Abelian and Non-Abelian Quantum Geometric Tensor

Abstract

We propose a generalized quantum geometric tenor to understand topological quantum phase transitions, which can be defined on the parameter space with the adiabatic evolution of a quantum many-body system. The generalized quantum geometric tenor contains two different local measurements, the non-Abelian Riemannian metric and the non-Abelian Berry curvature, which are recognized as two natural geometric characterizations for the change of the ground-state properties when the parameter of the Hamiltonian varies. Our results show the symmetry-breaking and topological quantum phase transitions can be understood as the singular behavior of the local and topological properties of the quantum geometric tenor in the thermodynamic limit.

5 pages, 2 figures

References in corpus (15)

• Topological entanglement entropy
• Detecting topological order in a ground state wave function
• Ground state overlap and quantum phase transitions
• Quantum Anomalous Hall Effect in Hg${}_{1-y}$Mn${}_y$Te Quantum Wells
• Fidelity, dynamic structure factor, and susceptibility in critical phenomena
• Quantum critical scaling of the geometric tensors
• A General Theorem Relating the Bulk Topological Number to Edge States in Two-dimensional Insulators
• Scaling of geometric phases close to quantum phase transition in the XY chain
• Geometric phases and criticality in spin chain systems
• 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
• Fidelity analysis of topological quantum phase transitions
• A Characterization of Topological Insulators: Chern Numbers for a Ground State Multiplet
• Fidelity in topological quantum phases of matter
• Geometric phase and quantum phase transition in an inhomogeneous periodic XY spin-1/2 model

Cited by in corpus (79)

• Fidelity approach to quantum phase transitions
• Low-frequency divergence and quantum geometry of the bulk photovoltaic effect in topological semimetals
• Experimental Measurement of the Quantum Metric Tensor and Related Topological Phase Transition with a Superconducting Qubit
• Classifying and measuring the geometry of the quantum ground state manifold
• Consequences of Time-reversal-symmetry Breaking in the Light-Matter Interaction: Berry Curvature, Quantum Metric and Diabatic Motion
• Topological invariants and phase diagrams for one-dimensional two-band non-Hermitian systems without chiral symmetry
• Geometry and response of Lindbladians
• Extracting the quantum metric tensor through periodic driving
• How to Measure the Quantum Geometry of Bloch Bands
• Quantum geometry induced second harmonic generation
• Topological quantum phase transitions retrieved through unsupervised machine learning
• Experimental Observation of Tensor Monopoles with a Superconducting Qudit
• Fundamental bound on topological gap
• Quantum Metric and Correlated States in Two-dimensional Systems
• Momentum space metric, non-local operator, and topological insulators
• Nonabelian magnonics in antiferromagnets
• Characterization of topological phase transitions via topological properties of transition points
• Relating the topology of Dirac Hamiltonians to quantum geometry: When the quantum metric dictates Chern numbers and winding numbers
• Winding numbers of phase transition points for one-dimensional topological systems
• The Euler Number of Bloch States Manifold and the Quantum Phases in Gapped Fermionic Systems
• Nontrivial quantum geometry of degenerate flat bands
• Steady-state Hall response and quantum geometry of driven-dissipative lattices
• Geometric characterization of anomalous Landau levels of isolated flat bands
• Conventional quantum phase transition driven by complex parameter in non-Hermitian PT-symmetric Ising model
• Time-Reversal Even Charge Hall Effect from Twisted Interface Coupling
• Geometric Critical Exponents in Classical and Quantum Phase Transitions
• Floquet-engineering of nodal rings and nodal spheres and their characterization using the quantum metric
• Quantum Kinetic Theory of Nonlinear Optical Currents: Finite Fermi surface and Fermi sea contributions
• Relating the entanglement spectrum of noninteracting band insulators to their quantum geometry and topology
• Dual Haldane sphere and quantized band geometry in chiral multifold fermions
• Direct measurement of quantum Fisher information
• Optimal design strategy for non-Abelian geometric phases using Abelian gauge fields based on quantum metric
• Revealing Chern number from quantum metric
• Geometric phase and phase diagram for non-Hermitian quantum XY model
• Extracting non-Abelian quantum metric tensor and its related Chern numbers
• Analytic approach to quantum metric and optical conductivity in Dirac models with parabolic mass in arbitrary dimensions
• Quantum distance and the Euler number index of the Bloch band in a 1D spin model
• Identifying non-Hermitian critical points with quantum metric
• Intrinsic nonlinear conductivity induced by quantum geometry in altermagnets and measurement of the in-plane Néel vector
• Classical analog of the quantum metric tensor
• Phase space formulation of the Abelian and non-Abelian quantum geometric tensor
• Quantum Geometry Probed by Chiral Excitonic Optical Response of Chern Insulators
• Non-Abelian quantum geometric tensor in degenerate topological semimetals
• Effective description of Chern insulators
• Band geometry from position-momentum duality at topological band crossings
• Geometry of the parameter space of a quantum system: Classical point of view
• Polaron spectra and edge singularities for correlated flat bands
• Detection of quantum geometric tensor by nonlinear optical response
• Diagnosing Topological Phase Transitions in 1D Superconductors using Berry Singularity Markers
• Quantum geometry and elliptic optical dichroism in $p$-wave magnets
• Quantum phase transition without gap opening
• Nonlinear Hall effects with an exceptional ring
• Geometric bound on structure factor
• Geometrical Rabi oscillations and Landau-Zener transitions in non-Abelian systems
• Quantum response theory and momentum-space gravity
• Quantum geometry and $X$-wave magnets with $X=p,d,f,g,i$
• Bounds for nonadiabatic transitions
• Quantum geometric tensor and the topological characterization of the extended Su-Schrieffer-Heeger model
• Higher-order bulk photovoltaic effects, quantum geometry and application to $p$-wave magnets
• Quantum geometry in many-body systems with precursors of criticality
• Enhancing the Hyperpolarizability of Crystals with Quantum Geometry
• From quantum geometry to non-linear optics and gerbes: Recent advances in topological band theory
• Orbital anomalous Hall effect in the few-layer Weyl semimetal TaIrTe4
• Geometric semimetals and their simulation in synthetic matter
• Universal intrinsic orbital dynamics from Berry curvature in electronic two-band systems
• Direct measurement of the quantum geometric tensor in pseudo-Hermitian systems
• Band topology of pseudo-Hermitian phases through tensor Berry connections and quantum metric
• Quantum geometric tensors from sub-bundle geometry
• Quantum simulation of Abelian Wu-Yang monopoles in spin-1/2 systems
• Uhlmann and scalar Wilczek-Zee phases of degenerate quantum systems
• A Brillouin torus decomposition for two-dimensional topological insulators
• Group Structure of Wilson Loops in 2D Models with 2- and 4-Band Energy Spectra
• Singular flat bands in three dimensions: Landau level spreading, quantum geometry, and Weyl reconstruction
• Exploring Many-Body Quantum Geometry Beyond the Quantum Metric with Correlation Functions: A Time-Dependent Perspective
• Quantum Geometry of Finite XY Chains: A Comparison of Neveu-Schwarz and Ramond Sectors
• Third-harmonic generation in superconductors: Role of quantum geometry in the competition between Higgs mode and quasiparticles
• Quantum Geometric Helical Superconductivity
• Theory of Berry Singularity Markers: Diagnosing Topological Phase Transitions via Lock-In Tomography
• Distinct topological excitonic insulators characterized by quantum geometry