A critical coupling for the magnetic instability of holographic SU(2) plasmas

Abstract

We show that gravity can remove the linear magnetic instability of a holographic plasma with an $SU(2)$ global symmetry. In the probe limit, charged vector bosons become tachyonic in their lowest Landau level above a critical field and condense into a vortex lattice. In Einstein-$SU(2)$ Yang-Mills theory in AdS$_5$ we include the backreaction of the field exactly, expanding only in the condensate around the D'Hoker-Kraus magnetic brane. The critical field diverges as the coupling $\alpha = \kappa^2/\hat g^2$ approaches $\alpha_\star = 2/3$, above which the normal phase has no linear charged instability at any field or temperature: we show this at zero temperature and establish it numerically at finite temperature. At zero temperature the field cancels from the mass of the lowest Landau level in the brane's AdS$_3 \times \mathbb{R}^2$ throat, and the instability is a violation of the throat's Breitenlohner-Freedman bound. The onset temperature vanishes at $\alpha_\star$ with an essential singularity of Berezinskii-Kosterlitz-Thouless type. For two-derivative Einstein-Yang-Mills duals in $d = 4$, 5 and 6 dimensions the normal phase has a charged instability if and only if $C_J/C_T<8/(d^2(d+1))$, one tenth in $d = 4$. Above $\alpha_L = 0.372$ the gravitational attraction between vortices makes the uniform lattice unstable and the transition first order, so $B_c$ is only a spinodal; whether a condensed phase survives above $\alpha_\star$ is left open.