Geometric tensor and the topological characterization of the Bloch band in a two-band lattice model

We investigate the quantum Riemannian metric and the Euler characteristic number of the Bloch states manifold in a two-band lattice model, where a topological phase transition from the normal to the Chern insulator occurs. We derive the topological Euler number of the band from the Gauss–Bonnet theorem on the closed Bloch states manifold in the first Brillouin zone, where the Riemannian metric of the states manifold is established by the real part of the quantum geometric tensor in the 2D quasi-momentum space. Meanwhile, we show that the imaginary part of the geometric tensor corresponds to the Berry curvature which leads to the Chern number characterization of the band insulator. We discuss the topological numbers induced by the geometric tensor analytically in the case of two-band Hamiltonian and characteristic the zero-temperature phase diagram by the Euler number and first Chern number, respectively.