# Majorana Excitons in a Kitaev Chain of Semiconductor Quantum Dots in a Nanowire

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## Abstract

**:**

## 1. Introduction

## 2. Kitaev Chain in a Semiconductor Nanowire

## 3. Majorana and Bond Fermions in Kitaev Hamiltonian

#### 3.1. Exact Diagonalization in Normal Fermion Basis

#### 3.2. Bond Fermions

#### 3.3. Energy Spectrum

## 4. Kitaev Chain and a Light Induced Valence Hole

#### 4.1. Exact Diagonalization of Electron–Hole System

#### 4.2. Energy Spectrum of the Electron–Hole System

#### 4.3. Absorption Spectrum

#### 4.3.1. Analytic Result for Localized Hole

#### 4.3.2. Absorption for Mobile Hole

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Exact Diagonalization for Chain of Length Three

## Appendix B. Analytic Calculation of Absorption Spectrum for Localized Hole

## References

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**Figure 1.**(

**a**) Schematic of the system and the light absorption experiment. (

**b**) Schematic of the Hamiltonian terms between two adjacent dots according to Equation (1), where conduction (valence) levels are labeled by ${c}_{i}\left({h}_{i}\right)$ operators. The conduction level is the reference of energy, hence the downward arrow indicates negative $\mu $. For TEM image and an atomistic description of the quantum dot nanowire system, see Refs. [30,34,36].

**Figure 2.**Schematic of Kitaev chain in the Majorana and bond representation, with non-zero bond Fermions in purple, and the nonlocal zero mode, ${a}_{N}$, living on the two ends of the chain.

**Figure 3.**Energy spectra of Kitaev chain in normal (

**left**) and bond (

**right**) basis, where $\Delta =t<0$ and $\mu =0$. Energy is normalized to $\left|t\right|$.

**Figure 4.**Energy spectra of the full Hamiltonian Equation (1) with one hole in the even subspace, as a function of electron–hole interaction, V, for $N=3$ dots, $\Delta =t$, and $\mu =0$. (

**left**) for the case of localized hole, $\tau =0$, (

**right**) for a mobile hole with $\tau =0.3\left|t\right|$. The overlap of transparent markers makes the degenerate levels look darker. The peak energies, ${E}_{0}$ and ${E}_{\pm}$, discussed in Section 4.3.1 are also shown according to Equation (18a,b).

**Figure 5.**An electron created by ${c}_{i}^{\u2020}$ is a superposition of creation and annihilation operators of two bond Fermions, ${a}_{i}^{(\u2020)}$ and ${a}_{i-1}^{(\u2020)}$, according to Equation (A6). The interaction $-V{n}_{i}^{e}{n}_{i}^{h}$ mixes up the two bond Fermions according to Equation (A8). Note that when i is one of the two ends, then one of the bond Fermions is the zero mode ${a}_{N}$ (see Appendix B for more details).

**Figure 6.**(

**left**) The averaged absorption spectrum, $\overline{A}\left(E\right)$, and (

**middle**and

**right**) spatially resolved absorption, ${\overline{A}}_{i}\left(E\right)$, for $\Delta =t,\mu =0$, and for $N=3$ dots: (

**top**) for a mobile hole with $\tau =0.1\left|t\right|$, (

**bottom**) for a localized hole, $\tau =0$, according to the analytic results in Equations (17) and (18). The spectra are plotted against $(E-\eta )/\left|t\right|$ while changing $V/\left|t\right|$ on the y-axis. The bright curves show the location of the peaks as V changes, and the color scale shows their heights. Gaussian profile was used for the peaks with the width $\sigma =0.025\left|t\right|$. The maximum value of each peak shows the magnitude of the corresponding matrix element.

**Figure 7.**Absorption spectrum for a chain of length $N=9$, and for $\Delta =t,\mu =0,V=10\left|t\right|$, and changing $\tau $. (

**left**) The full averaged spectrum $\overline{A}$, (

**middle**) the spatially resolved spectrum for the first dot ${\overline{A}}_{1}$, and (

**right**) the spatially resolved spectrum for the second dot ${\overline{A}}_{2}$. The bright curves show the location of the peaks as $\tau $ changes, and the colorscale shows their heights. Gaussian profile was used for the peaks with the width $\sigma =0.025\left|t\right|$. The maximum value of each peak shows the magnitude of the corresponding matrix element.

