Abstract
We show that the Wannier obstruction and the fragile topology of the nearly flat bands in twisted bilayer graphene at magic angle are manifestations of the nontrivial topology of two-dimensional real wave functions characterized by the Euler class. To prove this, we examine the generic band topology of two-dimensional real fermions in systems with space-time inversion symmetry. The Euler class is an integer topological invariant classifying real two-band systems. We show that a two-band system with a nonzero Euler class cannot have an -symmetric Wannier representation. Moreover, a two-band system with the Euler class has band crossing points whose total winding number is equal to . Thus the conventional Nielsen-Ninomiya theorem fails in systems with a nonzero Euler class. We propose that the topological phase transition between two insulators carrying distinct Euler classes can be described in terms of the pair creation and annihilation of vortices accompanied by winding number changes across Dirac strings. When the number of bands is bigger than two, there is a topological invariant classifying the band topology, that is, the second Stiefel Whitney class (). Two bands with an even (odd) Euler class turn into a system with () when additional trivial bands are added. Although the nontrivial second Stiefel-Whitney class remains robust against adding trivial bands, it does not impose a Wannier obstruction when the number of bands is bigger than two. However, when the resulting multiband system with the nontrivial second Stiefel-Whitney class is supplemented by additional chiral symmetry, a nontrivial second-order topology and the associated corner charges are guaranteed.
- Received 28 August 2018
DOI:https://doi.org/10.1103/PhysRevX.9.021013
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
Electrons in certain crystals such as graphene mimic massless particles that move at relativistic speeds. These “Dirac quasiparticles” carry a quantum number called handedness, which comes in two types: left handed and right handed. The conventional expectation is that the number of left- and right-handed Dirac quasiparticles should be equal. However, recent studies of twisted bilayer graphene, in which the two layers sit slightly askew to one another, suggest that all the Dirac quasiparticles have the same handedness. Here, we provide a general theory that identifies the topological origin of the failure of the naive expectation.
Our theory shows that the mismatch between the number of left- and right-handed Dirac quasiparticles can occur when their wave functions possess discontinuities of topological origin. This is surprising because the handedness of Dirac quasiparticles is a local property of the band structure in momentum space, whereas the continuity or discontinuity of wave functions reflects the global topological property of the band structure.
Our results resolve a fundamental issue in the electronic band structure of twisted bilayer graphene and provide new insight for identification and characterization of new topological phases of matter. We also expect that this work will stimulate future research activities to uncover the fundamental relationships between the local properties of quasiparticle excitations in momentum space and the global topology of wave functions in other crystals.
