The Euler Number of Bloch States Manifold and the Quantum Phases in   Gapped Fermionic Systems
  We propose a topological Euler number to characterize nontrivial topological phases of gapped fermionic systems, which originates from the Gauss-Bonnet theorem on the Riemannian structure of Bloch states established by the real part of the quantum geometric tensor in momentum space. Meanwhile, the imaginary part of the geometric tensor corresponds to the Berry curvature which leads to the Chern number characterization. We discuss the topological numbers induced by the geometric tensor analytically in a general two-band model. As an example, we show that the zero-temperature phase diagram of a transverse field XY spin chain can be distinguished by the Euler characteristic number of the Bloch states manifold in a (1+1)-dimensional Bloch momentum space. 
Yu-Quan Ma, Shi-Jian Gu, Shu Chen, Heng Fan and Wu-Ming Liu
cond-mat.str-el hep-th quant-ph