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Flat band, spin-1 Dirac cone, and Hofstadter diagram in the fermionic square kagome model

Tomonari Mizoguchi, Yoshihito Kuno, and Yasuhiro Hatsugai
Phys. Rev. B 104, 035161 – Published 30 July 2021
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  • INTRODUCTION
  • MODEL AND BAND STRUCTURE
  • HOFSTADTER PROBLEM
  • SUMMARY
  • ACKNOWLEDGMENTS
    • References

    Abstract

    We study characteristic band structures of the fermions on a square kagome lattice, one of the two-dimensional lattices hosting a corner-sharing network of triangles. We show that the band structures of the nearest-neighbor tight-binding model exhibit many characteristic features, including a flat band which is ubiquitous among frustrated lattices. On the flat band, we elucidate its origin by using the molecular-orbital representation and also find localized exact eigenstates called compact localized states. In addition to the flat band, we also find two spin-1 Dirac cones with different energies. These spin-1 Dirac cones are not described by the simplest effective Dirac Hamiltonian because the middle band is bended and the energy spectrum is particle-hole asymmetric. We also investigated the Hofstadter problem on a square kagome lattice in the presence of an external field and find that the profile of the Chern numbers around the modified spin-1 Dirac cones coincides with the conventional one.

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    • Received 5 March 2021
    • Revised 24 June 2021
    • Accepted 12 July 2021

    DOI:https://doi.org/10.1103/PhysRevB.104.035161

    ©2021 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Electronic structure
    1. Physical Systems
    Kagome latticeTopological materials
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Tomonari Mizoguchi*, Yoshihito Kuno, and Yasuhiro Hatsugai

    • Department of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan

    • *mizoguchi@rhodia.ph.tsukuba.ac.jp

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    Issue

    Vol. 104, Iss. 3 — 15 July 2021

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    Images

    • Figure 1
      Figure 1

      A square kagome lattice. A black dashed square denotes the unit cell.

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    • Figure 2
      Figure 2

      The band structures for the tight-binding model for (a) t1=1, t2=1, (b) t1=1, t2=1.2, and (c) t1=1.2, t2=1. For (a), we show the zoom-up of the spin-1 Dirac cones. Pink lines correspond to the directions where the middle band has flat dispersion.

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    • Figure 3
      Figure 3

      Schematic figure of (a) MOs and (b) a CLS on a square plaquette for any t1 and t2. The panels (c) and (d) are MOs and an additional CLS on an octagonal plaquette, respectively, which exist only for t1=t2.

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    • Figure 4
      Figure 4

      Schematics of the loop states. The loop is along (a) the x direction and (b) the y direction. The numbers beside the sites denote the values of the wave function (up to the normalization constant). The wave function is zero on the sites without the number.

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    • Figure 5
      Figure 5

      Duality between the square kagome lattice and the square octagon lattice. The sites on the square octagon lattice are depicted as gray dots.

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    • Figure 6
      Figure 6

      Schematic of the flux distribution considered in this paper. The gray shade denotes the area of the unit cell (in the absence of the magnetic flux).

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    • Figure 7
      Figure 7

      (a) The Hofstadter diagram of the square kagome model. The zoom-up in the low field for (b) near E=0 and (c) E=2. The numbers at the blanks are the Chern numbers.

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