Topological hinge modes in Dirac semimetals

doi:10.1007/s11467-022-1221-y

Front. Phys.

2023, Vol. 18

Issue (1) : 13308 https://doi.org/10.1007/s11467-022-1221-y

RESEARCH ARTICLE

Topological hinge modes in Dirac semimetals

Xu-Tao Zeng¹, Ziyu Chen¹, Cong Chen^1,², Bin-Bin Liu¹, Xian-Lei Sheng^1,³(

), Shengyuan A. Yang^4,⁵

¹. School of Physics, Beihang University, Beijing 100191, China
². Department of Physics, The University of Hong Kong, Hong Kong, China
³. Peng Huanwu Collaborative Center for Research and Education, Beihang University, Beijing 100191, China
⁴. Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, Singapore
⁵. Center for Quantum Transport and Thermal Energy Science, School of Physics and Technology, Nanjing Normal University, Nanjing 210023, China

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Abstract

Dirac semimetals (DSMs) are an important class of topological states of matter. Here, focusing on DSMs of band inversion type, we investigate their boundary modes from the effective model perspective. We show that in order to properly capture the boundary modes, k-cubic terms must be included in the effective model, which would drive an evolution of surface degeneracy manifold from a nodal line to a nodal point. Sizable k-cubic terms are also needed for better exposing the topological hinge modes in the spectrum. Using first-principles calculations, we demonstrate that this feature and the topological hinge modes can be clearly exhibited in β-CuI. We extend the discussion to magnetic DSMs and show that the time-reversal symmetry breaking can gap out the surface bands and hence is beneficial for the experimental detection of hinge modes. Furthermore, we show that magnetic DSMs serve as a parent state for realizing multiple other higher-order topological phases, including higher-order Weyl-point/nodal-line semimetals and higher-order topological insulators.

Keywords topological hinge Dirac semimetals

Corresponding Author(s): Xian-Lei Sheng

Issue Date: 17 November 2022

Cite this article:

Xu-Tao Zeng,Ziyu Chen,Cong Chen, et al. Topological hinge modes in Dirac semimetals[J]. Front. Phys. , 2023, 18(1): 13308.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-022-1221-y
https://academic.hep.com.cn/fop/EN/Y2023/V18/I1/13308

Fig.1 DSM effective model without the

k

-cubic terms [Eq. (1)]. (a) We discretize the model on a 3D hexagonal lattice. (b) The corresponding BZ. (c) Bulk band structure. Here, each band is twofold degenerate. (d) Surface band dispersion on a side surface. There is a surface nodal line form by the crossing of surface bands, which connects the projections of two bulk Dirac points. (e) The corresponding surface spectrum along high symmetry path and (f) the constant energy slice at Fermi level. Here, we take the model parameters as

C 0 = 1, C 1 = 0.2, C 2 = 0, M 0 = 1, M 1 = 0.5, M 2 = 0.5, and A = 1

Fig.2 DSM effective model with cubic terms included [Eq. (2)

+

Eq. (1)]. (a) Bulk band structure. (b) Surface band dispersion on a side surface. There is a Dirac cone at the surface BZ center. (c) Surface spectrum and (d) constant energy slice at Fermi level for the side surface. (e) Spectrum of a 1D hexagonal tube geometry (with 30 cell length of an edge) as shown in (f). The hinge modes are indicated by the red lines. (f) Spatial distribution of the hinge mode marked by the star in (e). Here, we take the parameters as

C 0 = 1, C 1 = 0.2, C 2 = 0, M 0 = 1, M 1 = 0.5, M 2 = 0.5, and A = 1

Fig.3 Results for the effective 2D Hamiltonian

H ~ λ (k x, k y) = H ~ 0 λ + H ~ 1 λ

when

λ = π / 3

. (a) Evolution of the Wannier centers for the occupied bands. (b) Nested Berry phase when

λ

varies along

k z

. The system has a nontrivial second-order topology for

λ

between the two Dirac points. (c, d) Spectra for the nanodisk geometry (c) without and (d) with

H ~ 1 λ

. The insets show the distribution of the states marked in red in the spectra. Here, we take the parameters as

C 0 = 1, C 1 = 0.2, C 2 = 0, M 0 = 1, M 1 = 0.5, M 2 = 0.5, and A = 1

Fig.4 (a) Crystal structure of hexagonal

β

-CuI. (b) The first BZ of

β

-CuI and its projected surface BZ on (010) planes. (c, d) Band structure of

β

-CuI (c) without and (d) with spin-orbit coupling.

