Quantum distance and the Euler number index of the Bloch band in a 1D spin model

Abstract

We study the Riemannian metric and the Euler characteristic number of the Bloch band in a 1D spin model with multi-site spins exchange interactions. The Euler number of the Bloch band originates from the Gauss-Bonnet theorem on the topological characterization of the closed Bloch states manifold in the first Brillouin zone. We study this approach analytically in a transverse field XY spin chain with three-site spin coupled interactions. We define a class of cyclic quantum distance on the Bloch band and on the ground state, respectively, as a local characterization for quantum phase transitions. Specifically, we give a general formula for the Euler number by means of the Berry curvature in the case of two-band models, which reveals its essential relation to the first Chern number of the band insulators. Finally, we show that the ferromagnetic-paramagnetic phases transition in zero-temperature can be distinguished by the Euler number of the Bloch band.

10 pages, 11 figures, some typos are corrected, the published version

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• Measurement of Topological Order based on Metric-Curvature Correspondence
• Mapping quantum geometry and quantum phase transitions to real space by a fidelity marker
• Cross-dimensional universality classes in static and periodically driven Kitaev models
• Quantum geometry of singlet superconductors
• Quantum geometrical properties of topological materials
• Classical description of the parameter space geometry in the Dicke and Lipkin-Meshkov-Glick models
• Quantum geometric tensor and the topological characterization of the extended Su-Schrieffer-Heeger model
• Adiabatic charge transport in extended SSH models
• Real space quantum metric of solids
• Absorbance marker: Detection of quantum geometry and spread of Wannier function in disordered 2D semiconductors