The ordered phase of charged N=4 SYM plasma

Recently is has been shown \cite{Gladden:2024ssb} that strongly coupled ${\cal N}=4$ supersymmetric Yang-Mills plasma with diagonal $U(1)$ $R$-charge chemical potential is unstable at $\frac{\mu}{2\pi T}=\sqrt{2}$. We construct a new homogeneous and isotropic phase of this plasma that dominates in the microcanonical ensemble (but not in the grand canonical one). The new phase extends to arbitrary high temperatures, \ie as $T\gg \mu$, and is characterized by an expectation value of a dimension-2 operator, with ${\cal O}_2\propto T^2$ -- it is the {\it ordered conformal phase}. In the limit $\frac\mu T\to 0$, this ordered phase has vanishing energy density $\frac{{\cal E}}{T^4}\propto \frac{\mu^2}{T^2}$ and is a low entropy density state $\frac{{\cal S}}{T^3}\propto \frac{\mu^2}{T^2}$.