The impossibility of exactly flat non-trivial Chern bands in strictly local periodic tight binding models

Chen, Li; Mazaheri, Tahereh; Seidel, Alexander; Tang, Xiang

doi:10.1088/1751-8113/47/15/152001

Cited by 85 publications

(91 citation statements)

References 33 publications

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“…As a result, the conductivity vanishes, at least on the noninteracting level. This accords with the vanishing Chern number of strictly flat single particle bands where dε k /dk = 0 , throughout the Brillouin zone, as proven recently [7] for tight binding lattices.On-site Hubbard interaction seems to delocalize vicinally caged carriers, which was considered as indication for nonzero conductivity [8]. However, short range interactions cannot impair the huge flat band degeneracy at low fillings [9].…”

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confidence: 87%

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Flat-band conductivity properties at long-range Coulomb interactions

Häusler

2015

Phys. Rev. B

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Dispersionless (flat) electronic bands are investigated regarding their conductance properties. Due to "caging" of carriers these bands are usually insulating at partial filling, at least on the non-interacting level. Considering the specific example of a T3-lattice we study long-range Coulomb interactions. A non-trivial dependence of the conductivity on flat band filling is obtained, exhibiting an infinite number of zeros. Near these zeros, the conductivity rises linearly with carrier density. At densities half way in between adjacent conductivity-zeros, strongly enhanced conductivity is predicted, accompanying a solid-solid phase transition.PACS numbers: 71.10. Fd,73.20.Qt,73.20.Jc,73.90.+f Recently, electronic flat bands in periodic lattices have received considerable attention [1], where single particle energies ε k of at least one tight binding band stay non-or weakly dispersing, throughout the Brillouin zone. One focus of interest in two spatial dimensions, similar to Landau levels, is the effect of interactions which in the absence of kinetic energy always has to be treated nonperturbatively. Fractional Chern insulator (FCI) phases have been identified [2,3], some of which exhibit Chern numbers larger than unity [4] and thereby generalize fractional quantum Hall states. Meanwhile, many lattices have been detected to host flat bands. Band flatness [5] arises due to localization by local quantum interferences, coined [6] as "caging" of carriers. As a result, the conductivity vanishes, at least on the noninteracting level. This accords with the vanishing Chern number of strictly flat single particle bands where dε k /dk = 0 , throughout the Brillouin zone, as proven recently [7] for tight binding lattices.On-site Hubbard interaction seems to delocalize vicinally caged carriers, which was considered as indication for nonzero conductivity [8]. However, short range interactions cannot impair the huge flat band degeneracy at low fillings [9]. Competing charge density wave phases have been studied [3]. Here, we investigate long range Coulomb interactions which at any filling will lift the flat band degeneracy. In quantum Hall systems, when kinetic energy is quenched, they cause Wigner crystallization at fillings ν < 1 /5 [10] (in graphene at ν < 0.28 [11]) while without magnetic field crystallization appears when the Coulomb energy E C exceeds the kinetic energy E K by a sufficiently big factor [12], E C /E K > 37 . The longitudinal conductivityof clean systems at zero temperature diverges [13] and is then due to sliding of the hexagonal crystal as a whole [14]. Its Drude weight D = e 2 n/m is determined by carrier density n and particle masses m, just as in the absence of interactions [15]; in this work we do not con- sider crystalline disorder. In flat bands, where no kinetic energy competes, we always expect a Wigner crystallized phase. Conductance properties, quantified again by the Drude weight D, cannot simply be proportional to m −1 since m is a priori meaningless to parameterize kinetic energy [5]. In ...

