Global Symmetry and Maximal Chaos
Indranil Halder
TL;DR
This work extends the universal chaos bound to quantum systems with a global symmetry in a thermodynamic ensemble with chemical potential $\\mu$. By analyzing out-of-time-ordered correlators and exploiting analyticity in an effective half-strip with $\\beta_{eff}=\\beta(1-\\left|\\mu/\\mu_c\\right|)$, the authors derive two regimes: (i) fixed-charge excitations yield the standard bound $\\lambda_L\\leq 2\\pi/\\beta$, independent of $\\mu$, and (ii) arbitrarily high-charge excitations yield a $\\mu$-dependent bound $\\lambda_L\\leq 2\\pi T/((1-\\left|\\mu/\\mu_c\\right|)\\hbar)$. They illustrate these bounds with internal-symmetry-conforming CFTs (where $\\mu_c\to\infty$) and with 2D holographic CFTs dual to rotating BTZ black holes (where the bound is saturated as $\\lambda_L=2\\pi/[\\beta(1-\\Omega_H)]$). The results connect chaos growth to global charges and holographic rotations, outlining regimes of validity and suggesting avenues for exploring chaos near critical chemical potentials and in higher dimensions.
Abstract
In this note we study chaos in generic quantum systems with a global symmetry generalizing seminal work [arXiv : 1503.01409] by Maldacena, Shenker and Stanford. We conjecture a bound on instantaneous chaos exponent in a thermodynamic ensemble at temperature $T$ and chemical potential $μ$ for the continuous global symmetry under consideration. For local operators which could create excitation up to some fixed charge, the bound on chaos (Lyapunov) exponent is independent of chemical potential $λ_L \leq \frac{2 πT}{ \hbar} $. On the other hand when the operators could create excitation of arbitrary high charge, we find that exponent must satisfy $λ_L \leq \frac{2 πT}{(1-|\fracμ{μ_c}|) \hbar} $, where $μ_c$ is the maximum value of chemical potential for which the thermodynamic ensemble makes sense. As specific examples of quantum mechanical systems we consider conformal field theories. In a generic conformal field theory with internal $U(1)$ symmetry living on a cylinder the former bound is applicable, whereas in more interesting examples of holographic two dimensional conformal field theories dual to Einstein gravity, we argue that later bound is saturated in presence of a non-zero chemical potential for rotation.