We proposed an action of neutral fermions interacting with external electromagnetic fields to construct a 3+1 dimensional topological field theory as the effective action attained by integrating out the fermionic fields in the related path integral.
These neutral quasiparticles are assumed to emerge from the collective behavior of the original physical particles and holes (antiparticles). Although our construction is general it is particularly useful to formulate effective actions of the time reversal invariant topological insulators.
A semiclassical constrained Hamiltonian system which was established to study dynamical systems of matrix valued non-Abelian gauge fields is employed to formulate spin Hall effect in noncommuting coordinates at the first order in the constant noncomm
utativity parameter theta . The method is first illustrated by studying the Hall effect on the noncommutative plane in a gauge independent fashion. Then, the Drude model type and the Hall effect type formulations of spin Hall effect are considered in noncommuting coordinates and theta deformed spin Hall conductivities which they provide are acquired. It is shown that by adjusting theta different formulations of spin Hall conductivity are accomplished. Hence, the noncommutative theory can be envisaged as an effective theory which unifies different approaches to similar physical phenomena.
