Subdimensional topologies, indicators and higher order phases

Abstract

The study of topological band structures have sparked prominent research interest the past decade, culminating in the recent formulation of rather prolific classification schemes that encapsulate a large fraction of phases and features. Within this context we recently reported on a class of unexplored topological structures that thrive on the concept of {\it sub-dimensional topology}. Although such phases have trivial indicators and band representations when evaluated over the complete Brillouin zone, they have stable or fragile topologies within sub-dimensional spaces, such as planes or lines. This perspective does not just refine classification pursuits, but can result in observable features in the full dimensional sense. In three spatial dimensions (3D), for example, sub-dimensional topologies can be characterized by non-trivial planes, having general topological invariants, that are compensated by Weyl nodes away from these planes. As a result, such phases have 3D stable characteristics such as Weyl nodes, Fermi arcs and edge states that can be systematically predicted by sub-dimensional analysis. Within this work we further elaborate on these concepts. We present refined representation counting schemes and address distinctive bulk-boundary effects, that include momentum depended (higher order) edge states that have a signature dependence on the perpendicular momentum. As such, we hope that these insights might spur on new activities to further deepen the understanding of these unexplored phases.

11+4pages; 8+3 figures

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• Manipulation and braiding of Weyl nodes using symmetry-constrained phase transitions
• Fragile topologically flat band in the checkerboard antiferromagnetic monolayer FeSe
• Symmetry-enforced topological band crossings in orthorhombic crystals: Classification and materials discovery
• Topological continuum charges of acoustic phonons in 2D
• Nodal higher-order topological superconductivity from a ${\mathcal{C}}_4$-symmetric Dirac semimetal
• Three-dimensional $\mathcal{P}\mathcal{T}$-symmetric topological phases with Pontryagin index
• Disorder-induced topological quantum phase transitions in multi-gap Euler semimetals
• Higher-order topological semimetals and nodal superconductors with an order-two crystalline symmetry
• Magnetic band representations, Fu-Kane-like symmetry indicators and magnetic topological materials
• Projected spin texture as a bulk indicator of fragile topology
• Dynamical Fragile Topology in Floquet Crystals
• Andreev reflection in Euler materials
• Magnetic wallpaper Dirac fermions and topological magnetic Dirac insulators
• Classification of Weyl points and nodal lines based on magnetic point groups for spin-$\frac{1}{2}$ quasiparticles
• Higher-order topological corner states induced solely by onsite potentials with mirror symmetry
• The topological invariants of rotationally symmetric crystals
• Fragile topological phase on the triangular kagome lattice and its bulk-boundary correspondence
• Two-dimensional orbital-obstructed insulators with higher-order band topology
• Second Euler number in four dimensional synthetic matter
• Selective branching, quenching, and converting of topological modes
• High-chirality and multiquaternion Weyl nodes in hexagonal ReO${}_3$
• Network model for magnetic higher-order topological phases
• Hinge modes of three-dimensional Euler insulators