Local Topological Markers in Odd Dimensions

Abstract

Local topological markers have proven to be a valuable tool for investigating systems with topologically non-trivial bands. Due to their local nature, such markers can treat translationally invariant systems and spatially inhomogeneous systems on an equal footing. Among the most prevalent of these is the so-called Chern marker, which is available for systems in two spatial dimensions. In this paper, we describe how to generalize this marker to 1d and 3d systems, by showing that the relevant expressions accurately describe the phenomenon of topological pumping given by the first and second Chern numbers in 1d and 3d respectively. In addition to providing general derivations, we verify the markers by numerically considering model Hamiltonians. These results will open the door for future studies including the influence of disorder on topological pumping and topological phase transitions in odd-dimensional systems.

14 pages. Minor edits. Comments welcome

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• Tutorial: Classifying Photonic Topology Using the Spectral Localizer and Numerical $K$-Theory
• Phase transitions and scale invariance in topological Anderson insulators
• Optimally Localized Wannier Functions for 2D Chern Insulators
• 1D quasicrystals and topological markers
• Effects of Boundary on Orbital Magnetization for a Bilayer System with Different Chern Numbers
• Orbital Magnetization under an Electric Field and Orbital Magnetoelectric Polarizabilty for a Bilayer Chern System
• Flow of unitary matrices: Real-space winding numbers in one and three dimensions
• Evaluation of real-space second Chern number using the kernel polynomial method
• Tensor network method for real-space topology in quasicrystal Chern mosaics
• Coarse geometric approach to topological phases: Invariants from real-space representations