Geometric approach to fragile topology beyond symmetry indicators

Abstract

We present a framework to systematically address topological phases when finer partitionings of bands are taken into account, rather than only considering the two subspaces spanned by valence and conduction bands. Focusing on $C_2\mathcal{T}$-symmetric systems that have gained recent attention, for example in the context of layered van-der-Waals graphene heterostructures, we relate these insights to homotopy groups of Grassmannians and flag varieties, which in turn correspond to cohomology classes and Wilson-flow approaches. We furthermore make use of a geometric construction, the so-called Plücker embedding, to induce windings in the band structure necessary to facilitate non-trivial topology. Specifically, this directly relates to the parametrization of the Grassmannian, which describes partitioning of an arbitrary band structure and is embedded in a better manageable exterior product space. From a physical perspective, our construction encapsulates and elucidates the concepts of fragile topological phases beyond symmetry indicators as well as non-Abelian reciprocal braiding of band nodes that arises when the multiple gaps are taken into account. The adopted geometric viewpoint most importantly culminates in a direct and easily implementable method to construct model Hamiltonians to study such phases, constituting a versatile theoretical tool.

22 pages, 9 figures, 9 pages of appendix; new version has updated discussion

References in corpus (16)

• Classification of topological insulators and superconductors in three spatial dimensions
• Topological Crystalline Insulators
• The space group classification of topological band insulators
• Topology of crystalline insulators and superconductors
• Bulk Topological Invariants in Noninteracting Point Group Symmetric Insulators
• Wannier representation of Z_2 topological insulators
• Periodic Table for Topological Bands with Non-Hermitian Bernard-LeClair Symmetries
• Building Blocks of Topological Quantum Chemistry: Elementary Band Representations
• Topological surface states in three-dimensional magnetic insulators
• Topological Euler class as a dynamical observable in optical lattices
• $PT$ Symmetric Real Dirac Fermions and Semimetals
• Topology of density matrices
• Current inversion at the edges of a chiral $\mathbf{p}$-wave superconductor
• Homotopy Theory of Strong and Weak Topological Insulators
• Bulk topology of line-nodal structures protected by space group symmetries in class AI
• Eigenbundles, Quaternions, and Berry's Phase

Cited by in corpus (3)

• PAI-graphene: a new topological semimetallic two-dimensional carbon allotrope with highly tunable anisotropic Dirac cones
• General construction of flat bands with and without band crossings based on wave function singularity
• Floquet Non-Abelian Topological Insulator and Multifold Bulk-Edge Correspondence