  Mn$_3$Sn, a metallic antiferromagnet with an anti-chiral 120$^\circ$ spin
structure, generates intriguing magneto-transport signatures such as a large
anomalous Hall effect, spin-polarized current with novel symmetries, anomalous
Nernst effect, and magneto-optic Kerr effect. When grown epitaxially as
MgO(110)[001]$\parallel$ Mn$_3$Sn($0\bar{1}\bar{1}0$)[0001], Mn$_3$Sn
experiences a uniaxial tensile strain, which changes the bulk six-fold
anisotropy landscape to a perpendicular magnetic anisotropy with two stable
states. In this work, we investigate the field-assisted spin orbit-torque
(SOT)-driven response of the order parameter in single-domain Mn$_3$Sn with
uniaxial tensile strain. We find that for a non-zero external magnetic field,
the order parameter can be switched between the two stable states if the
magnitude of the input current is between two field-dependent critical
currents. Below the lower critical current, the order parameter exhibits a
stationary state in the vicinity of the initial stable state. On the other
hand, above the higher critical current, the order parameter shows oscillatory
dynamics which could be tuned from the 100's of megahertz to the gigahertz
range. We obtain approximate expressions of the two critical currents and find
them to agree very well with the numerical simulations for experimentally
relevant magnetic fields. We also obtain unified functional form of the
switching time versus the input current for different magnetic fields. Finally,
we show that for lower values of Gilbert damping ($\alpha \leq 2\times
10^{-3}$), the critical currents and the final steady states depend
significantly on the damping constant. The numerical and analytic results
presented in our work can be used by both theorists and experimentalists to
understand the SOT-driven order dynamics in PMA Mn$_3$Sn and design future
experiments and devices.