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confidence: 87%

supporting

confidence: 87%

Flat-band conductivity properties at long-range Coulomb interactions

Häusler

2015

Phys. Rev. B

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“…[8] as a consequence of the no-go theorem mentioned in the preceding subsection. First, we notice that the eigenvalue problem for the flat band (or degenerate flat bands) is given by…”

Section: Flat-band Hamiltoniansmentioning

confidence: 83%

“…When more than one flat band is present, the case of interest is usually that in which there are flat bands that all have the same energy eigenvalue (regardless of whether or not the remaining bands are also flat, with different energy). A particular question that has appeared [8] is whether the Hamiltonian that has the flat band or bands can have strictly-short-range matrix elements. (This paper appeared earlier than the published version of Ref.…”

Section: Flat-band Hamiltoniansmentioning

confidence: 99%

“…(We consider only systems that are invariant under a discrete group of translations on a lattice in real [position] space, and have in mind tightbinding models that have only a finite-dimensional space of orbitals [single-particle states] available at each lattice site.) The various approaches just mentioned are (i) tensor network states [1,2]-the extension of matrix product states [3] to more than one dimension-which can be applied to interacting [4] as well as to non-interacting particles; (ii) compactly-supported Wannier-type functions [5,6], an extreme example of Wannier functions [7], of interest in electronic structure calculations; and (iii) flat-band Hamiltonians with strictly short-range hopping [8], an extreme form of the flat-band approach popular in constructions of topological insulator states, including ones with interactions.…”

Section: Introductionmentioning

confidence: 99%

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Compactly supported Wannier functions and algebraicK-theory

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2017

Phys. Rev. B

109

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In a tight-binding lattice model with n orbitals (single-particle states) per site, Wannier functions are n-component vector functions of position that fall off rapidly away from some location, and such that a set of them in some sense span all states in a given energy band or set of bands; compactlysupported Wannier functions are such functions that vanish outside a bounded region. They arise not only in band theory, but also in connection with tensor-network states for non-interacting fermion systems, and for flat-band Hamiltonians with strictly short-range hopping matrix elements. In earlier work, it was proved that for general complex band structures (vector bundles) or general complex Hamiltonians-that is, class A in the ten-fold classification of Hamiltonians and band structures-a set of compactly-supported Wannier functions can span the vector bundle only if the bundle is topologically trivial, in any dimension d of space, even when use of an overcomplete set of such functions is permitted. This implied that, for a free-fermion tensor network state with a nontrivial bundle in class A, any strictly short-range parent Hamiltonian must be gapless. Here, this result is extended to all ten symmetry classes of band structures without additional crystallographic symmetries, with the result that in general the non-trivial bundles that can arise from compactlysupported Wannier-type functions are those that may possess, in each of d directions, the non-trivial winding that can occur in the same symmetry class in one dimension, but nothing else. The results are obtained from a very natural usage of algebraic K-theory, based on a ring of polynomials in e ±ikx , e ±iky , . . . , which occur as entries in the Fourier-transformed Wannier functions.

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Magnetic‐Flux‐Induced Tunable Completely Flat Band and Topological Nearly Flat Band in Square Kagome Lattice

Chen

et al. 2023

Annalen der Physik

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The flat bands in a square kagome lattice containing square and triangle plaquettes, are respectively investigated, and the origin of the doubly degenerated completely flat bands and the corresponding compact localized states are elucidated. It is found that the introduction of external magnetic flux enriches the modulation parameters, making the system present many interesting results. In the case that the magnetic flux penetrates through each square plaquette, the tunable completely flat band has one more tunable parameter. And when the external magnetic flux penetrates through each triangle plaquette, excepting a completely flat band, the band dispersions of the system present the topological nearly flat band, which is very useful to realize the interesting fractional quantum Hall physics. The average density of states is also calculated to corroborate the completely FB generated by the highly localized eigenstates. Furthermore, the implementation of the square kagome lattice system based on the current photonic waveguide network technology is demonstrated. The scheme opens up a way to generate the tunable completely flat band and topological nearly flat band in square kagome lattice under multi-parameter variable conditions.

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The impossibility of exactly flat non-trivial Chern bands in strictly local periodic tight binding models

Cited by 85 publications

References 33 publications

Flat-band conductivity properties at long-range Coulomb interactions

Flat-band conductivity properties at long-range Coulomb interactions

Compactly supported Wannier functions and algebraicK-theory

Magnetic‐Flux‐Induced Tunable Completely Flat Band and Topological Nearly Flat Band in Square Kagome Lattice

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