  The quantum phase transition in iron-based superconductors with 'half-Dirac'
node at the electron Fermi surface as a $T=0$ structural phase transition
described in terms of nematic order is discussed. An effective low energy
theory that describes half-Dirac nodal Fermions and their coupling to Ising
nematic order that describes the phase transition is derived and analyzed using
renormalization group (RG) study of the large-$N_f$ version of the theory. The
inherent absence of Lorentz invariance of the theory leads to RG flow structure
where the velocities $v_F$ and $v_\Delta$ at the paired half-Dirac nodes
($1\overline{1}$ and $2\overline{2}$) in general flow differently under RG,
implying that the nodal electron gap is deformed and the $C_4$ symmetry is
broken, explaining the structural (orthogonal to orthorhombic) phase transition
at the quantum critical point (QCP). The theory is found to have Gaussian fixed
point $\lambda^*=0, (v_{\Delta}/v_F)^*=0$ with stable flow lines toward it,
suggesting a second order nematic phase transition. Interpreting the
fermion-Ising nematic boson interaction as a decay process of nematic Ising
order parameter scalar field fluctuations into half-Dirac nodal fermions, I
find that the theory surprisingly behaves as systems with dynamical critical
exponent $z = 1$, reflecting undamped quantum critical dynamics and emergent
fully relativistic field theory arising from the non(fully)-relativistic field
theory and is direct consequence of $(v_{\Delta}/v_F)^*=0$ fixed point. The
nematic critical fluctuations lead to remarkable change to the spectral
function peak where at a critical point $\lambda_c$, directly related to
nematic QCP, the central spectral peak collapses and splits into satellite
spectral peaks around nodal point. The vanishing of the zero modes density of
states leads to the undamped $z=1$ quantum critical dynamics.
