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Experimental realization of artificial gauge field has made it possible to simulate important models with electromagnetic field or spin-orbit interaction in condensed matter physics, which opens a new avenue to engineer novel quantum states and phenomena. The spin-orbit coupled system reveals many significant phenomena in condensed matter physics, such as quantum spin Hall effect, topological insulator and topological superconductor. The combined effect of Zeeman interaction and spin-orbit coupling leads to a nontrivial topological phase. The analytic solution of few-body system provides an in-depth insight into the physical phenomena, which has been studied extensively. Through the analytic study of two-body physics, we show new quantum phenomena for various gauge field parameters. We investigate the two-body interacting fermionic gas with spin-orbit coupling and Zeeman interaction in a ring trap. Through the plane wave expansion method, two-body fermionic system is solved analytically. In the absence of Zeeman interaction, the total momentum of the ground state is zero. With the increase of Zeeman interaction, an energy level crossing occurs between the lowest energy levels for different total momentum spaces and the ground state changes from zero total momentum space to non-zero total momentum space. Considering the Zeeman interaction, the total momentum of the ground state changes from zero to finite value. The single particle analysis shows that the ground energy level transition is induced by Zeeman energy level splitting. The momentum distributions of the ground state are given to provide an intuitive physical picture. This work can be further extended to the exploration of the heteroatom system, lattice system and higher spin system.
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Keywords:
- two-body Fermionic system /
- spin-orbit coupling /
- Zeeman interaction
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Google Scholar
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Google Scholar
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图 1 最低能量
$ {E_{\rm{0}}}$ 随着自旋轨道耦合参量$ \alpha$ 的变化. 插图为$ {E_0} + {\alpha ^2}$ 随$ \alpha$ 的变化,$ n = 0$ ,$ g = 1$ ,$ \beta = 0$ Figure 1. The lowest energy
${E_{\rm{0}}}$ versus spin-orbit coupling parameter$\alpha $ . Insert is${E_0} + {\alpha ^2}$ versus$\alpha $ .$n = 0$ ,$g = 1$ ,$\beta = 0$ .
图 2 在弱相互作用条件下最低能量
$ {E_{\rm{0}}}$ 和$ {E_1}$ 随塞曼系数$ \beta $ 的变化,$ g = 1$ ,$ \alpha = 1.2{\text{π}}$ Figure 2. The lowest energies
${E_{\rm{0}}}$ and${E_1}$ versus Zeeman interaction parameter$\beta $ in the weak interaction condition.$g = 1$ ,$\alpha = 1.2{\text{π}}$ .
图 3 能量
${E_0}$ 和${E_1}$ 随着接触相互作用g的变化,$\alpha = 1.2{\text{π}}$ (a)$\beta = 0$ ; (b)$\beta = {\text{π}}$ ; (c)$\beta = 2{\text{π}}$ Figure 3. The energies
${E_{\rm{0}}}$ and${E_1}$ versus contact interaction parameter g,$\alpha = 1.2{\text{π}}$ : (a)$\beta = 0$ ; (b)$\beta = {\text{π}}$ ; (c)$\beta = 2{\text{π}}$ .
图 4 单粒子能级e–和e+ (a)
$ \alpha = 0.2{\text{π}}$ ,$ \beta = 0.1{\text{π}}$ ; (b)$ \alpha = 1.2{\text{π}}$ ,$ \beta = {\text{π}}$ ; (c)$ \alpha = 1.2{\text{π}}$ ,$ \beta = 2{\text{π}}$ Figure 4. The single particle eigenenergies with two branches e– and e+: (a)
$\alpha = 0.2{\text{π}}$ ,$\beta = 0.1{\text{π}}$ ; (b)$\alpha = 1.2{\text{π}}$ ,$\beta = {\text{π}}$ ; (c)$\alpha = 1.2{\text{π}}$ ,$\beta = 2{\text{π}}$ .
