Pseudospin-$\frac{3}{2}$ fermions, type-II Weyl semimetals, and critical Weyl semimetals in tricolor cubic lattices

Pseudospin- $\frac{3}{2}$ fermions, type-II Weyl semimetals, and critical Weyl semimetals in tricolor cubic lattices

Motohiko Ezawa

Phys. Rev. B 94, 195205 – Published 16 November 2016

Abstract

Multiband touchings, such as three-band, six-band, and eight-band touchings, together with the emergence of high-pseudospin fermions were predicted recently at high-symmetry points in three-dimensional space. In this paper, we propose a simple cubic model whose unit cell contains three atoms. There are six bands in the system due to the spin degrees of freedom. The four-band and two-band touchings are realized at high-symmetry points, where we derive low-energy theories, demonstrating the emergence of pseudospin-3/2 fermions and Weyl fermions, respectively. Away from the high-symmetry points, we find critical Weyl fermions present exactly at the boundary between the type-I and type-II Weyl fermions. This critical Weyl fermion transforms into the type-I or type-II Weyl fermion once the magnetic field is applied.

Received 11 September 2016
Revised 24 October 2016

DOI:https://doi.org/10.1103/PhysRevB.94.195205

Physics Subject Headings (PhySH)

Spin-orbit coupling Topological phases of matter Weyl fermions

Weyl semimetal

Tight-binding model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Motohiko Ezawa

Department of Applied Physics, University of Tokyo, Hongo 7-3-1, 113-8656, Japan

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Vol. 94, Iss. 19 — 15 November 2016

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Images

Figure 1
Bird's eye view of the band structure near the $P$ point. The horizontal plane is spanned by the $k_{z}$ and $k_{x}$ axes. The vertical axis is the energy $E = E (k_{x}, π, k_{z})$ . (a) Two-band (cyan circle) and four-band (red circle) touchings are observed at the $P$ point. There appear other two-band touchings (green circle) off the $P$ point. (b),(c) An enlarged portion of the two-band (four-band) touching indicates the emergence of (one) two Weyl cones at the $P$ point.
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Figure 2
Lattice structure and the Brillouin zone. (a) The lattice structure of a tricolor cubic lattice. (b) The unit cell contains three atoms, which are colored by magenta, cyan, and yellow. Each colored atom forms the body-center-cubic lattice. (c) The Brillouin zone and the high-symmetry points $Γ, P, H, N$ , and $Δ$ . The green bold lines show the cut along the $Γ − Δ − H − G − N − Σ − Γ − Λ − P − F − H − F − P − D − N$ line.
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Figure 3
Band structures of the six-band model. The band structure along the $Γ − Δ − H − G − N − Σ − Γ − Λ − P$ - $F − H − F − P − D − N$ line for typical values of parameters $(t, λ)$ as indicated in the figures. (a) All energy bands are degenerate with respect to up and down spins when $λ = 0$ . The fourfold degenerate points are marked by magenta circles. (b) The spin degeneracy is resolved by the SOI $(λ \neq 0)$ except for the high-symmetry points (indicated by circles) where the SOI vanishes. Pseudospin-3/2 fermions (red) and Weyl fermions (cyan) emerge at various points.
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Figure 4
Band structures in the vicinity of the $P$ point. The horizontal axes are the $k_{z}$ and $k_{x}$ axes, while the vertical axis is the energy $E$ . (a1)–(d1) The band structure without and with magnetic field $B$ in the [001] direction. The value of $B$ is given in the figures. Four-band touching (red circle) and two two-band touchings (green) are observed. (a2)–(d2) Four-band and two-band touchings occur along the $k_{z}$ axis $(k_{x} = k_{y} = 0)$ . The cross sections of various surfaces are well approximated by parabolic curves. (b3) and (b4) show enlarged portions of the vicinity of the four-band touching in (b1) and (b2), respectively. Symbols C, I, and II stand for the critical, type-I, and type-II Weyl points, respectively. The points indicated by II* are identical to those in Figs. 5 and 6. (b2) shows that the four-band touching (red circle) is broken but the two two-band touchings (green) are not. The critical Weyl points in (a2) turn out to be type-I and type-II Weyl points in (b2). See also Fig. 5 for details. The pair of the type-II Weyl points (II*) in (c2) and (b4) are annihilated in (d2). See Fig. 6 for details.
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Figure 5
Transition from the critical Weyl fermions to the type-I and type-II Weyl fermions. (a1),(a2) Critical Weyl points. (b1) Type-I Weyl point. (b2) Type-II Weyl point. They represent bird's eye views of the corresponding points in Figs. 4(a2) and 4(b2).
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Figure 6
Pair annihilation of two type-II Weyl fermions. (a) A bird's eye view of the type-II points marked II* in Figs. 4(b2) and 4(b4). (b) These two Weyl fermions merge at a critical magnetic field as in Fig. 4(c2). (c) They are annihilated beyond the critical field.
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