In this work, we develop a systematic method of constructing flat-band models with and without band crossings. Our construction scheme utilizes the symmetry and spatial shape of a compact localized state (CLS) and also the singularity of the flat-ban
d wave function obtained by a Fourier transform of the CLS (FT-CLS). In order to construct a flat-band model systematically using these ingredients, we first choose a CLS with a specific symmetry representation in a given lattice. Then, the singularity of FT-CLS indicates whether the resulting flat band exhibits a band crossing point or not. A tight-binding Hamiltonian with the flat band corresponding to the FT-CLS is obtained by introducing a set of basis molecular orbitals, which are orthogonal to the FT-CLS. Our construction scheme can be systematically applied to any lattice so that it provides a powerful theoretical framework to study exotic properties of both gapped and gapless flat bands arising from their wave function singularities.
Flat bands have band crossing points with other dispersive bands in many systems including the canonical flat band models in the Lieb and kagome lattices. Here we show that some of such band degeneracy points are unavoidable because of the symmetry r
epresentation (SR) of the flat band under unitary symmetry. We refer to such a band degeneracy point of flat bands as a SR-enforced band crossing. SR-enforced band crossing is distinct from the conventional band degeneracy protected by symmetry eigenvalues or topological charges in that its protection requires both specific symmetry representation and band flatness of the flat band, simultaneously. Even $n$-fold rotation $C_n$ ($n=2,3,4,6$) symmetry, which cannot protect band degeneracy without additional symmetries due to its abelian nature, can protect SR-enforced band crossings in flat band systems. In two-dimensional flat band systems with $C_n$ symmetry, when the degeneracy of a SR-enforced band crossing is lifted by a $C_n$ symmetry-preserving perturbation, we obtain a nearly flat Chern band. Our theory not only explains the origin of the band crossing points of FBs existing in various models, but also gives a strict no-go theorem for isolated FBs in a given lattice arising from the SR.
Landau levels of a band are conventionally believed to be bounded by its upper and lower band edges at zero magnetic field so that no Landau level is expected to appear in the gapped regions of the original band structure. Two notable examples evadin
g this expectation are topological bands with quantized invariants and flat bands with singular band crossing. Here we introduce a distinct class of flat band systems in which an isolated flat band exhibits anomalous Landau level spreading (LLS) outside the zero-field energy bounds. In particular, we demonstrate that the LLS is determined by the cross-gap Berry connection measuring the wave function geometry of multi-bands. Moreover, we find that symmetry puts strong constraints on the LLS of flat bands. Namely, the LLS hows a quadratic magnetic field dependence in systems with space-time-inversion symmetry while it completely vanishes when the system respects chiral symmetry. Also, when the system respects time-reversal or reflection symmetries at zero magnetic field, the maximum and minimum values of the LLS have the same magnitude but with opposite signs. Our general mechanism for the LLS of flat bands reveals the fundamental role of wave function geometry in describing anomalous magnetic responses in solids.
Geometry of the wave function is a central pillar of modern solid state physics. In this work, we unveil the wave-function geometry of two-dimensional semimetals with band crossing points (BCPs). We show that the Berry phase of BCPs are governed by t
he quantum metric describing the infinitesimal distance between quantum states. For generic linear BCPs, we show that the corresponding Berry phase is determined either by an angular integral of the quantum metric, or equivalently, by the maximum quantum distance of Bloch states. This naturally explains the origin of the $pi$-Berry phase of a linear BCP. In the case of quadratic BCPs, the Berry phase can take an arbitrary value between 0 and $2pi$. We find simple relations between the Berry phase, maximum quantum distance, and the quantum metric in two cases: (i) when one of the two crossing bands is flat; (ii) when the system has rotation and/or time-reversal symmetries. To demonstrate the implication of the continuum model analysis in lattice systems, we study tight-binding Hamiltonians describing quadratic BCPs. We show that, when the Berry curvature is absent, a quadratic BCP with an arbitrary Berry phase always accompanies another quadratic BCP so that the total Berry phase of the periodic system becomes zero. This work demonstrates that the quantum metric plays a critical role in understanding the geometric properties of topological semimetals.