Fig.5 (a, b) Projected spectrum and the Fermi contours for the (010) surface. (c) Spectrum for the 1D tube geometry of

β

-CuI as shown in (d). Here, each side of the tube cross section has a length of 60 unit cells. The hinge modes are highlighted by the red lines. (d) Spatial distribution of two hinge modes marked in (c).

Fig.6 (a) Illustration of the lattice model for magnetic DSM. (b) Bulk band structure. (c) Surface spectrum of the model [Eq. (9)] along high symmetry paths for the side surface normal to

y

. (d) Surface band dispersion. (e) Spectrum of a 1D tube geometry shown in (f). Each side of the cross section has a length of 40 cells. The hinge modes are highlighted by the red lines. (f) Spatial distribution of the model marked by star in (e). Here, we take the parameters as

m 0 = 2,

m 1 = m 2 = w = 0.5, v = 1, m 3 = 0.2

, and all other parameters are set to zero.

Fig.7 Magnetic higher-order Weyl semimetal. (a, b) Illustration of transition from 2 Dirac points to 4 Weyl points (c) Bulk band structure. (d) Constant energy slice at Fermi level for the side surface. (e) Spectrum of a 1D square tube geometry (with 60 cell length of an edge) as shown in (f). The hinge modes are indicated by the red lines. (f) Spatial distribution of the hinge mode of the hinge Dirac cone at

Γ

point in (e). Here, we take the parameters as

C 0 = 0,

C 1 = 0, C 2 = 0, M 0 = 1, M 1 = 0.5, M 2 = 0.5, A = 1

and

D = 0.5

Fig.8 Magnetic higher-order nodal-line semimetal. (a) Bulk band structure. (b) Two nodal rings are illustrated. Each ring features a nontrivial winding number. (c) The hinge modes are indicated by the red lines. (d) Spatial distribution of the hinge mode marked by the star in (c). Here, we take the parameters as

C 0 = 0, C 1 = 0, C 2 = 0, M 0 = 1, M 1 = 0.5,

M 2 = 0.5, A = 1

and

D = 0.5

Fig.9 Magnetic higher-order topological insulator. (a) Bulk band structure. (b) Surface spectrum are totally gapped by introducing

C 4 z

breaking term

D k z σ x s y

. (c) Spectrum of a 1D square tube geometry (with 20 cell length of an edge) as shown in (d). The hinge modes are indicated by the red(diagonal) and blue(off-diagonal) lines. (d) Spatial distribution of the hinge mode of the hinge Dirac cone at

Γ

point in (c). Here, we take the parameters as

C 0 = 0, C 1 = 0,

C 2 = 0, M 0 = 1, M 1 = 0.5, M 2 = 0.5, A = 1

and

D = 0.5

Fig. B1 (a, b) Projected spectrum and the Fermi contour on the (010) surface. The projection of bulk Dirac points are indicated by two white points. (c) Spectrum of a 1D tube geometry as shown in (d). Each side of the cross section has a width of 40 cells. The hinge modes are highlighted by the red color. (d) Spatial distribution of the hinge mode marked by star in (c). In the calculation, we take the parameters as

C 0 = 1, C 1 = 0.25, C 2 = 0, M 0 = 1, M 1 = 0.5, M 2 = 0.5, A 0 = 1, A 1 =

A 2 = 0, B 1 = B 2 = 0.5

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