馬約拉那零能量模態 (Majorana zero mode) 的編織 (braiding) 特徵在實現拓樸量子計算中有著重要的作用。在這篇碩士論文當中，我們將以數值方式模擬兩個在一維無自旋的P-波超導線中的馬約拉那零能量模態交換的整個過程。兩種用相位扭折 (kink) 來交換馬約拉那零能量模態的程序將會被檢驗。

當兩個馬約拉那零能量模態交換時，特定的幾何相位將作為特徵出現。這個相位會呈現為在具備 $
m \epsilon$ 混成能量由兩個零能量準粒子態形成的兩個混成態中的 $
m
i/2$ 相位或偶宇稱與奇宇稱多粒子基態之間的 $
m
i/2$ 相位差。利用波戈留波夫-德堅內 (Bogoliubov-de Gennes) 表述以及基於喬登-維格納 (Jordan-Wigner) 轉換的多體表示法，我們分別得到準粒子激發態以及宇稱本徵態。借助緩慢時間演化的分段近似，可以在完整的哈米頓量 (Hamiltonian) 下演化在編織程序執行當中被追蹤的狀態。

在第一個程序中，相位扭折緩慢的從左邊的馬約拉那零能量模態移動到右邊的馬約拉那零能量模態。我們發現幾何相位在兩種表示法中都為零且在不同的演化時間下是不變的。這證明用於交換兩個馬約拉那零能量模態的扭折程序是失敗的。另一方面，在第二個程序中相位扭折被固定在中心且一半的超導線上的相位被緩慢地從零轉到 $2
i$。我們發現計算出的幾何相位確實是 $
m
i/2$ ，證實了預期的特徵。

The braiding signature of Majorana zero modes plays an essential role in realizing topological quantum computation. In this thesis, we will numerically simulate the whole process of exchanging two Majorana zero modes supported in a one-dimensional spinless p-wave superconducting wire. Two protocols by using phase kinks for exchanging Majorana zero modes will be examined.

When two Majorana zero modes exchange, particular geometric phases will arise as the signature. These phases will exhibit as $
m
i/2$ in the two hybridized states formed by two zero-energy quasi-particle states with hybridization energy $
m \epsilon$ or $
m
i/2$ phase difference between the even and odd parity degenerate many-particle ground states. Using the Bogoliubov-de Gennes formulation and the many-body representation based on the Jordan-Wigner transformation, we obtain the quasi-particle excitation states and the parity eigenstates separately. With the help of the piecewise approximation of adiabatic time evolution, the states that are tracked during the execution of braiding protocols can be evolved under the complete Hamiltonian.

In the first protocol, the phase kink moves adiabatically from the left Majorana zero mode to the right Majorana zero mode. We find that the geometric phase is zero and is invariant under different evolution times in both representations. It proves the failure of the kink protocol for exchanging two Majorana zero modes. On the other hand, in the second protocol, the phase kink is fixed at the center, and the phase for one-half of the superconducting wire is rotated adiabatically from zero to $2
i$. We find that the computed geometric phases are $
m
i/2$ indeed, confirming the expected signature.

摘要 .................................................................................. I
Abstract ............................................................................. II
Acknowledgements .................................................................... III
List of Figures ...................................................................... VI

1 Introduction ........................................................................ 1

2 Theory .............................................................................. 5
2.1 Majorana Operators ................................................................ 5
2.2 Kitaev Toy Model (One-Dimensional Spinless P-Wave Superconducting Wire) ........... 8
2.3 Majorana zero modes - Quasi-particle State ....................................... 15
2.4 Braiding of Majorana Zero Modes .................................................. 17
2.5 Effective Hamiltonian of Majorana Operators ...................................... 21

3 Methods ............................................................................ 23
3.1 Bogoliubov-de Gennes ............................................................. 24
3.2 Time-Dependent Bogoliubov-De Gennes .............................................. 27
3.3 Many-Body Representation ......................................................... 28
3.4 Adiabatic Evolution .............................................................. 31
3.5 Berry Phase and Geometric Phase .................................................. 34

4 Results ............................................................................ 36
4.1 Kink Protocol .................................................................... 36
4.2 Pairing Phase Rotation Protocol .................................................. 46

5 Conclusion ......................................................................... 56

References ........................................................................... 57

[1] A. Y. Kitaev, “Unpaired Majorana fermions in quantum wires”, Physics–Uspekhi 44, 131 (2001).

[2] S. Ran, C. Eckberg, Q.-P. Ding, Y. Furukawa, T. Metz, S. R. Saha, I.-L. Liu, M. Zic, H. Kim, J. Paglione, and N. P. Butch, “Nearly ferromagnetic spin-triplet superconductivity”, Science 365, 684–687 (2019).

[3] L. Jiao, S. Howard, S. Ran, Z. Wang, J. O. Rodriguez, M. Sigrist, Z. Wang, N. P. Butch, and V. Madhavan, “Chiral superconductivity in heavy-fermion metal UTe2”, Nature 579, 523–527 (2020).

[4] M. Kayyalha, D. Xiao, R. Zhang, J. Shin, J. Jiang, F. Wang, Y.-F. Zhao, R. Xiao, L. Zhang, K. M. Fijalkowski, P. Mandal, M. Winnerlein, C. Gould, Q. Li, L. W. Molenkamp, M. H. W. Chan, N. Samarth, and C.-Z. Chang, “Absence of evidence for chiral Majorana modes in quantum anomalous Hallsuperconductor devices”, Science 367, 64–67 (2020).

