Three-dimensional spin–orbit coupled Fermi gases: Fulde–Ferrell pairing, Majorana fermions, Weyl fermions, and gapless topological superfluidity

doi:10.1088/1674-1056/24/5/050502

Abstract We theoretically investigate a three-dimensional Fermi gas with Rashba spin–orbit coupling in the presence of both out-of-plane and in-plane Zeeman fields. We show that, driven by a sufficiently large Zeeman field, either out-of-plane or in-plane, the superfluid phase of this system exhibits a number of interesting features, including inhomogeneous Fulde–Ferrell pairing, gapped or gapless topological order, and exotic quasi-particle excitations known as Weyl fermions that have linear energy dispersions in momentum space (i.e., massless Dirac fermions). The topological superfluid phase can have either four or two topologically protected Weyl nodes. We present the phase diagrams at both zero and finite temperatures and discuss the possibility of their observation in an atomic Fermi gas with synthetic spin–orbit coupling. In this context, topological superfluid phase with an imperfect Rashba spin–orbit coupling is also studied.

Keywords: Fulde– Ferrell superfluid topological superfluid spin– orbit coupling

Received: 14 November 2014
Revised: 02 December 2014
Accepted manuscript online:

PACS:	05.30.Fk	(Fermion systems and electron gas)
	03.75.Hh	(Static properties of condensates; thermodynamical, statistical, and structural properties)
	03.75.Ss	(Degenerate Fermi gases)
	67.85.-d	(Ultracold gases, trapped gases)

Fund: Project supported by the ARC Discovery Projects (Grant Nos. FT140100003, FT130100815, DP140103231, and DP140100637), the National Basic Research Program of China (Grant No. 2011CB921502), the US National Science Foundation, and the Welch Foundation (Grant No. C-1669).

Corresponding Authors: Han Pu
E-mail: hpu@rice.edu

About author: 05.30.Fk; 03.75.Hh; 03.75.Ss; 67.85.-d

Cite this article:

Xia-Ji Liu, Hui Hu, Han Pu Three-dimensional spin–orbit coupled Fermi gases: Fulde–Ferrell pairing, Majorana fermions, Weyl fermions, and gapless topological superfluidity 2015 Chin. Phys. B 24 050502

