Global Symmetry and Maximal Chaos

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 $\mu$ 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 $\lambda_L \leq \frac{2 \pi T}{ \hbar}$. On the other hand when the operators could create excitation of arbitrary high charge, we find that exponent must satisfy $\lambda_L \leq \frac{2 \pi T}{(1-|\frac{\mu}{\mu_c}|) \hbar}$, where $\mu_c$ 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.