  Using lattice simulations, we show that there is a phase of thermal QCD,
where the spectral density $\rho(\lambda)$ of Dirac operator changes as
$1/\lambda$ for the infrared eigenvalues $\lambda<T$. This behavior persists
over the entire low energy band we can resolve accurately, over three orders of
magnitude on our largest volumes. We propose that in this "IR phase", the
well-known non-interacting scale invariance at very short distances (UV,
$\lambda \rightarrow \infty$, asymptotic freedom), coexists with very different
interacting type of scale invariance at long distances (IR, $\lambda<T$). Such
dynamics may be responsible for the unusual fluidity properties of the medium
observed at RHIC and LHC. We point out its connection to the physics of
Banks-Zaks fixed point, leading to the possibility of massless glueballs in the
fluid. Our results lead to the classification of thermal QCD phases in terms of
IR scale invariance. The ensuing picture naturally subsumes the standard chiral
crossover feature at $"\!T_c\!" \,\approx 155$ MeV. Its crucial new aspect is
the existence of temperature $T_{IR}$ (200 MeV $< T_{IR} < $ 250 MeV) marking
the onset of IR phase and possibly a true phase transition.
