我們研究了具有兩種自旋軌道耦合（Dreselhaus 耦合和Rashba 耦合）的
三維反鐵磁絕緣體的拓撲相。在前人的工作中，具有自旋軌道相互作用的
反鐵磁絕緣體的拓撲性質在假定的Neel order 的情況下來分類的。然而，自
旋軌道的存在可能會改變Neel order 的自旋模式。我們自洽地計算磁序。在
Dreselhaus 自旋軌道耦合的情況下，自旋的排列是反鐵磁性的，並且自旋
可以指向任何方向。在Rashba 相互作用的情況下，自旋反鐵磁位於xy 平
面上。此外，在小的最近臨跳躍的情況下，自旋是傾斜的，自旋在xy 平面
上是反鐵磁性的，但在z 方向上是鐵磁性的。我們對這兩個磁序的相的拓
撲性質進行了分類。對於純反鐵磁序，系統仍舊保有修正的時間反演對稱
性，使系統仍以Z2 指標來刻劃拓撲相; 而對於傾斜反鐵磁序，系統則是保
有滑動反射對稱和點反演對稱，可以用Z4 指標來描述。此外，我們檢查了
bulk-edge
correspondence，當表面與塊材內部具有相同的對稱性時，表面狀
態是無間隙的；否則，表面狀態是有間隙的。我們的結果對具有自旋軌道
耦合的反鐵磁絕緣體提供了完整的拓撲描述。

We investigate topological phases of a 3D anferromagnetic insulator with two
kinds of spinorbit
coupling, the Dresselhaus coupling and the Rashba coupling.
In the previous work, the topological property of antiferromagnetic insulators with
spinorbit
interaction is classified by preassumed Neel order. The presence of spinorbit,
however, may change the pattern of spins in Neel order. We compute the
magnetic order selfconsistently.
In the case of Dresselhaus spinorbit
coupling,
the arrangement of spin is antiferromagnetic with the spin pointing to any direction.
In the case of Rashba interaction, spins antiferromagneticlly lie on the xy plane.
Furthermore, in the presence of small interaction controled by hopping parameter,
spins are tilted so that spins are antiferromagnetic in xy plane but are ferromagnetic
in z direction. The topological properties of these two phases are classified.
For pure antiferromagentic order, the system has modified timereversal
symmetry
so that the system is still characterized by Z2, while for tilted antiferromagnetic
orders, the system is characterized by a glidemirror
symmetry and inversion symmetry
and can be described by a Z4 index. Furthermore, we check the bulkedge
correspondence such that when the surface has the same symmetry as that for the
bulk state, the surface state is gapless; otherwise, the surface state is gapped. Our
results provide a complete topological characterization for anferromagnetic insulators
with spinorbit
couplings.

Contents
Acknowledgements
摘要i
Abstract ii
1 Introduction 1
2 Previous research:Antiferromagnetic topological insulator 3
3 The spin arrangement of 3D cubic lattice with spinorbit
interaction 9
3.1 Dresselhaus spinorbit
interaction . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1.1 Canonical Transformation . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1.2 Mean Field Theory in Spin Model . . . . . . . . . . . . . . . . . . . . 13
3.2 Rashba spinorbit
interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 subconclusion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4 The Topological Property of 3D Cubic Lattice with Antiferromagnetization and Ferromagnetization
27
4.1 Introduction to Time Reversal Symmetry and Z2 Index . . . . . . . . . . . . . 27
4.2 Introduction to Inversion Symmetry and Z4 Index . . . . . . . . . . . . . . . . 31
4.2.1 Chern Simons Term and Inversion Symmetry . . . . . . . . . . . . . . 31
4.2.2 Z4 Index and Bulkhinge
Correspondence . . . . . . . . . . . . . . . . 31
4.3 Topological Property of Antiferromagnetic Insulator . . . . . . . . . . . . . . 34
4.3.1 Z2 Index of Antiferromagnetic 3D Topological Insulator . . . . . . . . 34
4.3.2 Antiferromagnetic Topological Insulator and Z4 Index With Inversion
Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.4 Subconclusion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5 Conclusion 51
A Canonical Transformation 53
A.1 Finding Generator S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
A.2 Transforming to Spin Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
B Inversion Symmetry and Glide Symmetry 63
B.1 Ih(k)I−1 = h(−k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
B.2 Gz(kx)HGz(kx)−1 = H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
References 77

