Yu-Quan Ma, Shi-Jian Gu, Shu Chen, Heng Fan, Wu-Ming Liu
We propose a new 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, we
show the familiar Chern number is also contained in this approach due to the
Berry curvature is given by the imaginary part of the geometric tensor. We
discuss this approach analytically in a general two-band model, and as an
example, we show the quantum phases of a transverse field XY spin chain can be
characterized by the Euler characteristic number and the $Z_2$ number in 1+1
dimensional momentum space, respectively.
View original:
http://arxiv.org/abs/1202.2397
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