Instability in ${\cal N}=4$ supersymmetric Yang-Mills theory at finite density

Equilibrium states of ${\cal N}=4$ supersymmetric Yang-Mills theory can be characterized by the temperature and three chemical potentials, corresponding to the ${\rm U}(1)^3$ subgroup of the $R$-symmetry group. We investigate the phase diagram of the theory at strong coupling, in the grand canonical ensemble in flat space, using its holographic description via the five-dimensional model of Behrnd, Cveti\v{c}, and Sabra. The bulk action includes the metric, three Abelian gauge fields, and two neutral scalar fields. The equilibrium state described by the charged black brane is always thermodynamically unstable at low temperature. Relativistic hydrodynamics with multiple conserved charges predicts that thermodynamic instability is accompanied by a dynamical instability, with the eigenvalues and eigenvectors of the corresponding Hessian playing a key role in identifying the unstable modes. We explicitly demonstrate this for three equal chemical potentials, finding unstable quasinormal modes that describe $R$-charge diffusion. Consequently, the low-temperature phase of ${\cal N}=4$ supersymmetric Yang-Mills theory with equal chemical potentials is not described by the AdS-Reissner-Nordstr\"om black brane.