  In this paper, we shall perform a detailed analysis of the Exact Holographic
Mapping first introduced in arXiv:1309.6282, which was proposed as an explicit
example of holographic duality between quantum many-body systems and
gravitational theories. We obtain analytic results for free-fermion systems
that not only confirm previous numerical results, but also elucidate the exact
relationships between the various physical properties of the bulk and boundary
systems. Our analytic results allow us to study the asymptotic properties that
are difficult to probe numerically, such as the near horizon regime of the
black hole geometry. We shall also explore a few interesting but hitherto
unexplored bulk geometries, such as that corresponding to a boundary critical
fermion with nontrivial dynamic critical exponent. Our analytic framework also
allows us to study the holographic mapping of some of these boundary theories
in dimensions 2+1 or higher.
