Black string spectrum of shift-symmetric Horndeski theories

In the present work, we study four-dimensional black strings in Horndeski models with translation invariance. Imposing that the scalar field depends on the string-generator coordinate, the Klein-Gordon equation admits a linear profile as a solution. This relaxation allows finding rotating, asymptotically AdS$_3\times \mathbb{R}$ black strings, dressed with an effective cosmological constant. In this regard, we show that in the full spectrum of shift-symmetric Horndeski theories with Einstein limit, the scalar charge needs to be fixed in terms of the parameter space. This method is employed to concrete examples to illustrate the scheme we go along with. Regarding the conserved charges, we compute them via the Euclidean method and show the fulfillment of the associated Smarr Law. Finally, we exhibit that our AdS strings are locally and globally stable under small fluctuations around the equilibrium.