Holographic superfluids as duals of rotating black strings

We study the breaking of an Abelian symmetry close to the horizon of an uncharged rotating Anti-de Sitter black string in 3+1 dimensions. The boundary theory living on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb{R}^{2} \times S^{1} $$\end{document} has no rotation, but a magnetic field that is aligned with the axis of the black string. This boundary theory describes non-rotating (2+1)-dimensional holographic superfluids with non-vanishing superfluid velocity. We study these superfluids in the grand canonical ensemble and show that for sufficiently small angular momentum of the dual black string and sufficiently small superfluid velocity the phase transition is 2nd order, while it becomes 1st order for larger superfluid velocity. Moreover, we observe that the phase transition is always 1st order above a critical value of the angular momentum independent of the choice of the superfluid velocity.