Orbital design of flat bands in non-line-graph lattices via line-graph wave functions

Hang Liu, Gurjyot Sethi, Sheng Meng, and Feng Liu

Phys. Rev. B 105, 085128 – Published 17 February 2022

Abstract

Line-graph (LG) lattices are known for having flat bands (FBs) from the destructive interference of Bloch wave functions encoded in only lattice symmetry. Here, we develop a generic atomic/molecular $o r b i t a l$ design principle for FBs in non-LG lattices. Based on linear combination of atomic orbital theory, we demonstrate that the underlying wave-function symmetry of FBs in a LG lattice can be transformed into the atomic/molecular orbital symmetry in a non-LG lattice. We illustrate such orbital-designed topological FBs in three 2D non-LG, square, trigonal, and hexagonal lattices, where the designed orbitals faithfully reproduce the corresponding lattice symmetries of checkerboard, kagome, and diatomic-kagome lattices, respectively. Interestingly, systematic design of FBs with a high Chern number is also achieved based on the same principle. Fundamentally our theory enriches the FB physics; practically, it significantly expands the scope of FB materials, since most materials have multiple atomic/molecular orbitals at each lattice site, rather than a single $s$ orbital mandated in graph theory and generic lattice models.

Received 17 April 2021
Accepted 1 February 2022

DOI:https://doi.org/10.1103/PhysRevB.105.085128

Physics Subject Headings (PhySH)

Topological materials

2-dimensional systems

First-principles calculations

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Hang Liu ^1,2,3, Gurjyot Sethi¹, Sheng Meng^2,3,*, and Feng Liu ^1,†

¹Department of Materials Science and Engineering, University of Utah, Salt Lake City, Utah 84112, USA
²Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, People's Republic of China
³Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, People's Republic of China

^*smeng@iphy.ac.cn
^†fliu@eng.utah.edu

Article Text (Subscription Required)

Click to Expand

Supplemental Material (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 105, Iss. 8 — 15 February 2022

Reuse & Permissions

Access Options

Article Available via CHORUS

Download Accepted Manuscript

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
(a) Illustration of lattice wave-function symmetries, viewed on two sites ( $A$ and $B$ , shaded) in a $1 \times 1$ checkerboard primitive cell (black thin lines), to be transformed into two orbitals on one site (blue dot) in a square lattice: $| s 〉 \sim | A 〉 + | B 〉, | p 〉 \sim | A 〉 - | B 〉$ . (b) Same as (a) but viewed on four sites ( $A_{1}, A_{2}, B_{1}$ , and $B_{2}$ , shaded) in a $\sqrt{2} \times \sqrt{2}$ checkerboard supercell (black thin lines). The two orbitals become $| s 〉 \sim | A_{1} 〉 + | A_{2} 〉 + | B_{1} 〉 + | B_{2} 〉, | d 〉 \sim | A_{1} 〉 + | A_{2} 〉 - | B_{1} 〉 - | B_{2} 〉$ .
Reuse & Permissions
Figure 2
TFB in a square lattice. (a) Γ-point wave functions of a checkerboard lattice (gray lines), exhibiting s- (upper) and $p$ -orbital symmetry (lower). Red and green dots represent, respectively, positive and negative wave-function nodes. (b) Square lattice with ( $s, p$ ) orbitals. (c) Band structure of (b) with hopping integrals in Eq. (2). (d) The CLS of FB in (b) on a square plaquette, illustrating overall zero outward hoppings. Red, green, and gray arrows represent positive, negative, and zero hopping integrals, respectively. (e)–(h) Same as (a)–(d) with ( $s, d_{x y}$ ) orbitals. The bands in (g) can be viewed as folded from (c); the inset of (g) shows the FBZs before (solid line) and after folding (dashed line).
Reuse & Permissions
Figure 3
Dual TFBs in a square lattice. (a) Three Γ-point wave functions of a diamond-octagon lattice, with s- (left), $p_{x}$ - (middle), and $p_{y}$ -orbital symmetry (right), respectively. (b) Square lattice with ( $s$ , $p_{x}, p_{y}$ ) orbitals. (c) Band structure of (b) with NN hopping integrals in Eq. (3). (d) The CLS of lower (left) and upper FB (right) in (c) on a square plaquette. Gray arrows indicate vanishing outward hoppings.
Reuse & Permissions
Figure 4
SOC-induced evolution of band structure and Berry curvature Ω of TFB in ${s p}^{2}$ square lattice with hopping parameters of Eq. (3). The strength of onsite SOC is (a) $λ = 10^{- 6} t$ ; the inset shows a $100 \times magnification$ of the tiny circle around Γ, (b) $λ = 10^{- 4} t$ , (c) $λ = 10^{- 2} t$ , and (d) $λ = 5 \times 10^{- 2} t$ . The calculated Ω is for upper TFB, in units of $a^{2}$ ( $a$ is lattice constant).
Reuse & Permissions
Figure 5
TFB in a trigonal lattice. (a) Three Γ-point wave functions of a kagome lattice, with s- (left), $p_{x}$ - (middle), and $p_{y}$ -orbital symmetry (right), respectively. (b) Trigonal lattice with ( $s, p_{x}, p_{y}$ ) orbitals. (c) Band structure of (b) with NN hopping integrals in Eq. (4). (d)–(f) Same as (a)–(c) with ( $s, d_{x y}, d_{x^{2} - y^{2}}$ ) orbitals. The bands in (f) can be viewed as folded from (c); the inset of (f) shows the FBZs before and after folding.
Reuse & Permissions
Figure 6
Dual TFBs in a hexagonal lattice. (a) Four Γ-point wave functions of a diatomic-kagome lattice, all with $d$ -orbital symmetry. (b) Hexagonal lattice with ( $d_{x y}, d_{x^{2} - y^{2}}$ ) orbitals on site $A$ and $B$ . (c) Band structure of (b) with a NN hopping integral $t_{d d σ} = - \frac{8}{9}$ .
Reuse & Permissions

Physical Review B

covering condensed matter and materials physics