作為具有自旋—軌道耦合的莫特-赫巴德模型系統，三氯化釕作為基塔耶夫材料的候選材料引起了巨大的興趣。實驗表明，三氯化釕在低溫下有序，但在施加有限磁場時會出現無序相。在熱霍爾訊號和拉曼數據中的低能激發行為也表明可能存在馬約拉納邊緣模式。然而，對於實驗的解釋及其實際的底層機制仍然存在許多爭論，因為三氯化釕是范德華材料，其可能存在堆疊層錯，而確切的基塔耶夫自旋液體在可解極限外只有微小的穩定區域。最近一份關於熱霍爾導率的實驗數據暗示，除了磁振子之外，我們還應該考慮聲子的效應。我們首先嘗試建立單層三氯化釕的最小模型，並使用線性自旋波理論（LSWT）來計算不同極限下的磁振子能帶結構。我們計算熱導率並確保它們滿足不同極限下的物理圖像。我們將聲子的貢獻留給未來的工作。本論文的結果有助於闡明微觀物理如何對磁振子熱輸運等宏觀現像做出貢獻，並為凝聚態物質的其他研究，特別是關於具有強自旋軌道耦合的強關聯磁性材料的研究奠定基礎。
i

As a Mott-Hubbard model system with strong spin-orbit coupling, α-RuCl3 is of huge interest as a candidate for the Kitaev material. Experiments show that α-RuCl3 orders at low temperature but a disordered phase occurs when a finite magnetic field is applied. The behaviors of its low energy excitation in thermal Hall signals and Raman data also suggest that there might exist Majorana edge modes. However, there are still lots of debates on the interpretations of the experiments and their actual underlying mechanism since α-RuCl3 is a Van der Waals material with the possible presence of stacking faults and the exact Kitaev spin liquid has only a tiny regime of stability beyond the solvable limit. Recent experimental data of α-RuCl3 on thermal Hall conductivity suggests that, apart from magnon, we should also consider the effect of phonons. We first try building a minimal model of monolayer α-RuCl3 and using linear spin wave theory (LSWT) to calculate the magnon band structures from different limits. We calculate the thermal conductivities and make sure they satisfy the physical pictures in different limits. We leave the contribution from phonons to future works. The results in this thesis help elucidate how microscopic physics contributes to macroscopic phenomena like magnon thermal transport and serve as a general foundation for other research in condensed matter, especially studies about strongly correlated magnetic materials with strong spin-orbit coupling.

Contents
Acknowledgements
摘要 i
Abstract ii
1 Introduction 1
1.1 A Kitaev material candidate: α-RuCl3 . . . . . . . . . . . . . . . . . . 1
1.2 Obstacles in recent experiments . . . . . . . . . . . . . . . . . . . . . 2
1.3 The work in this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 The spin Hamiltonian for the low energy excitations of α-RuCl3 5
2.1 Single-electron states in the tetrahedrons RuCl6 . . . . . . . . . . . . . 6
2.1.1 Solve H0i in the Hilbert space H0,i . . . . . . . . . . . . . . . . 8
2.2 The many-body ground states of H0 . . . . . . . . . . . . . . . . . . . 15
2.3 The hopping matrix H1 resulting from orbital overlaps . . . . . . . . . 16
2.3.1 The second-order effective Hamiltonian . . . . . . . . . . . . . 18
2.3.2 The effective Hamiltonian with energy ± t2
(1)
U . . . . . . . . . . 20
2.4 The mapping of electron operators to local spin operators . . . . . . . . 22
2.5 General spin model for α − RuCl3 . . . . . . . . . . . . . . . . . . . . 24
2.5.1 Inversion symmetry . . . . . . . . . . . . . . . . . . . . . . . 25
2.5.2 C2 rotation symmetry . . . . . . . . . . . . . . . . . . . . . . . 25
3 Magnon bands structures 31
3.1 Holstein-Primakoff transformation and magnon Hamiltonian . . . . . . 31
3.2 Fourier transform and BdG transformation . . . . . . . . . . . . . . . . 34
3.3 Different choices of magnetic order . . . . . . . . . . . . . . . . . . . 35
3.4 Berry curvature of magnon bands . . . . . . . . . . . . . . . . . . . . . 36
3.5 Examples: Models with FM ground states . . . . . . . . . . . . . . . . 37
3.6 Examples: Models with AFM ground states . . . . . . . . . . . . . . . 42
3.6.1 a Neel state on a honeycomb lattice . . . . . . . . . . . . . . . 42
3.6.2 a zigzag state on a honeycomb lattice . . . . . . . . . . . . . . 44
4 Thermal transport signals 53
4.1 General formulation of thermal transport . . . . . . . . . . . . . . . . . 53
4.2 Naive derivation of thermal conductivity . . . . . . . . . . . . . . . . . 54
4.3 thermal Hall transport contributed by magnons . . . . . . . . . . . . . 55
4.3.1 temperature dependence with linear dispersion relation . . . . . 57
4.3.2 temperature dependence with quadratic dispersion relation . . . 58
4.4 Examples: Models with FM ground states . . . . . . . . . . . . . . . . 58
5 Discussion and Future works 61
5.1 Symmetry analysis of the magnon bands . . . . . . . . . . . . . . . . . 61
5.1.1 Hamiltonian under time-reversal symmetry operation . . . . . . 61
5.1.2 rules of finding degeneracy in spin-wave system . . . . . . . . 63
5.1.3 sublattice, time-reversal and particle-hole symmetry . . . . . . 65
5.2 Physical meaning of the thermal transport signals . . . . . . . . . . . . 65
5.3 future works on magnon band analysis . . . . . . . . . . . . . . . . . . 65
5.4 future works on phonon analysis . . . . . . . . . . . . . . . . . . . . . 66
A Appendix 69
A.1 Atomic orbitals and crystal field theory . . . . . . . . . . . . . . . . . 69
A.2 Degenerate perturbation theory . . . . . . . . . . . . . . . . . . . . . . 71
A.3 Fourier transform of tight-binding couplings . . . . . . . . . . . . . . . 77
A.4 Bogoliubov-de-Gennes Transformation . . . . . . . . . . . . . . . . . 81
A.4.1 Example: 2 × 2 matrix . . . . . . . . . . . . . . . . . . . . . . 81
A.4.2 generalized BdG transformaiton . . . . . . . . . . . . . . . . . 82
A.5 Example: Spin wave on a 1D spin chain . . . . . . . . . . . . . . . . . 83
A.5.1 1D FM Heisenberg model . . . . . . . . . . . . . . . . . . . . 83
A.5.2 1D AFM Heisenberg model . . . . . . . . . . . . . . . . . . . 84
A.6 Berry curvature and transport phenomena . . . . . . . . . . . . . . . . 87
A.6.1 gauge transformation and Berry curvature . . . . . . . . . . . . 88
A.6.2 the bands for semi-infinite honeycomb lattices with zigzag bound-
aries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
references 91

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