[1] J.T.
Kao, “Note of canonical transformation for strong coupling hubbard model,”
[2] X.L.
Qi, T. L. Hughes, and S.C.
Zhang, “Topological field theory of timereversal
invariant
insulators,” Phys. Rev. B, vol. 78, p. 195424, Nov 2008.
[3] R. S. K. Mong, A. M. Essin, and J. E. Moore, “Antiferromagnetic topological insulators,”
Phys. Rev. B, vol. 81, p. 245209, Jun 2010.
[4] A. Sekine and K. Nomura, “Axion electrodynamics in topological materials,” Journal of
Applied Physics, vol. 129, p. 141101, Apr 2021.
[5] A. M. Turner, Y. Zhang, R. S. K. Mong, and A. Vishwanath, “Quantized response
and topology of magnetic insulators with inversion symmetry,” Phys. Rev. B, vol. 85,
p. 165120, Apr 2012.
[6] S. Ono and H. Watanabe, “Unified understanding of symmetry indicators for all internal
symmetry classes,” Phys. Rev. B, vol. 98, p. 115150, Sep 2018.
[7] M. Mogi, M. Kawamura, R. Yoshimi, A. Tsukazaki, Y. Kozuka, N. Shirakawa, K. Takahashi,
M. Kawasaki, and Y. Tokura, “A magnetic heterostructure of topological insulators
as a candidate for an axion insulator,” Nature materials, vol. 16, no. 5, pp. 516–521, 2017.
[8] D. Zhang, M. Shi, T. Zhu, D. Xing, H. Zhang, and J. Wang, “Topological axion states
in the magnetic insulator mnbi2te4 with the quantized magnetoelectric effect,” Phys. Rev.
Lett., vol. 122, p. 206401, May 2019.
[9] Y. Xu, Z. Song, Z. Wang, H. Weng, and X. Dai, “Higherorder
topology of the axion
insulator euin2as2,” Phys. Rev. Lett., vol. 122, p. 256402, Jun 2019.
[10] J. Li, Y. Li, S. Du, Z. Wang, B.L.
Gu, S.C.
Zhang, K. He, W. Duan, and Y. Xu, “Intrinsic
magnetic topological insulators in van der waals layered mnbi2te4family
materials,”
Science Advances, vol. 5, no. 6, p. eaaw5685, 2019.
[11] B. A. Bernevig, “Topological insulators and topological superconductors,” Princeton university
press, 2013.
[12] H. Kim, K. Shiozaki, and S. Murakami, “Glidesymmetric
magnetic topological crystalline
insulators with inversion symmetry,” Phys. Rev. B, vol. 100, p. 165202, Oct 2019.
[13] P. Hosur, S. Ryu, and A. Vishwanath, “Chiral topological insulators, superconductors, and
other competing orders in three dimensions,” Physical Review B, vol. 81, no. 4, p. 045120,
2010.
[14] C. L. Kane and E. J. Mele, “Z 2 topological order and the quantum spin hall effect,”
Physical review letters, vol. 95, no. 14, p. 146802, 2005.
[15] L. Fu and C. L. Kane, “Time reversal polarization and a Z2 adiabatic spin pump,” Phys.
Rev. B, vol. 74, p. 195312, Nov 2006.
[16] L. Fu, C. L. Kane, and E. J. Mele, “Topological insulators in three dimensions,” Physical
review letters, vol. 98, no. 10, p. 106803, 2007.
[17] L. Fu and C. L. Kane, “Topological insulators with inversion symmetry,” Physical Review
B, vol. 76, no. 4, p. 045302, 2007.
[18] S.Q.
SHEN, “Topological insulators: Dirac equation in condensed matter,” Springer,
2018.
[19] M. Chang, “Lecture notes on topological insulators,” 2017.
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