Diagnostics for generalized power-law torsion-matter coupling $f(T)$
  model
  The currently accelerated expansion of our Universe is unarguably one of the
most intriguing problems in today's physics research. Two realistic non-minimal
torsion-matter coupling $f(T)$ models have been established and studied in our
previous papers [Phys. Rev. D92, 104038(2015) and Eur. Phys. J. C77, 504(2017)]
aiming to explain this "dark energy" problem. In this paper, we study the
generalized power-law torsion-matter coupling $f(T)$ model. Dynamical system
analysis shows that the three expansion phases of the Universe, i.e. the
radiation dominated era, the matter dominated era and the dark energy dominated
era, can all be reproduced in this generalized model. By using the statefinder
and $Om$ diagnostics, we find that the different cases of the model can be
distinguished from each other and from other dark energy models such as the two
models in our previous papers, $\Lambda$CDM, quintessence and Chaplygin gas.
Furthermore, the analyses also show that all kinds of generalized power-law
torsion-matter coupling model are able to cross the $w=-1$ divide from below to
above, thus the decrease of the energy density resulting from the crossing of
$w$ will make the catastrophic fate of the Universe avoided and a de Sitter
expansion fate in the future will be approached.
