State Dependence of Krylov Complexity in $2d$ CFTs

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.

24 pages, 5 figures, minor corrections

References in corpus (19)

• Complexity and Shock Wave Geometries
• Krylov complexity from integrability to chaos
• Krylov Localization and suppression of complexity
• Operator growth in open quantum systems: lessons from the dissipative SYK
• Quantum complexity and topological phases of matter
• Probing quantum scars and weak ergodicity-breaking through quantum complexity
• Krylov complexity and orthogonal polynomials
• Krylov Complexity in Free and Interacting Scalar Field Theories with Bounded Power Spectrum
• Spread Complexity and Topological Transitions in the Kitaev Chain
• A universal approach to Krylov State and Operator complexities
• Floquet conformal field theories with generally deformed Hamiltonians
• Scars from protected zero modes and beyond in $U(1)$ quantum link and quantum dimer models
• Phases of scrambling in eigenstates
• Spread complexity as classical dilaton solutions
• Entanglement and geometry from subalgebras of the Virasoro algebra
• Chaos and operator growth in 2d CFT
• Fast Scrambling of Mutual Information in Kerr-AdS${}_4$
• On Quantum Complexity
• Fast Scrambling of mutual information in Kerr-AdS${}_{\text{5}}$

Cited by in corpus (20)

• Quantum Dynamics in Krylov Space: Methods and Applications
• Krylov complexity and chaos in quantum mechanics
• Spectral and Krylov Complexity in Billiard Systems
• Spread and Spectral Complexity in Quantum Spin Chains: from Integrability to Chaos
• Krylov construction and complexity for driven quantum systems
• Operator growth and Krylov Complexity in Bose-Hubbard Model
• Krylov Complexity and Spectral Form Factor for Noisy Random Matrix Models
• Spread complexity for measurement-induced non-unitary dynamics and Zeno effect
• Krylov complexity for non-local spin chains
• Thermalization in Krylov Basis
• Krylov Complexity of Open Quantum Systems: From Hard Spheres to Black Holes
• Krylov complexity of deformed conformal field theories
• Brickwall One-Loop Determinant: Spectral Statistics & Krylov Complexity
• Operator size growth in Lindbladian SYK
• Krylov operator complexity in holographic CFTs: Smeared boundary reconstruction and the dual proper radial momentum
• Krylov Complexity in the Schrödinger Field Theory
• Information scrambling in quantum walks: Discrete-time formulation of Krylov complexity
• Higher-Order Krylov State Complexity in Random Matrix Quenches
• Krylov complexity and Wightman power spectrum with positive chemical potential in Schrödinger field theory
• Krylov Complexity Under Hamiltonian Deformations and Toda Flows