  We report a comprehensive electronic structure investigation of the
paramagnetic (PM), the large moment antiferromagnetic (LMAF), and the hidden
order (HO) phases of URu$_2$Si$_2$. We have performed relativistic
full-potential calculations on the basis of the density functional theory
(DFT), employing different exchange-correlation functionals to treat electron
correlations within the open $5f$-shell of uranium. Specifically, we
investigate---through a comparison between calculated and low-temperature
experimental properties---whether the $5f$ electrons are localized or
delocalized in URu$_2$Si$_2$. We also performed dynamical mean field theory
calculations (LDA+DMFT) to investigate the temperature evolution of the
quasi-particle states at 100~K and above, unveiling a progressive opening of a
quasi-particle gap at the chemical potential when temperature is reduced. A
detailed comparison of calculated properties with known experimental data
demonstrates that the LSDA and GGA approaches, in which the uranium $5f$
electrons are treated as itinerant, provide an excellent explanation of the
available low-temperature experimental data of the PM and LMAF phases. We show
furthermore that due to a materials-specific Fermi surface instability a large,
but partial, Fermi surface gapping of up to 750 K occurs upon antiferromagnetic
symmetry breaking. The occurrence of the HO phase is explained through
dynamical symmetry breaking induced by a mode of long-lived antiferromagnetic
spin-fluctuations. This dynamical symmetry breaking model explains why the
Fermi surface gapping in the HO phase is similar but smaller than that in the
LMAF phase and it also explains why the HO and LMAF phases have the same Fermi
surfaces yet different order parameters. Suitable derived order parameters for
the HO are proposed to be the Fermi surface gap or the dynamic spin-spin
correlation function.