[1]	Bloch I, Dalibard J and Nascimbéne S 2012 Nat. Phys. 8 267
[2]	Chin C, Grimm R, Julienne P and Tiesinga E 2010 Rev. Mod. Phys. 82 1225
[3]	Lin Y J, Jiménez-Garciia K and Spielman I B 2011 Nature 471 83
[4]	Hasan M Z and Kane C L 2010 Rev. Mod. Phys. 82 3045
[5]	Qi X L and Zhang S C 2011 Rev. Mod. Phys. 83 1057
[6]	Xiao D, Chang M C and Niu Q 2010 Rev. Mod. Phys. 82 1959
[7]	Yip S K 2014 Annu. Rev. Condens. Matter Phys. 5 15
[8]	Zhang J, Liu X J, Hu H and Pu H 2014 Annual Review of Cold Atoms and Molecules Vol. 2 (Singapore: World Scientific Publishing) Chap. 2
[9]	Zhai H 2015 Rep. Prog. Phys. 78 026001
[10]	Xu Y and Zhang C 2015 Int. J. Mod. Phys. B 29 1530001
[11]	Vyasanakere j P, Zhang S and Shenoy V B 2011 Phys. Rev. B 84 014512
[12]	Zhu S L, Shao L B, Wang Z D and Duan L M 2011 Phys. Rev. Lett. 106 100404
[13]	Hu H, Jiang L, Liu X J and Pu H 2011 Phys. Rev. Lett. 107 195304
[14]	Yu Z Q and Zhai H 2011 Phys. Rev. Lett. 107 195305
[15]	Gong M, Tewari S and Zhang C 2011 Phys. Rev. Lett. 107 195303
[16]	Jiang L, Liu X J, Hu H and Pu H 2011 Phys. Rev. A 84 063618
[17]	Liu X J, Jiang L, Pu H and Hu H 2012 Phys. Rev. A 85 021603
[18]	Zhou K and Zhang Z 2012 Phys. Rev. Lett. 108 025301
[19]	He L and Huang X G 2012 Phys. Rev. Lett. 108 145302
[20]	Anderson B M, Juzeliūnas G, Galitski V M and Spielman I B 2012 , Phys. Rev. Lett. 108 235301
[21]	Gong M, Chen G, Jia S and Zhang C 2012 Phys. Rev. Lett. 109 105302
[22]	Seo K, Han L and Sá de Melo C A R 2012 Phys. Rev. Lett. 109 105303
[23]	Liu X J and Hu H 2012 Phys. Rev. A 85 033622
[24]	Wei R and Mueller E J 2012 Phys. Rev. A 86 063604
[25]	Hu H, Jiang L, Pu H, Chen Y and Liu X J 2013 Phys. Rev. Lett. 110 020401
[26]	He L and Huang X G 2013 Ann. Phys. 337 163
[27]	Zheng Z, Gong M, Zou X, Zhang C and Guo G C 2013 Phys. Rev. A 87 031602
[28]	Dong L, Jiang L, Hu H and Pu H 2013 Phys. Rev. A 87 043616
[29]	Wu F, Guo G C, Zhang W and Yi W 2013 Phys. Rev. Lett. 110 110401
[30]	Liu X J and Hu H 2013 Phys. Rev. A 87 051608
[31]	Dong L, Jiang L and Pu H 2013 New J. Phys. 15 075014
[32]	Zhou X F, Guo G C, Zhang W and Yi W 2013 Phys. Rev. A 87 063606
[33]	Hu H and Liu X J 2013 New J. Phys. 15 093037
[34]	Seo K, Zhang C and Tewari S 2013 Phys. Rev. A 87 063618
[35]	Iskin M 2013 Phys. Rev. A 88 013631
[36]	Xu Z F, You L and Ueda M 2013 Phys. Rev. A 87 063634
[37]	Anderson B M, Spielman I B and Juzeliūnas G 2013 Phys. Rev. Lett. 111 125301
[38]	Chen C 2013 Phys. Rev. Lett. 111 235302
[39]	Liu X J and Hu H 2013 Phys. Rev. A 88 023622
[40]	Qu C, Zheng Z, Gong M, Xu Y, Mao L, Zou X, Guo G C and Zhang C 2013 Nat. Comm. 4 2710
[41]	Zhang W and Yi W W 2013 Nat. Comm. 4 2711
[42]	Shenoy V B 2013 Phys. Rev. A 88 033609
[43]	Liu X J 2013 Phys. Rev. A 88 043607
[44]	Hu H, Dong L, Cao Y, Pu H and Liu X J 2014 Phys. Rev. A 90 033624
[45]	Xu Y, Chu R and Zhang C 2014 Phys. Rev. Lett. 112 136402
[46]	Cao Y, Zou S H, Liu X J, Yi S, Long G L and Hu H 2014 Phys. Rev. Lett. 113 115302
[47]	Devreese J P A, Tempere J and Sá de Melo C A R 2014 Phys. Rev. Lett. 113 165304
[48]	Jiang L, Tiesinga E, Liu X J, Hu H and Pu H 2014 Phys. Rev. A 90 053606
[49]	Zheng Z, Pu H, Zou X and Guo G C 2014 Phys. Rev. A 90 063623
[50]	Cao Y, Liu X J, He L, Long G L and Hu H 2014 arXiv:1410.0987
[51]	Wang P, Yu Z Q, Fu Z, Miao J, Huang L, Chai S, Zhai H and Zhang J 2012 Phys. Rev. Lett. 109 095301
[52]	Cheuk L W, Sommer A T, Hadzibabic Z, Yefsah Z, Bakr W S and Zwierlein M W 2012 Phys. Rev. Lett. 109 095302
[53]	Williams R A, Beeler M C, LeBlanc L J, Jiménez-García K and Spielman I B 2013 Phys. Rev. Lett. 111 095301
[54]	Fu F, Huang L, Meng Z, Wang P, Liu X J, Pu H, Hu H and Zhang J 2013 Phys. Rev. A 87 053619
[55]	Wilczek F 2009 Nat. Phys. 5 614
[56]	Sau J D and Tewari S 2012 Phys. Rev. B 86 104509
[57]	Fulde P and Ferrell R A 1964 Phys. Rev. 135 A550
[58]	Giorgini S, Pitaevskii L P and Stringari S 2008 Rev. Mod. Phys. 80 1215
[59]	Barzykin V and Gor'kov L P 2002 Phys. Rev. Lett. 89 227002
[60]	Yip S K 2002 Phys. Rev. B 65 144508
[61]	Agterberg D F and Kaur R P 2007 Phys. Rev. B 75 064511
[62]	Dimitrova O and Feigel'man M V 2007 Phys. Rev. B 76 014522
[63]	Michaeli K, Potter A C and Lee P A 2012 Phys. Rev. Lett. 108 117003
[64]	Wong C L M, Liu J, Law K T and Lee P A 2013 Phys. Rev. B 88 060504
[65]	This is a branch cut at kz=0 to a 3D surface plot of the energy spectrum, as shown, for example, in Figs. 2(b) and 2(c)
[66]	Ku M J H, Sommer A T, Cheuk L W and Zwierlein M W 2012 Science 335 563
[67]	Hu H, Liu X J and Drummond P D 2006 Europhys. Lett. 74 574
[68]	Hu H, Liu X J and Drummond P D 2008 Phys. Rev. A 77 061605

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Three-dimensional spin–orbit coupled Fermi gases: Fulde–Ferrell pairing, Majorana fermions, Weyl fermions, and gapless topological superfluidity

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