图 5 对于不同的
$ \alpha $ 和$ \beta $ , 基态的动量分布,$ g = 1$ Figure 5. The momentum distributions of ground states for different
$\alpha $ and$\beta $ ,$g = 1$ . -
[1] Zhai H 2015 Rep. Prog. Phys. 78 026001
Google Scholar
[2] Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045
Google Scholar
[3] 施婷婷, 汪六九, 王璟琨, 张威 2020 物理学报 69 016701
Google Scholar
Shi T T, Wang L J, Wang J K, Zhang W 2020 Acta Phys. Sin. 69 016701
Google Scholar
[4] Lin Y J, Garcis K J, Spielman I B 2011 Nature 83 471
[5] Wang P J, Yu Z Q, Fu Z K, Miao J, Huang L H, Chai S J, Zhai H, Zhang J 2012 Phys. Rev. Lett. 109 095301
Google Scholar
[6] Cheuk L W, Sommer A T, Hadzibabic Z, Yefsah T, Bakr W S, Zwierlein M W 2012 Phys. Rev. Lett. 109 095302
Google Scholar
[7] Huang L H, Meng Z M, Wang P J, Peng P, Zhang S L, Chen L C, Li D H, Zhou Q, Zhang J 2016 Nat. Phys. 12 540
Google Scholar
[8] Wu Z, Zhang L, Sun W, Xu X T, Wang B Z, Ji S C, Deng Y, Chen S, Liu X J, Pan J W 2016 Sicence 354 83
Google Scholar
[9] Zhou J, Zhang W, Yi W 2011 Phys. Rev. A 84 063603
Google Scholar
[10] Chen J, Hu H, Gao X L 2014 Phys. Rev. A 90 023619
Google Scholar
[11] Meng Z, Huang L, Peng P, Li D, Chen L, Xu Y, Zhang C, Wang P, Zhang J 2016 Phys. Rev. Lett. 117 235304
Google Scholar
[12] Dong L, Jiang L, Pu H 2013 New J. Phys. 15 075014
Google Scholar
[13] Chen C 2013 Phys. Rev. Lett. 111 235302
Google Scholar
[14] Qu C L, Zheng Z, Gong M, Xu Y, Mao L, Zou X B, Guo G C, Zhang C W 2013 Nat. Commun. 4 2710
Google Scholar
[15] Zhang W, Yi W 2013 Nat. Commun. 4 2711
Google Scholar
[16] Valdés-Curiel A, Trypogeorgos D, Liang Q Y, Anderson R P, Spielman I B arXiv: 1907.08637
[17] Liu X J, Hu H, Pu H 2015 Chin. Phys. B 24 050502
Google Scholar
[18] Cao Y, Liu X J, He L Y, Long G L, Hu H 2015 Phys. Rev. A 91 023609
Google Scholar
[19] Devreese J P A, Tempere J, Sá de Melo C A R 2014 Phys. Rev. Lett. 113 165304
Google Scholar
[20] Luo X B, Zhou K Z, Liu W M, Liang Z X, Zhang Z D 2014 Phys. Rev. A 89 043612
Google Scholar
[21] Xu Y, Zhang C W 2015 Phys. Rev. Lett. 114 110401
Google Scholar
[22] Zhou K Z, Zhang Z D 2019 J. Phys. Chem. Solids 128 207
Google Scholar
[23] Yang S, Wu F, Yi W, Zhang P 2019 Phys. Rev. A 100 043601
Google Scholar
[24] Yu Z Q, Zhai H 2011 Phys. Rev. Lett. 107 195305
Google Scholar
[25] Vyasanakere J P, Shenoy V B 2012 New J. Phys. 14 043041
Google Scholar
[26] Usui A, Fogarty T, Campbell S, Gardiner S A, Busch T 2020 New J. Phys. 22 013050
Google Scholar
[27] Li Q M, Callaway J 1991 Phys. Rev. B 43 3278
Google Scholar
[28] Cui X L, Yi W 2014 Phys. Rev. X 4 031026
[29] Wang J K, Yi W, Zhang W 2016 Front. Phys. 11 118102
Google Scholar
[30] Peng S G, Zhang C X, Tan S, Jiang K J 2018 Phys. Rev. Lett. 120 060408
Google Scholar
[31] Cui X L 2017 Phys. Rev. A 95 030701
Google Scholar
[32] Gong B H, Li S, Zhang X H 2019 Phys. Rev. A 99 012703
Google Scholar
[33] Chen X, Guan L M, Chen S 2011 Eur. Phys. J. D 64 459
Google Scholar
[34] Song B, He C D, Zhang S C, Hajiyev E, Huang W, Liu X J, Jo G B 2016 Phys. Rev. A 94 061604
Google Scholar
[35] Olshanii M 1998 Phys. Rev. Lett. 81 938
Google Scholar
[36] Busch T, Englert B G, Rzazewski K, Wilkens M 2001 J. Phys. B 34 4571
Google Scholar
[37] Chen X, Hu H P, Jiang Y Z, Chen S 2013 Eur. Phys. J. D 67 166
Google Scholar
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