Six-dimensional quantum Hall effect and three-dimensional topological pumps

Abstract

Modern technological advances allow for the study of systems with additional synthetic dimensions. Using such approaches, higher-dimensional physics that was previously deemed to be of purely theoretical interest has now become an active field of research. In this work, we derive from first principles using a semiclassical equation-of-motion approach the bulk response of a six-dimensional Chern insulator. We find that in such a system a quantized bulk response appears with a quantization originating from a six-dimensional topological index: the 3 rd Chern number. Alongside this unique six-dimensional response, we rigorously describe the lower even-dimensional Chern-type responses that can occur due to nonvanishing 1 st and 2 nd Chern numbers in subspaces of the sixdimensional space. Last, we propose how to realize such a bulk response using three-dimensional topological charge pumps in cold atomic systems.

Higher-order topological insulators, topological pumps and the quantum Hall effect in high dimensions

“…An even q is crucial for obtaining a low-energy theory corresponding to decoupled Dirac cones [20], thus, defining regions in the BZ that are separated by gap closures. Moreover, our results readily extend to 3D, explaining the appearance of hinge modes and relating corner states to a six-dimensional CI and its third Chern number [19,30]. These charges arise from a combination of xy (k) for a selected path ink, exhibiting regions with: (i) zero-energy states localized at the upper/lower left corner, (ii) zero-energy states localized at the upper/lower right corner, and (iii) no zero-energy solutions.…”

“…A paradigmatic example of a TI is the Chern insulator (CI), appearing in even dimensions. CIs exhibit quantized bulk transport responses [18,19] proportional to the topological indices (Chern numbers) characterizing their spectra. Interestingly, using dimensional reduction, a CI in d dimensions is mapped to a family of models in d − m dimensions, dubbed the "descendant pump family".…”

Electric circuit simulations of nth -Chern-number insulators in 2n -dimensional space and their non-Hermitian generalizations for arbitrary n

“…They are characterized by the second-Chern number. In the same way, the 6D quantum Hall effect is realized in 3D topological pumpings 31 . However, these phases are disentangled, where they are decomposed into two or three independent copies of 2D quantum Hall insulators.…”

Observation of Hofstadter butterfly and topological edge states in reconfigurable quasi-periodic acoustic crystals

“…A number of theoretical and experimental works have recently induced quasiperiodicity in a broad variety of systems 7,[21][22][23][24][25][26][27] to explore topological phase transitions and edge states. Even more, topological phases in higher dimensions 23,[28][29][30][31] have been predicted and the those characterized by second class Chern number in 2D quasiperiodic crystals observed experimentally 21,22 .…”