# Topological Properties in a Λ/V-Type Dice Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model and Hamiltonian

## 3. Results and Discussion

#### 3.1. Band Structures

#### 3.2. Chern Numbers and the Edge-State Spectrum

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Geometry of the dice lattice. R, B, and G are sublattice sites, and are marked with red, blue, and green dots respectively. The lattice constant a is taken as $a=1$. The vectors ${a}_{n}$ ($n=1,2,3$) link the nearest-neighbor sites that pertain to different sublattices. The vectors ${b}_{n}$ ($n=1,2,3$) link the nearest-neighbor R sites or B sites. The circle arrows represent the next-nearest-neighbor tunnelings accompanied by a phase ${e}^{i\varphi}$. (

**b**) The first Brillouin zone of the model. $\Gamma $, K, and M are high-symmetry points in the high-symmetry path, and are connected by three red dashed lines.

**Figure 2.**The metal-insulator phase diagrams in the cases of $1/3$ filling (

**a**) and $2/3$ filling (

**b**). M refers to the metal phase and I refers to the bulk insulating phase. In each case, there are four chosen typical points, a, b, c, and d, which are marked by green dots and are discussed in the main text.

**Figure 3.**Dispersions of the $\Lambda /V$-type dice model along the high-symmetry path $\Gamma $-$\mathrm{K}$-$\mathrm{M}$-$\Gamma $. (

**a**) ${\Delta}_{a}=-2$, ${\varphi}_{a}=-2.5$, (

**b**) ${\Delta}_{b}=2$, ${\varphi}_{b}=0.5$, (

**c**) ${\Delta}_{c}=-0.5$, ${\varphi}_{c}=0.1$, (

**d**) ${\Delta}_{d}=-2$, ${\varphi}_{d}=1.5$. The red, blue, and green solid lines show the dispersions. The lower Fermi energy (black dashed line) and higher Fermi energy (black solid line) show the cases of $1/3$ and $2/3$ filling, respectively.

**Figure 4.**Two full phase diagrams with (

**a**) $1/3$ filling and (

**b**) $2/3$ filling, respectively. The blue solid lines and axes surround the metallic region (M), and the bulk insulating region consists of several topological regions. The black solid lines separate the topological nontrivial phase from the topological trivial phase, and the red solid lines distinguish the topological nontrivial phase with various ${C}_{\frac{1}{3}}$ and ${C}_{\frac{2}{3}}$.

**Figure 5.**Two edge-state spectra of a cylindrical geometry with a zigzag edge. (

**a**) $\Delta =-2$, $\varphi =1.5$. There is a pair of edge modes in the case of $1/3$ filling, corresponding to ${C}_{\frac{1}{3}}=-1$, while there are no edge modes in the case of $2/3$ filling, which corresponds to ${C}_{\frac{2}{3}}=0$; (

**b**) $\Delta =1$, $\varphi =-1.3$. There are two pairs of edge modes in the case of $1/3$ filling, which corresponds to ${C}_{\frac{1}{3}}=2$, and there is only a pair of edge modes in the case of $2/3$ filling, which corresponds to ${C}_{\frac{2}{3}}=-1$.

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**MDPI and ACS Style**

Cheng, S.; Gao, X.
Topological Properties in a Λ/*V*-Type Dice Model. *Crystals* **2021**, *11*, 467.
https://doi.org/10.3390/cryst11050467

**AMA Style**

Cheng S, Gao X.
Topological Properties in a Λ/*V*-Type Dice Model. *Crystals*. 2021; 11(5):467.
https://doi.org/10.3390/cryst11050467

**Chicago/Turabian Style**

Cheng, Shujie, and Xianlong Gao.
2021. "Topological Properties in a Λ/*V*-Type Dice Model" *Crystals* 11, no. 5: 467.
https://doi.org/10.3390/cryst11050467