Classification of flat bands according to the band-crossing singularity of Bloch wave functions

We show that flat bands can be categorized into two distinct classes, that is, singular and nonsingular flat bands, by exploiting the singular behavior of their Bloch wave functions in momentum space. In the case of a singular flat band, its Bloch wave function possesses immovable discontinuities generated by the band-crossing with other bands, and thus the vector bundle associated with the flat band cannot be defined. This singularity precludes the compact localized states from forming a complete set spanning the flat band. Once the degeneracy at the band crossing point is lifted, the singular flat band becomes dispersive and can acquire a finite Chern number in general, suggesting a new route for obtaining a nearly flat Chern band. On the other hand, the Bloch wave function of a nonsingular flat band has no singularity, and thus forms a vector bundle. A nonsingular flat band can be completely isolated from other bands while preserving the perfect flatness. All one-dimensional flat bands belong to the nonsingular class. We show that a singular flat band displays a novel bulk-boundary correspondence such that the presence of the robust boundary mode is guaranteed by the singularity of the Bloch wave function. Moreover, we develop a general scheme to construct a flat band model Hamiltonian in which one can freely design its singular or nonsingular nature. Finally, we propose a general formula for the compact localized state spanning the flat band, which can be easily implemented in numerics and offer a basis set useful in analyzing correlation effects in flat bands.Comment: 23 pages, 13 figure

Searching for topological density wave insulators in multi-orbital square lattice systems

We study topological properties of density wave states with broken translational symmetry in two-dimensional multi-orbital systems with a particular focus on t$_{2g}$ orbitals in square lattice. Due to distinct symmetry properties of d-orbitals, a nodal charge or spin density wave state with Dirac points protected by lattice symmetries can be achieved. When an additional order parameter with opposite reflection symmetry is introduced to a nodal density wave state, the system can be fully gapped leading to a band insulator. Among those, topological density wave (TDW) insulators can be realized, when an effective staggered on-site potential generates a gap to a pair of Dirac points connected by the inversion symmetry which have the same topological winding numbers. We also present a mean-field phase diagram for various density wave states, and discuss experimental implications of our results.Comment: 15 pages, 10 figures, 7 table

Failure of Nielsen-Ninomiya theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: application to twisted bilayer graphene at magic angle

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 $I_{ST}$ 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 $I_{ST}$-symmetric Wannier representation. Moreover, a two-band system with the Euler class $e_{2}$ has band crossing points whose total winding number is equal to $-2e_2$. 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 $Z_{2}$ topological invariant classifying the band topology, that is, the second Stiefel Whitney class ($w_2$). Two bands with an even (odd) Euler class turn into a system with $w_2=0$ ($w_2=1$) 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 multi-band 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.Comment: 23 pages, 13 figure

Topological insulators and metal-insulator transition in the pyrochlore iridates

The possible existence of topological insulators in cubic pyrochlore iridates A$_{2}$Ir$_{2}$O$_{7}$ (A = Y or rare-earth elements) is investigated by taking into account the strong spin-orbit coupling and trigonal crystal field effect. It is found that the trigonal crystal field effect, which is always present in real systems, may destabilize the topological insulator proposed for the ideal cubic crystal field, leading to a metallic ground state. Thus the trigonal crystal field is an important control parameter for the metal-insulator changeover. We propose that this could be one of the reasons why distinct low temperature ground states may arise for the pyrochlore iridates with different A-site ions. On the other hand, examining the electron-lattice coupling, we find that softening of the $\textbf{q}$=0 modes corresponding to trigonal or tetragonal distortions of the Ir pyrochlore lattice leads to the resurrection of the strong topological insulator. Thus, in principle, a finite temperature transition to a low-temperature topological insulator can occur via structural changes. We also suggest that the application of the external pressure along [111] or its equivalent directions would be the most efficient way of generating strong topological insulators in pyrochlore iridates.Comment: 10 pages, 11 figures, 2 table