Charged Anti-de Sitter BTZ black holes in Maxwell-$f(T)$ gravity

Inspired by the BTZ formalism, we discuss the Maxwell-$f(T)$ gravity in (2+1)-dimensions. The main task is to derive exact solutions for a special form of $f(T)=T+\epsilon T^2$, with $T$ being the torsion scalar of Weitzenb$\ddot{\mbox{o}}$ck geometry. To this end, a triad field is applied to the equations of motion of charged $f(T)$ and sets of circularly symmetric non-charged and charged solutions have been derived. We show that, in the charged case, the monopole-like and the ${\it ln}$ terms are linked by a correlative constant despite of known results in teleparallel geometry and its extensions [39]. Furthermore, it is possible to show that the event horizon is not identical with the Cauchy horizon due to such a constant. The singularities and the horizons of these black holes are examined: they are new and have no analogue in literature due to the fact that their curvature singularities are soft. We calculate the energy content of these solutions by using the general vector form of the energy-momentum within the framework of $f(T) $ gravity. Finally, some thermodynamical quantities, like entropy and Hawking temperature, are derived.