  Unconventional high temperature superconductivity as well as
three-dimensional bulk Dirac cone quantum states arising from the unique
d-orbital topology has been a recent priority research area in physics. In iron
pnictide compounds, although transport phenomena arisen from this multiple band
Fermi surface are intriguing and scientifically important, they still do not
give an adequate matching to neither experimental observations on the band
picture nor theoretical calculations and a debate continues. Here we describe a
new analytical approach of mobility spectrum, in which the carrier number is
conveniently described as a function of mobility without any hypothesis about
the number of carriers, on both longitudinal and transverse transport of high
quality single crystal Ba(FeAs)$_2$ in a wide range of magnetic field. We show
that the major numbers of carriers reside in large parabolic hole and electron
pockets with very different topology as well as remarkably different mobility
spectra, while the minor number of Dirac carriers resides in both hole- and
electron- Dirac quantum states with the largest mobility as high as 70,000
cm$^2$(Vs)$^{-1}$.