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.
Corbino-geometry has well-known applications in physics, as in the design of graphene heterostructures for detecting fractional quantum Hall states or superconducting waveguides for illustrating circuit quantum electrodynamics. Here, we propose and d
emonstrate a photonic Kagome lattice in the Corbino-geometry that leads to direct observation of non-contractible loop states protected by real-space topology. Such states represent the missing flat-band eigenmodes, manifested as one-dimensional loops winding around a torus, or lines infinitely extending to the entire flat-band lattice. In finite (truncated) Kagome lattices, however, line states cannot preserve as they are no longer the eigenmodes, in sharp contrast to the case of Lieb lattices. Using a continuous-wave laser writing technique, we experimentally establish finite Kagome lattices with desired cutting edges, as well as in the Corbino-geometry to eliminate edge effects. We thereby observe, for the first time to our knowledge, the robust boundary modes exhibiting self-healing properties, and the localized modes along toroidal direction as a direct manifestation of the non-contractible loop states.
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 wa
ve 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.
Topological insulator nanowires with uniform cross section, combined with a magnetic flux, can host both a perfectly transmitted mode and Majorana zero modes. Here we consider nanowires with rippled surfaces---specifically, wires with a circular cros
s section with a radius varying along its axis---and calculate their transport properties. At zero doping, chiral symmetry places the clean wires (no impurities) in the AIII symmetry class, which results in a $mathbb{Z}$ topological classification. A magnetic flux threading the wire tunes between the topologically distinct insulating phases, with perfect transmission obtained at the phase transition. We derive an analytical expression for the exact flux value at the transition. Both doping and disorder breaks the chiral symmetry and the perfect transmission. At finite doping, the interplay of surface ripples and disorder with the magnetic flux modifies quantum interference such that the amplitude of Aharonov-Bohm oscillations reduces with increasing flux, in contrast to wires with uniform surfaces where it is flux-independent.
The bulk-boundary correspondence, a topic of intensive research interest over the past decades, is one of the quintessential ideas in the physics of topological quantum matter. Nevertheless, it has not been proven in all generality and has in certain
scenarios even been shown to fail, depending on the boundary profiles of the terminated system. Here, we introduce bulk numbers that capture the exact number of in-gap modes, without any such subtleties in one spatial dimension. Similarly, based on these 1D bulk numbers, we define a new 2D winding number, which we call the pole winding number, that specifies the number of robust metallic surface bands in the gap as well as their topological character. The underlying general methodology relies on a simple continuous extrapolation from the bulk to the boundary, while tracking the evolution of Greens functions poles in the vicinity of the bulk band edges. As a main result we find that all the obtained numbers can be applied to the known insulating phases in a unified manner regardless of the specific symmetries. Additionally, from a computational point of view, these numbers can be effectively evaluated without any gauge fixing problems. In particular, we directly apply our bulk-boundary correspondence construction to various systems, including 1D examples without a traditional bulk-boundary correspondence, and predict the existence of boundary modes on various experimentally studied graphene edges, such as open boundaries and grain boundaries. Finally, we sketch the 3D generalization of the pole winding number by in the context of topological insulators.
We have performed density functional theory calculation and tight binging analysis in order to investigate the mechanism for the giant Rashba-type spin splitting (RSS) observed in Bi/Ag(111). We find that local orbital angular momentum induces moment
um and spin dependent charge distribution which results in spin-dependent hopping. We show that the spin-dependent interatomic-hopping in Bi/Ag(111) works as a strong effective field and induces the giant RSS, indicating that the giant RSS is driven by hopping, not by a uniform electric field. The effective field from the hopping energy difference amounts to be ~18 V/{AA}. This new perspective on the RSS gives us a hint for the giant RSS mechanism in general and should provide a strategy for designing new RSS materials by controlling spin-dependence of hopping energy between the neighboring atomic layers.
