State Dependence of Krylov Complexity in $2d$ CFTs

TL;DR

The paper investigates whether Krylov complexity (K-complexity) of an operator in a quantum field theory depends on the quantum state, focusing on two-dimensional conformal field theories. By computing four-point functions in excited heavy states and mapping to a Krylov (Toda-chain) framework, the authors derive Lanczos data and analyze the resulting K-complexity for both large-$c$ (holographic) and integrable 2d CFTs. They find a state-dependent transition in large-$c$ CFTs: for light heavy states with $h_H<c/24$ the K-complexity oscillates and remains bounded, whereas for heavy states with $h_H>c/24$ it grows exponentially, reflecting a shift from area- to volume-law entanglement; integrable theories like the free scalar and Ising CFT show bounded, non-exponential K-complexity regardless of the heavy state. The results support the view that operator growth in 2d CFTs is intimately tied to entanglement structure and that K-complexity can serve as a state-sensitive diagnostic of quantum chaos versus integrability, with potential holographic interpretations.

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.

Figures (5)

• Figure 1: K-Complexity below (left) and above (right) the threshold dimension of the heavy operator $h_H=c/24$ for the values of parameters $\alpha =1$, $\tau_0=0.98$, $\Delta_{L}=5$.
• Figure 2: Lanczos Coefficients $a_n, b_n$ for a light operator in heavy state in the Ising model example with values of parameters $\alpha =1$, $\tau_0=0.88$, $\Delta_{H}=1$, $\Delta_{L}=1/8$.
• Figure 3: K-complexity for a light operator ($\sigma$) in the light state $\ket{\sigma}$ in the Ising model example for the values of parameters $\alpha =1$, $\tau_0=0.88$, $\Delta_{\sigma}=1/8$.
• Figure 4: K-complexity for a light operator ($\sigma$) in the heavy state $\ket{\epsilon}$ in the Ising model for the values of parameters $\alpha =1$, $\tau_0=0.88$, $\Delta_{\epsilon}=1/2$, $\Delta_{\sigma}=1/8$.
• Figure 5: K-complexity for the $\epsilon$ operator in $\ket{\epsilon}$ state in the Ising model for the values of parameters $\alpha =1$, $\tau_0=0.88$, $\Delta_{\epsilon}=1/2$.