The Euler number of Bloch states manifold and the quantum phases in gapped fermionic systems

¹ School of Applied Science, Beijing Information Science and Technology University - Beijing 100192, China
² Department of Physics and ITP, The Chinese University of Hong Kong - Hong Kong, China
³ Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences Beijing 100190, China

Received: 2 March 2013
Accepted: 1 July 2013

Abstract

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 -dimensional Bloch momentum space.