this feature, we hypothesize that the notions of operator growth and K-complexity for primary operators in 2d CFTs are closely related to the underlying entanglement structure of the state in which they are computed, thereby demonstrating explicitly their state-dependent nature. To provide further evidence for our hypothesis, we perform an analogous computation of K-complexity in a model of free massless scalar field theory in 2d, and in the integrable 2d Ising CFT, where there is no such transition in the spectrum of states.