Local Chern Marker for Periodic Systems

Abstract

Topological invariants are global properties of the ground-state wave function, typically defined as winding numbers in reciprocal space. Over the years, a number of topological markers in real space have been introduced, allowing to map topological order in heterogeneous crystalline and disordered systems. Notably, even if these formulations can be expressed in terms of lattice-periodic quantities, they can actually be deployed in open boundary conditions only, as in practice they require computing the position operator $\mathbf{r}$ in a form that is ill-defined in periodic boundary conditions. Here we derive a local Chern marker for infinite two-dimensional systems with periodic boundary conditions in the large supercell limit, where the electronic structure is sampled with one single point in reciprocal space. We validate our approach with tight-binding numerical simulations on the Haldane model, including trivial/topological superlattices made of pristine and disordered Chern insulators. The strategy introduced here is very general and could be applied to other topological invariants and quantum-geometrical quantities in any dimension.

7 pages, 3 figures + supplementary material (3 pages)

References in corpus (11)

• Entanglement Spectrum of a Disordered Topological Chern Insulator
• Toward Realistic Amorphous Topological Insulators
• Amorphous topological matter: theory and experiment
• Orbital magnetization and Chern number in a supercell framework: Single k-point formula
• Scaling and renormalization in the modern theory of polarization: application to disordered systems
• Topological invariants based on generalized position operators and application to the interacting Rice-Mele model
• Estimating bulk and edge topological indices in finite open chiral chains
• Probing Chern number by opacity and topological phase transition by a nonlocal Chern marker
• Single-point spin Chern number in a supercell framework
• A real-space many-body marker for correlated ${\mathbb{Z}}_2$ topological insulators
• Towards a theory of surface orbital magnetization

Cited by in corpus (14)

• The Wannier Function Software Ecosystem for Materials Simulations
• Tutorial: Classifying Photonic Topology Using the Spectral Localizer and Numerical $K$-Theory
• Phase transitions and scale invariance in topological Anderson insulators
• Theory of local ${\mathbb{Z}}_2$ topological markers for finite and periodic two-dimensional systems
• Topological characterization of modified Kane-Mele-Rashba models via local spin Chern marker
• Polarization textures in crystal supercells with topological bands
• Quantum geometric bounds in spinful systems with trivial band topology
• Global and Local Topological Crystalline Markers for Rotation-Symmetric Insulators
• Fluctuations and Correlations of Local Topological Order Parameters in Disordered Two-dimensional Topological Insulators
• Regularized universal topological markers for Dirac systems
• Tensor network method for real-space topology in quasicrystal Chern mosaics
• Chirality reversal at finite magnetic impurity strength and local signatures of a topological phase transition
• Real-space formulation of the Chern invariant and topological phases in a disordered Chern insulator
• Topological transition induced by selective random defects on a honeycomb lattice