Quantization Of Physical Parameters

The dimensional reduction of the 3+1 system with Fermi points brings the anomaly to the (2+1)-dimensional systems with fully gapped fermionic spectrum. The most pronounced phenomena in these systems are related to spin and quantum statistics of the topological objects in 2+1 systems — skyrmions and to the quantisation of physical parameters, such as Hall and spin-Hall conductivity and fermionic charge. This quantization is determined by the integer valued momentum-space topological invariant expressed via the Green's function, which is discussed in Chapter 11, and demonstrates the interplay of p-space and r-space topologies. The Chern–Simons term is the r-space invariant (such as the Hopf invariant), while its prefactor is expressed in terms of p-space invariant. In general case the mixed Chern–Simons term composed of different effective gauge fields has the prefactor in terms p-space invariant supported by symmetry. Such Chern–Simons terms give rise to different types of intrinsic Hall and spin-Hall effect. The fully gapped vacua with different values of physical parameters are separated by topological quantum phase transition with gapless intermediate states.