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

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.

References in corpus (12)

• Fidelity, dynamic structure factor, and susceptibility in critical phenomena
• Quantum critical scaling of the geometric tensors
• Geometrical Description of the Fractional Quantum Hall Effect
• Geometric phases and criticality in spin chain systems
• Fidelity susceptibility and long-range correlation in the Kitaev honeycomb model
• Abelian and Non-Abelian Quantum Geometric Tensor
• Entanglement, fidelity and topological entropy in a quantum phase transition to topological order
• Fidelity analysis of topological quantum phase transitions
• Intrinsic geometry of quantum adiabatic evolution and quantum phase transitions
• Topological Classification of Gapped Spin Chains :Quantized Berry Phase as a Local Order Parameter
• Fidelity in topological quantum phases of matter
• Geometric phase and quantum phase transition in an inhomogeneous periodic XY spin-1/2 model

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