Thouless-Kohmoto-Nightingale-den Nijs formula for a general Hamiltonian
Hidenori Fukaya (Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan); Tetsuya Onogi (Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan); Satoshi Yamaguchi (Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan); Xi Wu (Physics Department, Ariel University, Ariel)
Topological insulators in odd dimensions are characterized by topological numbers. We prove the well-known relation between the topological number given by the Chern character of the Berry curvature and the Chern-Simons level of the low energy effective action for a general class of Hamiltonians bilinear in the fermion with general U(1) gauge interactions including nonminimal couplings by an explicit calculation. A series of Ward-Takahashi identities are crucial to relate the Chern-Simons level to a winding number, which could then be directly reduced to Chern character of Berry curvature by carrying out the integral over the temporal momenta.
