Classification of Dirac points with higher-order Fermi arcs

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Classification of Dirac points with higher-order Fermi arcs

Yuan Fang and Jennifer Cano

Phys. Rev. B 104, 245101 – Published 1 December 2021

Abstract

Dirac semimetals lack a simple bulk-boundary correspondence. Recently, Dirac materials with fourfold rotation symmetry have been shown to exhibit a higher-order bulk-hinge correspondence: they display “higher-order Fermi arcs,” which are localized on hinges where two surfaces meet and connect the projections of the bulk Dirac points. In this paper, we classify higher-order Fermi arcs for Dirac semimetals protected by a rotation symmetry and the product of time-reversal and inversion. Such Dirac points can be either linear in all directions or linear along the rotation axis and quadratic in other directions. By computing the filling anomaly for momentum-space planes on either side of the Dirac point, we find that all linear Dirac points exhibit higher-order Fermi arcs terminating at the projection of the Dirac point, while the Dirac points that are quadratic in two directions lack such higher-order Fermi arcs. When higher-order Fermi arcs do exist, they obey either a $Z_{2}$ (fourfold rotation axis) or $Z_{3}$ (three- or sixfold rotation axis) group structure. Finally, we build two models with sixfold symmetry to illustrate the cases with and without higher-order Fermi arcs. We predict higher-order Fermi arcs in ${Na}_{3} Bi$ .

Received 21 September 2021
Accepted 12 November 2021

DOI:https://doi.org/10.1103/PhysRevB.104.245101

Physics Subject Headings (PhySH)

Symmetry protected topological states Topological insulators

Dirac semimetal

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Yuan Fang ¹ and Jennifer Cano ^1,2

¹Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11974, USA
²Center for Computational Quantum Physics, Flatiron Institute, New York, New York 10010, USA

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Vol. 104, Iss. 24 — 15 December 2021

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Images

Figure 1
Schematic diagram showing HOFAs. (a) Hexagonal unit cell of space group $P 6 / m$ . (b) Each plane with fixed $k_{z} \neq 0, π$ is regarded as an effective 2D system with the symmetry of the magnetic layer group $p 6 / m^{'}$ . After this dimensional reduction, interlayer hopping between atoms separated by a unit cell is viewed as a $k_{z}$ -dependent onsite potential: $t^{'} = t e^{i k_{z}} + t^{†} e^{- i k_{z}}$ . Other interlayer hopping terms are similarly projected to in-plane hopping terms. (c) When the crystal is terminated in a $C_{6}$ -symmetric rod geometry, the 2D planes with nontrivial filling anomaly contribute corner states to HOFAs of the 3D model. Red lines indicate HOFAs that could appear between two Dirac points (whose projection onto the hinges is labeled by crosses).
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Figure 2
(a) The unit cell of $P 6 / m m m$ . $a_{3}$ is in the $z$ direction and is perpendicular to $a_{1}$ and $a_{2}$ . (b) The cross section of a rod which is $C_{6}$ symmetric, finite in the $a_{1}$ and $a_{2}$ directions and infinite in $a_{3}$ direction.
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Figure 3
(a) Rod spectrum of the tight-binding model described by Eq. (24) (see Sec. 5a for connection to ${Na}_{3} Bi$ .) There are HOFAs between $k_{z} = 0$ and $k_{z} = k_{0}$ , the projection of the bulk Dirac point. There are also gapless surface states projected to $k_{z} = 0$ . (b) Energy of states at $0 < k_{z} = π / 4 < k_{0}$ for the same model. The dashed red line indicates charge neutrality. The nontrivial filling anomaly is indicated by the charge neutrality point residing in the middle of six degenerate corner states. (c) Rod spectrum of the tight-binding model described by Eq. (31). There are no HOFA. (d) Energy of states at $0 < k_{z} = π / 4 < k_{0}$ for the second model. The dashed red line indicates charge neutrality. The lack of filling anomaly is indicated by the charge neutrality point residing in between two groups of degenerate states. There are gapless surface states projecting to $k_{z} = 0$ . For both models, the side length of the hexagon cross section [see Fig. 2] is 15. The parameters used to generate the plots are listed in Appendix pp5.
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Figure 4
In this schematic diagram, we take space group $P 6_{3} / m$ (with $T^{2} = - 1$ ) as an example of a case with screw symmetry. (a) Hexagonal unit cell of space group $P 6_{3} / m$ . (b) A fixed $k_{z}$ slice with $k_{z} \neq 0, π$ forms an effective 2D system described by the magnetic layer group $p 6 / m^{'}$ . The atoms project to $z = 0$ in the unit cell of this 2D system. The interlayer hopping term $t^{'} = t_{1} e^{i k_{z} / 2} + t_{2} e^{- i k_{z} / 2}$ becomes an in-plane hopping term. (c) Terminating the 2D models with boundaries that preserve $C_{6}$ symmetry, yields the spectrum of the rod states for the 3D model. Red lines indicate possible HOFAs that could appear between two Dirac points (whose projection onto the hinges is labeled by crosses).
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Figure 5
The projected 2D unit cell and Wyckoff positions of (a) $p 4 / m^{'}$ and $p 4 / m 1^{'}$ and (b) $p \bar{3}, p 6 / m^{'}, p \bar{3} 1^{'}$ , and $p 6 / m 1^{'}$ .
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Figure 6
Bulk and surface band structures for models of Dirac points in $P 6 / m m m$ . (a) The bulk BZ. (b) The surface BZ. This is a side surface with the normal $a_{1} = (1, 0, 0)$ . The $L$ and $M$ points of the bulk BZ are projected to $\bar{R}$ and $\bar{X}$ , respectively. (c) Bulk spectrum of the first model Eq. (24). (d) Surface spectrum of the first model. (e) Bulk spectrum of the second model. (f) Surface spectrum of the second model Eq. (31).
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