  Quantum chromodynamics is a fundamental non-abelian gauge theory of strong
interactions. The physical quantum chromodynamics vacuum state,
$|\theta\rangle$, is a linear superposition of the $n$-vacua states with
different topological numbers. Because of the configuration of the gauge
fields, the tunneling events can induce the local parity-odd domains. Those
interactions that occur in these domains can be affected by these effects.
Considering the hadron (nucleon) system, we introduce the parity-odd parton
distribution functions in order to describe the parity-odd structures inside
the hadron in this paper. We obtain 8 parity-odd parton distribution functions
at leading twist for spin-1/2 hadrons and present properties of these parton
distribution functions. By introducing the parity-odd quark-quark correlator,
we find the parity-odd effects vanish from the macroscopic point of view. Since
the parity-odd effects are confined in small domains, we consider the high
energy semi-inclusive deeply inelastic scattering process to investigate these
effects by calculating the single spin asymmetries.