[5] Q. L. He, L. Pan, A. L. Stern, E. C. Burks, X. Che, G. Yin, J. Wang, B. Lian, Q. Zhou, E. S. Choi, K. Murata, X. Kou, Z. Chen, T. Nie, Q. Shao, Y. Fan, S.-C. Zhang, K. Liu, J. Xia, and K. L. Wang, “Chiral Majorana fermion modes in a quantum anomalous Hall insulator–superconductor structure”, Science 357, 294–299 (2017).

[6] N. Read and D. Green, “Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect”, Physical Review B 61, 10267–10297 (2000).

[7] D. Sherman, J. S. Yodh, S. M. Albrecht, J. Nygård, P. Krogstrup, and C. M. Marcus, “Normal, superconducting and topological regimes of hybrid double quantum dots”, Nature Nanotechnology 12, 212–217 (2017).

[8] D. M. T. v. Zanten, D. Sabonis, J. Suter, J. I. Väyrynen, T. Karzig, D. I. Pikulin, E. C. T. O＇Farrell, D. Razmadze, K. D. Petersson, P. Krogstrup, and C. M. Marcus, “Photon-assisted tunnelling of zero modes in a Majorana wire”, Nature Physics 16, 663–668 (2020).

[9] M.-T. Deng, S. Vaitiekėnas, E. Prada, P. San-Jose, J. Nygård, P. Krogstrup, R. Aguado, and C. M. Marcus, “Nonlocality of Majorana modes in hybrid nanowires”, Physical Review B 98, 085125 (2018).

[10] D. I. Pikulin, B. van Heck, T. Karzig, E. A. Martinez, B. Nijholt, T. Laeven, G. W. Winkler, J. D. Watson, S. Heedt, M. Temurhan, V. Svidenko, R. M. Lutchyn, M. Thomas, G. de Lange, L. Casparis, and C. Nayak, Protocol to identify a topological superconducting phase in a three-terminal device, 2021.

[11] D. Aasen, M. Hell, R. V. Mishmash, A. Higginbotham, J. Danon, M. Leijnse, T. S. Jespersen, J. A. Folk, C. M. Marcus, K. Flensberg, and J. Alicea, “Milestones Toward Majorana-Based Quantum Computing”, Physical Review X 6, 031016 (2016).

[12] J. Alicea, Y. Oreg, G. Refael, F. v. Oppen, and M. P. A. Fisher, “Non-Abelian statistics and topological quantum information processing in 1D wire networks”, Nature Physics 7, 412–417 (2011).

[13] Q.-B. Cheng, J. He, and S.-P. Kou, “Verifying non-Abelian statistics by numerical braiding Majorana fermions”, Physics Letters A 380, 779–782 (2016).

[14] A. Więckowski, M. Mierzejewski, and M. Kupczyński, “Majorana phase gate based on the geometric phase”, Physical Review B 101, 014504 (2020).

[15] M. Sekania, S. Plugge, M. Greiter, R. Thomale, and P. Schmitteckert, “Braiding errors in interacting Majorana quantum wires”, Physical Review B 96, 094307 (2017).

[16] X.-M. Zhao, J. Yu, J. He, Q.-B. Cheng, Y. Liang, and S.-P. Kou, “The simulation of non-Abelian statistics of Majorana fermions in Ising chain with Z2 symmetry”, Modern Physics Letters B 31, 1750123 (2017).

[17] C. S. Amorim, K. Ebihara, A. Yamakage, Y. Tanaka, and M. Sato, “Majorana braiding dynamics in nanowires”, Physical Review B 91, 174305 (2015).

[18] T. Sanno, S. Miyazaki, T. Mizushima, and S. Fujimoto, “Ab initio simulation of non-Abelian braiding statistics in topological superconductors”, Physical Review B 103, BdG and zero mode., 054504 (2021).

[19] C.-K. Chiu, M. M. Vazifeh, and M. Franz, “Majorana fermion exchange in strictly one-dimensional structures”, EPL (Europhysics Letters) 110, 10001 (2015).

[20] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, “Classification of topological quantum matter with symmetries”, Rev. Mod. Phys. 88, 035005 (2016).

[21] V. Kremenetski, C. Mejuto-Zaera, S. J. Cotton, and N. M. Tubman, “Simulation of adiabatic quantum computing for molecular ground states”, The Journal of Chemical Physics 155, 234106 (2021).

[22] S. R. Manmana, A. Muramatsu, and R. M. Noack, “Time evolution of onedimensional Quantum Many Body Systems”, AIP Conference Proceedings 789, 269–278 (2005).

[23] N. Mukunda and R. Simon, “Quantum Kinematic Approach to the Geometric Phase. I. General Formalism”, Annals of Physics 228, 205–268 (1993).

[24] J.-J. Miao, H.-K. Jin, F.-C. Zhang, and Y. Zhou, “Majorana zero modes and long range edge correlation in interacting Kitaev chains: analytic solutions and density-matrix-renormalization-group study”, Scientific Reports 8, 488 (2018).