The Euler Number of Bloch States Manifold and the Quantum Phases in Gapped Fermionic Systems

TL;DR

This paper introduces a topological Euler number derived from the quantum geometric tensor to characterize topological phases in gapped fermionic systems, providing a new perspective beyond the Chern number.

Contribution

It proposes a novel topological invariant based on the Euler number from the Gauss-Bonnet theorem for Bloch states, expanding topological characterization methods.

Findings

Euler number distinguishes phases in a transverse field XY spin chain.

The geometric tensor's real part defines the Euler characteristic, while the imaginary part relates to the Chern number.

Analytical discussion of topological numbers in a general two-band model.

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 (1+1)-dimensional Bloch momentum space.