Topological Euler Number and the Characterization of Topological Phases in Gapped Fermionic Systems

Details Topological Euler Number and the Characterization of Topological Phases in Gapped Fermionic Systems Topological Euler Number and the Characterization of Topological Phases in Gapped Fermionic Systems

2012-02-11

arxiv.org/abs/1202.2397v2

Physics

Condensed Matter

Strongly Correlated Electrons

Paper is modified: several sections rewritten, this version submitted to PRL. Comments and suggestions are welcome

Scientific paper

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.

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