Abstract

Nonsymmorphic symmetries can enforce band connectivity that obstructs a single-band Wannier description. We show that a fractional translation L connecting distinct high-symmetry Wyckoff positions generically renders the Wannier center of an individual band gauge ill-defined, requiring a symmetry-enforced multiband object—a Wannier complex. We formulate a real-space topological classification of Wannier complexes and show that, when L is combined with certain point-group symmetries (notably C₄ and C₃), all symmetry-allowed Wannier-complex configurations carry a nontrivial quantized total electric polarization. This yields boundary phenomena that persist across symmetry-preserving deformations of the Hamiltonian, including parameter regimes with and without bulk gaps. We demonstrate the mechanism in minimal tight-binding models exhibiting Möbius-twisted Wilson-loop structures and higher-order corner modes, and propose experimental signatures in a dielectric photonic crystal and a first-principles electronic platform octa-graphene, accompanied by a three-dimensional extension.

Abstract

Nonsymmorphic symmetries can enforce band connectivity that obstructs a single-band Wannier description. We show that a fractional translation L connecting distinct high-symmetry Wyckoff positions generically renders the Wannier center of an individual band gauge ill-defined, requiring a symmetry-enforced multiband object—a Wannier complex. We formulate a real-space topological classification of Wannier complexes and show that, when L is combined with certain point-group symmetries (notably C₄ and C₃), all symmetry-allowed Wannier-complex configurations carry a nontrivial quantized total electric polarization. This yields boundary phenomena that persist across symmetry-preserving deformations of the Hamiltonian, including parameter regimes with and without bulk gaps. We demonstrate the mechanism in minimal tight-binding models exhibiting Möbius-twisted Wilson-loop structures and higher-order corner modes, and propose experimental signatures in a dielectric photonic crystal and a first-principles electronic platform octa-graphene, accompanied by a three-dimensional extension.

Abstract

Nonsymmorphic symmetries can enforce band connectivity that obstructs a single-band Wannier description. We show that a fractional translation L connecting distinct high-symmetry Wyckoff positions generically renders the Wannier center of an individual band gauge ill-defined, requiring a symmetry-enforced multiband object—a Wannier complex. We formulate a real-space topological classification of Wannier complexes and show that, when L is combined with certain point-group symmetries (notably C₄ and C₃), all symmetry-allowed Wannier-complex configurations carry a nontrivial quantized total electric polarization. This yields boundary phenomena that persist across symmetry-preserving deformations of the Hamiltonian, including parameter regimes with and without bulk gaps. We demonstrate the mechanism in minimal tight-binding models exhibiting Möbius-twisted Wilson-loop structures and higher-order corner modes, and propose experimental signatures in a dielectric photonic crystal and a first-principles electronic platform octa-graphene, accompanied by a three-dimensional extension.

Abstract

Nonsymmorphic symmetries can enforce band connectivity that obstructs a single-band Wannier description. We show that a fractional translation L connecting distinct high-symmetry Wyckoff positions generically renders the Wannier center of an individual band gauge ill-defined, requiring a symmetry-enforced multiband object—a Wannier complex. We formulate a real-space topological classification of Wannier complexes and show that, when L is combined with certain point-group symmetries (notably C₄ and C₃), all symmetry-allowed Wannier-complex configurations carry a nontrivial quantized total electric polarization. This yields boundary phenomena that persist across symmetry-preserving deformations of the Hamiltonian, including parameter regimes with and without bulk gaps. We demonstrate the mechanism in minimal tight-binding models exhibiting Möbius-twisted Wilson-loop structures and higher-order corner modes, and propose experimental signatures in a dielectric photonic crystal and a first-principles electronic platform octa-graphene, accompanied by a three-dimensional extension.