State Dependence of Krylov Complexity in $2d$ CFTs

Arnab Kundu; Vinay Malvimat; Ritam Sinha

doi:10.1007/JHEP09(2023)011

paper

State Dependence of Krylov Complexity in $2 d$ CFTs

arXiv:2303.03426 · doi:10.1007/JHEP09(2023)011

Abstract

We compute the Krylov Complexity of a light operator $O_{L}$ in an eigenstate of a $2 d$ 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 $O_{H}$ is below or above the critical dimension $h_{H} = c /24$ , marked by the $1 s t$ order Hawking-Page phase transition point in the dual $A d S_{3}$ 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.

arXiv comment24 pages, 5 figures, minor corrections

State Dependence of Krylov Complexity in $2 d$ CFTs

Abstract

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State Dependence of Krylov Complexity in 2d CFTs

Abstract

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Cited by in corpus (20)

State Dependence of Krylov Complexity in $2 d$ CFTs