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arXiv · 2303.03426

State Dependence of Krylov Complexity in $2d$ CFTs

Arnab Kundu·Vinay Malvimat·Ritam Sinha
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Abstract

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. 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.

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Arnab Kundu, Vinay Malvimat, Ritam Sinha. 2023-03-06. State Dependence of Krylov Complexity in $2d$ CFTs. https://doi.org/10.1007/jhep09(2023)011

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