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We review recent progresses in the study of flat band systems, especially focusing on the fundamental physics related to the singularity of the flat bands Bloch wave functions. We first explain that the flat bands can be classified into two classes: singular and nonsingular flat bands, based on the presence or absence of the singularity in the flat bands Bloch wave functions. The singularity is generated by the band crossing of the flat band with another dispersive band. In the singular flat band, one can find special kind of eigenmodes, called the non-contractible loop states and the robust boundary modes, which exhibit nontrivial real space topology. Then, we review the experimental realization of these topological eigenmodes of the flat band in the photonic lattices. While the singularity of the flat band is topologically trivial, we show that the maximum quantum distance around the singularity is a bulk invariant representing the strength of the singularity which protects the robust boundary modes. Finally, we discuss how the maximum quantum distance or the strength of the singularity manifests itself in the anomalous Landau level spreading of the singular flat band when it has a quadratic band-crossing with another band.
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Recently microcavities with anisotropic materials are shown to be able to create novel bands with non-zero local Berry curvature. The anisotropic refractive index of the cavity layer is believed to be critical in opening an energy gap at the tilted Dirac points. In this work, we show that an anticrossing between a cavity mode and a Bragg mode can also form within an empty microcavity without any birefringent materials. Flat bands are observed within the energy gap due to the particular refractive index distribution of the sample. The intrinsic TE-TM splitting and XY splitting induce the squeezing of the cavity modes in momentum space, so that the flat bands are spin-dependently tilted. Our results pave the way to investigate the spin orbit coupling of photons in a simple microcavity without anisotropic cavity layers.
We demonstrate that the concept of moire flat bands can be generalized to achieve electronic band engineering in all three spatial dimensions. For many two dimensional van der Waals materials, twisting two adjacent layers with respect to each other leads to flat electronic bands in the two corresponding spatial directions -- a notion sometimes referred to as twistronics as it enables a wealth of physical phenomena. Within this two dimensional plane, large moire patterns of nanometer size form. The basic concept we propose here is to stack multiple twisted layers on top of each other in a predefined pattern. If the pattern is chosen such that with respect to the stacking direction of layers, the large spatial moire features are spatially shifted from one twisted layer to the next, the system exhibits twist angle controlled flat bands in all of the three spatial directions. With this, our proposal extends the use of twistronic to three dimensions. We exemplify the general concept by considering graphitic systems, boron nitride and WSe$_2$ as candidate materials, but the approach is applicable to any two-dimensional van der Waals material. For hexagonal boron nitride we develope an ab initio fitted tight binding model that captures the corresponding three dimensional low-energy electronic structure. We outline that interesting three dimensional correlated phases of matter can be induced and controlled following this route, including quantum magnets and unconventional superconducting states.
For a general class of conducting polymers with arbitrary large unit cell and different on-site Coulomb repulsion values on different type of sites, I demonstrate in exact terms the emergence possibility of an upper, interaction created effective flat band. This last appears as a consequence of a kinetic energy quench accompanied by a strong interaction energy decrease, and leads to a non-saturated ferromagnetic state. This ordered state clearly differs from the known flat-band ferromagnetism. This is because it emerges in a system without bare flat bands, requires inhomogeneous on-site Coulomb repulsions values, and possesses non-zero lower interaction limits at the emergence of the ordered phase.
Twisted graphene bilayers provide a versatile platform to engineer metamaterials with novel emergent properties by exploiting the resulting geometric moir{e} superlattice. Such superlattices are known to host bulk valley currents at tiny angles ($alphaapprox 0.3 ^circ$) and flat bands at magic angles ($alpha approx 1^circ$). We show that tuning the twist angle to $alpha^*approx 0.8^circ$ generates flat bands away from charge neutrality with a triangular superlattice periodicity. When doped with $pm 6$ electrons per moire cell, these bands are half-filled and electronic interactions produce a symmetry-broken ground state (Stoner instability) with spin-polarized regions that order ferromagnetically. Application of an interlayer electric field breaks inversion symmetry and introduces valley-dependent dispersion that quenches the magnetic order. With these results, we propose a solid-state platform that realizes electrically tunable strong correlations.
The energy spectra for the tight-binding models on the Lieb and kagome lattices both exhibit a flat band. We present a model which continuously interpolates between these two limits. The flat band located in the middle of the three-band spectrum for the Lieb lattice is distorted, generating two pairs of Dirac points. While the upper pair evolves into graphene-like Dirac cones in the kagome limit, the low energy pair evolves until it merges producing the band-bottom flat band. The topological characterization of the Dirac points is achieved by projecting the Hamiltonian on the two relevant bands in order to obtain an effective Dirac Hamiltonian. The low energy pair of Dirac points is particularly interesting in this respect: when they emerge, they have opposite winding numbers, but as they merge, they have the same winding number. This apparent paradox is due to a continuous rotation of their states in pseudo-spin space, characterized by a winding vector. This simple, but quite rich model, suggests a way to a systematic characterization of two-band contact points in multiband systems.
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