State dependence of Krylov complexity in \(2d\) CFTs

DOI10.1007/JHEP09(2023)011arXiv2303.03426OpenAlexW4386459208MaRDI QIDQ6055099

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Publication date: 25 October 2023

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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 $Ad{S}_3$ 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.

Full work available at URL: https://arxiv.org/abs/2303.03426

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