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2023

State Dependence of Krylov Complexity in $2d$ CFTs

Kundu, Arnab, Malvimat, Vinay, Sinha, Ritam

Understand

We compute the Krylov Complexity of a light operator $\mathcal{O}_L$ in an eigenstate of a $2d$ CFT at large central charge $c$.

The eigenstate corresponds to a primary operator $\mathcal{O}_H$ 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 $\mathcal{O}_H$ 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.