**Table 1.**Describing the spectra plotted in Figure 3. Configurations of bond Fermions are the eigenstates of Kitaev Hamiltonian when $\Delta =t$ and $\mu =0$.

Index | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|

Configuration | $\left|\overline{111}\right.\u232a$ | $\left|\overline{110}\right.\u232a$ | $\left|\overline{010}\right.\u232a$ | $\left|\overline{011}\right.\u232a$ | $\left|\overline{100}\right.\u232a$ | $\left|\overline{101}\right.\u232a$ | $\left|\overline{001}\right.\u232a$ | $\left|\overline{000}\right.\u232a$ |

Label | $\left|\mathrm{GS}\right.\u232a$ | $\left|\overline{\mathrm{GS}}\right.\u232a$ | $\left|{\overline{a}}_{1}\right.\u232a$ | $\left|{a}_{1}\right.\u232a$ | $\left|{\overline{a}}_{2}\right.\u232a$ | $\left|{a}_{2}\right.\u232a$ | $\left|{a}_{1}{a}_{2}\right.\u232a$ | $\left|\overline{{a}_{1}{a}_{2}}\right.\u232a$ |

Parity | odd | even | odd | even | odd | even | odd | even |

Excitation | ||||||||

Energy | 0 | 0 | $2\left|t\right|$ | $2\left|t\right|$ | $2\left|t\right|$ | $2\left|t\right|$ | $4\left|t\right|$ | $4\left|t\right|$ |

Even | Odd | ||||||
---|---|---|---|---|---|---|---|

$\left|\overline{\mathrm{GS}};1\right.\u232a$ | $\left|{a}_{1};1\right.\u232a$ | $\left|{a}_{2};1\right.\u232a$ | $\left|\overline{{a}_{1}{a}_{2}};1\right.\u232a$ | $\left|\mathrm{GS};1\right.\u232a$ | $\left|{\overline{a}}_{1};1\right.\u232a$ | $\left|{\overline{a}}_{2};1\right.\u232a$ | $\left|{a}_{1}{a}_{2};1\right.\u232a$ |

$\left|\overline{\mathrm{GS}};2\right.\u232a$ | $\left|{a}_{1};2\right.\u232a$ | $\left|{a}_{2};2\right.\u232a$ | $\left|\overline{{a}_{1}{a}_{2}};2\right.\u232a$ | $\left|\mathrm{GS};2\right.\u232a$ | $\left|{\overline{a}}_{1};2\right.\u232a$ | $\left|{\overline{a}}_{2};2\right.\u232a$ | $\left|{a}_{1}{a}_{2};2\right.\u232a$ |

$\left|\overline{\mathrm{GS}};3\right.\u232a$ | $\left|{a}_{1};3\right.\u232a$ | $\left|{a}_{2};3\right.\u232a$ | $\left|\overline{{a}_{1}{a}_{2}};3\right.\u232a$ | $\left|\mathrm{GS};3\right.\u232a$ | $\left|{\overline{a}}_{1};3\right.\u232a$ | $\left|{\overline{a}}_{2};3\right.\u232a$ | $\left|{a}_{1}{a}_{2};3\right.\u232a$ |

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## Share and Cite

**MDPI and ACS Style**

Mohseni, M.; Allami, H.; Miravet, D.; Gayowsky, D.J.; Korkusinski, M.; Hawrylak, P.
Majorana Excitons in a Kitaev Chain of Semiconductor Quantum Dots in a Nanowire. *Nanomaterials* **2023**, *13*, 2293.
https://doi.org/10.3390/nano13162293

**AMA Style**

Mohseni M, Allami H, Miravet D, Gayowsky DJ, Korkusinski M, Hawrylak P.
Majorana Excitons in a Kitaev Chain of Semiconductor Quantum Dots in a Nanowire. *Nanomaterials*. 2023; 13(16):2293.
https://doi.org/10.3390/nano13162293

**Chicago/Turabian Style**

Mohseni, Mahan, Hassan Allami, Daniel Miravet, David J. Gayowsky, Marek Korkusinski, and Pawel Hawrylak.
2023. "Majorana Excitons in a Kitaev Chain of Semiconductor Quantum Dots in a Nanowire" *Nanomaterials* 13, no. 16: 2293.
https://doi.org/10.3390/nano13162293