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.

arXiv comment10 pages, 11 figures, some typos are corrected, the published version

References in corpus (55)

• Topological Insulators
• Topological insulators and superconductors
• Quantum Spin Hall Effect in Graphene
• $Z_2$ Topological Order and the Quantum Spin Hall Effect
• Berry Phase Effects on Electronic Properties
• Maximally-localized generalized Wannier functions for composite energy bands
• Topological Insulators in Three Dimensions
• Introducing Quantum Discord
• Classification of topological insulators and superconductors in three spatial dimensions
• Entanglement in quantum critical phenomena
• Topological entanglement entropy
• Topological insulators and superconductors: ten-fold way and dimensional hierarchy
• Detecting topological order in a ground state wave function
• Periodic table for topological insulators and superconductors
• Scaling of Entanglement close to a Quantum Phase Transitions
• Entanglement in a simple quantum phase transition
• Continuous Quantum Phase Transitions
• Quantum discord and the power of one qubit
• High temperature fractional quantum Hall states
• Fractional quantum Hall states at zero magnetic field
• Direct Measurement of the Zak phase in Topological Bloch Bands
• Nearly-flat bands with nontrivial topology
• Ground state overlap and quantum phase transitions
• Quantum phase transitions
• Fidelity approach to quantum phase transitions
• Fidelity, dynamic structure factor, and susceptibility in critical phenomena
• Quantum critical scaling of the geometric tensors
• Entanglement and quantum phase transition in the extended Hubbard model
• Local entanglement and quantum phase transition in the Hubbard model
• Diverging Entanglement Length in Gapped Quantum Spin Systems
• Geometrical Description of the Fractional Quantum Hall Effect
• Classification of Topological Insulators and Superconductors
• Topological invariants for spin-orbit coupled superconductor nanowires
• Scaling of geometric phases close to quantum phase transition in the XY chain
• Geometric phases and criticality in spin chain systems
• Classifying and measuring the geometry of the quantum ground state manifold
• Fidelity susceptibility and long-range correlation in the Kitaev honeycomb model
• Fidelity and Quantum phase transition for the Heisenberg chain with the next-nearest-neighbor interaction
• Abelian and Non-Abelian Quantum Geometric Tensor
• How to Measure the Quantum Geometry of Bloch Bands
• Topological Classification of Gapped Spin Chains :Quantized Berry Phase as a Local Order Parameter
• New Results for the Correlation Functions of the Ising Model and the Transverse Ising Chain
• Global quantum correlations in finite-size spin chains
• Momentum space metric, non-local operator, and topological insulators
• Fidelity susceptibility of the quantum Ising model in the transverse field: The exact solution
• Electron Localization in the Quantum-Hall Regime
• Many-body generalization of the Z2 topological invariant for the quantum spin Hall effect
• The Euler Number of Bloch States Manifold and the Quantum Phases in Gapped Fermionic Systems
• Scaling of ground state fidelity in the thermodynamic limit: XY model and beyond
• Symmetry protected Z2-quantization and quaternionic Berry connection with Kramers degeneracy
• Geometric phase and quantum phase transition in an inhomogeneous periodic XY spin-1/2 model
• Quantum phase transitions in a new exactly solvable quantum spin biaxial model with multiple spin interactions
• Fidelity susceptibility of one-dimensional models with twisted boundary conditions
• Quantization of non-Abelian Berry phase for time reversal invariant systems
• Criticalities of the transverse- and longitudinal-field fidelity susceptibilities for the d=2 quantum Ising model

Cited by in corpus (11)

• Semi-classical wave packet dynamics in non-uniform electric fields
• 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