Non-Abelian Hopf-Euler insulators

We discuss a class of three-band non-Abelian topological insulators in three dimensions that carry a single bulk Hopf index protected by spatiotemporal (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi mathvariant="script">PT</a:mi></a:math>) inversion symmetry. These phases may also host subdimensional topological invariants given by the Euler characteristic class, resulting in real Hopf-Euler insulators. Such systems naturally realize helical nodal structures in the three-dimensional Brillouin zone, providing a physical manifestation of the linking number described by the Hopf invariant. We show that, by opening a gap between the valence bands of these systems, one finds a fully-gapped “flag” phase, which displays a three-band multigap Pontryagin invariant. Unlike the previously reported <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi mathvariant="script">PT</c:mi></c:math>-symmetric four-band real Hopf insulator, which hosts a <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mi mathvariant="double-struck">Z</e:mi><e:mo>⊕</e:mo><e:mi mathvariant="double-struck">Z</e:mi></e:mrow></e:math> invariant, these phases are not unitarily equivalent to two copies of a complex two-band Hopf insulator. We show that such uncharted phases can be obtained through dimensional extension of two-dimensional Euler insulators, and that they support (i) an optical bulk integrated circular shift effect quantized by the Hopf invariant, (ii) quantum-geometric breathing in the real-space Wannier functions, and (iii) surface Euler topology on boundaries. Consequently, our findings pave the way for novel experimental realizations of real-space quantum geometry, as these systems may be directly simulated by utilizing synthetic dimensions in metamaterials or ultracold atoms. Published by the American Physical Society 2024