Mapping quantum geometry and quantum phase transitions to real space by a fidelity marker

Abstract

The quantum geometry in the momentum space of semiconductors and insulators, described by the quantum metric of the valence band Bloch state, has been an intriguing issue owing to its connection to various material properties. Because the Brillouin zone is periodic, the integration of quantum metric over momentum space represents an average distance between neighboring Bloch states, of which we call the fidelity number. We show that this number can further be expressed in real space as a fidelity marker, which is a local quantity that can be calculated directly from diagonalizing the lattice Hamiltonian. A linear response theory is further introduced to generalize the fidelity number and marker to finite temperature, and moreover demonstrates that they can be measured from the global and local optical absorption power against linearly polarized light. In particular, the fidelity number spectral function in 2D systems can be easily measured from the opacity of the material. Based on the divergence of quantum metric, a nonlocal fidelity marker is further introduced and postulated as a universal indicator of any quantum phase transitions provided the crystalline momentum remains a good quantum number, and it may be interpreted as a Wannier state correlation function. The ubiquity of these concepts is demonstrated for a variety of topological insulators and topological phase transitions in different dimensions.

11 pages, 5 figures

References in corpus (14)

• Universal Dynamic Conductivity and Quantized Visible Opacity of Suspended Graphene
• Classification of topological insulators and superconductors in three spatial dimensions
• Fidelity, dynamic structure factor, and susceptibility in critical phenomena
• Entanglement Spectrum of a Disordered Topological Chern Insulator
• Fidelity susceptibility, scaling, and universality in quantum critical phenomena
• Geometry of quantum phase transitions
• Fidelity susceptibility and long-range correlation in the Kitaev honeycomb model
• Correlation Length, Universality Classes, and Scaling Laws Associated with Topological Phase Transitions
• Universality classes of topological phase transitions with higher-order band crossing
• Universal topological marker
• Floquet-engineering of nodal rings and nodal spheres and their characterization using the quantum metric
• Quantum distance and the Euler number index of the Bloch band in a 1D spin model
• Quantum metric on the Brillouin Zone in correlated electron systems and its relation to topology for Chern insulators
• Optical absorption measurement of spin Berry curvature and spin Chern marker

Cited by in corpus (28)

• The quantum geometric origin of capacitance in insulators
• Topological properties of a non-Hermitian quasi-1D chain with a flat band
• Analytic approach to quantum metric and optical conductivity in Dirac models with parabolic mass in arbitrary dimensions
• Quantum geometry of singlet superconductors
• Dielectric and optical markers originating from quantum geometry
• Hofstadter spectrum of Chern bands in twisted transition metal dichalcogenides
• Corner Charge Fluctuation as an Observable for Quantum Geometry and Entanglement in Two-dimensional Insulators
• Intrinsic gyrotropic magnetic current from Zeeman quantum geometry
• Detecting the spread of valence band Wannier functions by optical sum rules
• Quantum geometrical properties of topological materials
• Quantum geometry and elliptic optical dichroism in $p$-wave magnets
• Quantum geometric origin of the Meissner effect and superfluid weight marker
• Quantum geometry and $X$-wave magnets with $X=p,d,f,g,i$
• Opacity of graphene independent of light frequency and polarization due to the topological charge of the Dirac points
• Quantum geometry in many-body systems with precursors of criticality
• Higher-order bulk photovoltaic effects, quantum geometry and application to $p$-wave magnets
• Quantum geometric bounds in spinful systems with trivial band topology
• Geometric semimetals and their simulation in synthetic matter
• Effects of the Hubbard interaction on the quantum metric
• Real space quantum metric of solids
• Quantum geometric tensors from sub-bundle geometry
• Berry curvature and quantum metric in copper-substituted lead phosphate apatite
• Non-Hermitian wave-packet dynamics and its realization within a non-Hermitian chiral cavity
• Absorbance marker: Detection of quantum geometry and spread of Wannier function in disordered 2D semiconductors
• Regularized universal topological markers for Dirac systems
• Quantum geometry, localization, and topological bounds of spin fluctuations
• Tensor network method for real-space topology in quasicrystal Chern mosaics
• Quantum Geometry Driven Finite-Momentum Exciton Fluctuations in Flat-Band Systems