Universal topological marker

Abstract

We elaborate that for topological insulators and topological superconductors described by Dirac models in any dimension and symmetry class, the topological order can be mapped to lattice sites by a universal topological marker. Deriving from a recently discovered momentum-space universal topological invariant, we introduce a topological operator that consists of alternating projectors to filled and empty lattice eigenstates and the position operators, multiplied by the Dirac matrices that are omitted in the Hamiltonian. The topological operator projected to lattice sites yields the topological marker, whose form is explicitly constructed for every topologically nontrivial symmetry class from 1D to 3D. The off-diagonal elements of the topological operator yields a nonlocal topological marker, which decays with a correlation length that diverges at topological phase transitions, and represents a Wannier state correlation function. Various prototype examples, including Su-Schrieffer-Heeger model, Majorana chain, Chern insulators, Bernevig-Hughes-Zhang model, 2D chiral and helical $p$-wave superconductors, lattice model of ${}^3$He B-phase, and 3D time-reversal symmetric topological insulators, etc, are employed to demonstrate the ubiquity of our formalism.

12 pages, 4 figures

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Cited by in corpus (27)

• Mapping quantum geometry and quantum phase transitions to real space by a fidelity marker
• Topological marker approach to an interacting Su-Schrieffer-Heeger model
• Robustness of topological order against disorder
• Tutorial: Classifying Photonic Topology Using the Spectral Localizer and Numerical $K$-Theory
• Probing Chern number by opacity and topological phase transition by a nonlocal Chern marker
• Optical absorption measurement of spin Berry curvature and spin Chern marker
• Dielectric and optical markers originating from quantum geometry
• Structural spillage: an efficient method to identify non-crystalline topological materials
• Theory of local ${\mathbb{Z}}_2$ topological markers for finite and periodic two-dimensional systems
• Quantum geometrical properties of topological materials
• Interacting Local Topological Markers: A one-particle density matrix approach for characterizing the topology of interacting and disordered states
• Coexistence of 1D and 2D topology and genesis of Dirac cones in the chiral Aubry-André model
• Fractal Surface States in Three-Dimensional Topological Quasicrystals
• Topological Phases in Fractals: Local Spin Chern Marker in the Sierpinski carpet Kane-Mele-Rashba Model
• Topological characterization of modified Kane-Mele-Rashba models via local spin Chern marker
• Quantized thermal and spin transports of dirty planar topological superconductors
• Topological zero-modes of the spectral localizer of trivial metals
• Orbital Magnetization under an Electric Field and Orbital Magnetoelectric Polarizabilty for a Bilayer Chern System
• Evaluation of real-space second Chern number using the kernel polynomial method
• Emergent fractals in dirty topological crystals
• Regularized universal topological markers for Dirac systems
• Phase-Space Topology in a Single-Atom Synthetic Dimension
• Tensor network method for real-space topology in quasicrystal Chern mosaics
• Absorbance marker: Detection of quantum geometry and spread of Wannier function in disordered 2D semiconductors
• Real-space topology and charge order in the Haldane-Holstein Model
• Topological transition induced by selective random defects on a honeycomb lattice
• Projected branes as platforms for crystalline, superconducting, and higher-order topological phases