  ${\it Ab}$ ${\it initio}$ electronic structure calculations reveal that
tetragonal distortion has a dramatic effect on the relative stability of the
various magnetic structures (C-, A-, G-, A$'$-AFM, and FM) of FeRh giving rise
to a wide range of novel stable/metastable structures and magnetic phase
transitions between these states. We predict that the ${\it cubic}$ G-AFM
structure, which was believed thus far to be the ground state, is metastable
and that the ${\it tetragonally}$ expanded G-AFM is the stable structure. The
low energy barrier separating these states suggests phase coexistence at room
temperature. We propose a novel A$'$-AFM phase to be the ${\it global}$ ground
state among all magnetic phases which arises from the strain-induced tuning of
the exchange interactions. The results elucidate the underlying mechanism for
the recent experimental findings of electric-field control of magnetic phase
transition driven via tetragonal strain. The novel magnetic phase transitions
open interesting prospects for exploiting strain engineering for the
next-generation memory devices.
