Operator growth in SU(2) Yang-Mills theory

TL;DR

The paper investigates quantum chaos in SU(2) Yang-Mills theory by mapping to a nonlinearly coupled harmonic oscillator and computing Krylov complexity and Lanczos coefficients. It demonstrates a chaotic transition manifested as $C_K(t)$ shifting from quadratic to linear growth, with Lanczos coefficients obeying $b_n\sim a n$ before saturation and enforcing the bound $\lambda_{\max}\le 2a$. A microcanonical analysis shows higher-energy modes dominate the chaotic growth, and the spectral form factor exhibits random-matrix-like features, hinting at a gravity-dual interpretation via operator growth. Overall, the work provides a concrete, computable framework for operator growth in a strongly interacting gauge theory and contributes to the broader link between chaos, complexity, and holography.

Abstract

Krylov complexity is a novel observable for detecting quantum chaos, and an indicator of a possible gravity dual. In this paper, we compute the Krylov complexity and the associated Lanczos coefficients in the SU(2) Yang-Mills theory, which can be reduced to a nonlinearly coupled harmonic oscillators (CHO) model. We show that there exists a chaotic transition in the growth of Krylov complexity. The Krylov complexity shows a quadratic growth in the early time stage and then grows linearly. The corresponding Lanczos coefficient satisfies the universal operator growth hypothesis, i.e., grows linearly first and then enters the saturation plateau. By the linear growth of Lanczos coefficients, we obtain an upper bound of the quantum Lyapunov exponent. Finally, we investigate the effect of different energy sectors on the K-complexity and Lanczos coefficients.

Figures (8)

• Figure 1: Poincaré sections of the CHO model for four different coupling constants $g_0 = 0.1,0.3,0.7,1$. As the coupling constant rises, the clear orbits with strong periodicity gradually become randomly scattered points.
• Figure 2: Poincaré sections in the CHO Hamiltonian system for four different energy $E=3, E=5, E=7, E=15$. When the energy is low, the Poincaré section shows the characteristics of integrable systems, i.e., clear periodic orbits. As the energy increases, the system gradually enters the chaotic phase.
• Figure 3:
• Figure 4: Spectral form factor $g(\beta, t)$ as a function of time, for $g_0 = 0.1$ and three different temperatures $T=1,T=2,t=10$. In the early stage, the spectral form factor holds 1. After that, it drops to a dip and then bounces back to a plateau with erratic fluctuations. The fluctuation in the spectral form factor reflects the discrete eigenvalues of the system.
• Figure 5: Lanczos coefficients for different coupling constants $g_0 = 0.1, 0.3, 0.7$ at temperature $\beta =1$. In the beginning, the Lanczos coefficients $b_n$ increase linearly with $n$. And after about n = 12, Lanczos coefficients enter the saturation plateau and fluctuate around a constant value.