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.

The Zak phase γ, the generalization of the Berry phase to Bloch wave functions in solids, is often used to characterize inversion-symmetric 1D topological insulators; however, since its value can depend on the choice of real-space origin and unit cell, only the difference between the Zak phase of two regions is believed to be relevant. Here, we show that one can extract an origin-independent part of γ, the so-called inter-cellular Zak phase γ inter , which can be used as a bulk quantity to predict the number of surface modes as follows: a neutral finite 1D tight-binding system has ns = γ inter /π (mod 2) number of in-gap surface modes below the Fermi level if there exists a commensurate bulk unit cell that respects inversion symmetry. We demonstrate this by first verifying that ±eγ inter /2π (mod e) is equal to the extra charge accumulation in the surface region for a general translationally invariant 1D insulator, while the remnant part of γ, the intra-cellular Zak phase γ intra , corresponds to the electronic part of the dipole moment of the bulk's unit cell. Second, we show that the extra charge accumulation can be related to the number of surface modes when the unit cell is inversion symmetric. This bulk-boundary correspondence using γ inter reduces to the conventional one using γ when the real-space origin is at the inversion center. Our work thereby clarifies the usage of γ in the bulk-boundary correspondence. We study several tight binding models to quantitatively check the relation between the extra charge accumulation and the inter-cellular Zak phase as well as the bulk-boundary correspondence using the inter-cellular Zak phase.

We investigate novel Landau level structures of semi-metals with nodal ring dispersions. When the magnetic field is applied parallel to the plane in which the ring lies, there exist almost nondispersive Landau levels at the Fermi level (EF = 0) as a function of the momentum along the field direction inside the ring. We show that the Landau levels at each momentum along the field direction can be described by the Hamiltonian for the graphene bilayer with fictitious inter-layer couplings under a tilted magnetic field. Near the center of the ring where the inter-layer coupling is negligible, we have Dirac Landau levels which explain the appearance of the zero modes. Although the inter-layer hopping amplitudes become finite at higher momenta, the splitting of zero modes is exponentially small and they remain almost flat due to the finite artificial in-plane component of the magnetic field. The emergence of the density of states peak at the Fermi level would be a hallmark of the ring dispersion. PACS numbers:Introduction.-Semi-metals, usually the reflection of unconventional electronic structures at the Fermi surface (FS), are related to various anomalous properties and/or exotic phases such as unconventional quantum Hall effect (QHE) in graphene systems [1][2][3][4], pressure induced anomalous Hall effect in the Weyl semi-metal (SM) [5] and non-Fermi liquid phase and peculiar quantum oscillations in the quadratic band-touching SM [6][7][8][9][10]. Also, in many cases, they are classified as topologically nontrivial metals involving surface states [12][13][14][15]18] which are generalizations of the concept of the topological insulator [16] to the metallic systems [17].

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