State dependence of Krylov complexity in 2d CFTs

A bstract We compute the Krylov Complexity of a light operator O O L in an eigenstate of a 2 d CFT at large central charge c. The eigenstate corresponds to a primary operator O 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 O O H is below or above the critical dimension h H = c/ 24, marked by the 1 st 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 2 d 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 2 d, and in the integrable 2 d Ising CFT, where there is no such transition in the spectrum of states.