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We investigate the interband coupling induced odd-frequency pairing state by solving the microscopic Ginzburg-Landau model for the two band superconductor magnesium diboride (MgB2). It is found that the interband coupling can induce a new domain structure and a heliacal spontaneous magnetic vortex-antivortex pair around the cyclical domain wall, which breaks down spin-rotational symmetry and supports a time-reversal violating bound state, allowing the coexistence of spin-singlet and spin-triplet state close to the spontaneous vortex core. The odd-frequency spin-triplet even parity pairing state occurs since a successive operation in the orbital parity (P) and the time-reversal (T) obeys PT = + 1(–1) for spin-singlet (spin-triplet) pairing amplitude. A general phase diagram is presented.
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图 1 当温度
$T\in(T_1,T_2)$ , 外磁场磁通涡量$m=2$ 时, 四度旋转对称的Abriksov涡旋格子图 (a)−(c)分别为${\text{π}}-$ 带和σ带序参量和磁场的空间分布图; (e)−(g)分别为对应的等高图; (d)和(h)分别为${\text{π}}$ 带和$\text{σ}$ 带波函数的相位分布图, 两者的相位绕向彼此相反Fig. 1. The fourfold rotational symmetric Abrikosov vortex for the superconductor with coupled
${\text{π}}-$ and$\text{σ}-$ pairings under the external magnetic flux of two quanta (m = 2) and at the temperature$T\in(T_1,T_2)$ : Panels (a)−(c) show the spatial distributions for$ {\text{π}}-$ and$\text{σ}-$ channel amplitude and magnetic field H, respectively; (e)−(g) show the contour plots, respectively; (d) and (h) show the phase distribution for the${\text{π}}-$ and$\text{σ}-$ channel, An opposite phase winding is indicated in the${\text{π}}-$ and$\text{σ}-$ channels.
图 2 当温度
$T=0.1{\rm K}$ , 外磁场磁通涡量$m=2$ 时,${\text{π}}-$ 和$\text{σ}-$ 带空间分离的两度旋转对称半涡旋图 (a)−(c)分别为${\text{π}}-$ 和$\text{σ}-$ 带序参量和磁场的空间分布图Fig. 2. The twofold rotational symmetric semi-vortex unit, consisting of spatially separated
${\text{π}}-$ wave and$\text{σ}-$ channel cores at the temperature$T=0.1{\rm K}$ and under the external magnetic flux of two quanta (m = 2), (a)−(c) show the contour plots for$ {\text{π}}-$ and$ {\text{σ}}-$ channel amplitude and magnetic field H, respectively.
图 3 当温度
$T=0.1{\rm K}$ 时自发场图 (a)−(c)分别为${\text{π}}-$ 带和$\text{σ}-$ 带序参量和磁场的空间分布图; (d)−(f)分别为对应的等高图Fig. 3. Spontaneous magnetic field at the temperature
$T=0.1{\rm K}$ , Panels (a))−(c) show the spatial distributions for${\text{π}}-$ and$\text{σ}-$ channel amplitude and spontaneous magnetic field H, respectively; (d))−(f) show the contour plots, respectively. -
[1] Berezinskii V L 1974 JETP Lett. 20 287
[2] Kirkpatrick T R, Belitz D 1991 Phys. Rev. Lett. 66 1533
Google Scholar
[3] Coleman P, Miranda E, Tsvelik A 1993 Phys. Rev. Lett. 70 2960
Google Scholar
[4] Balatsky A, Abrahams E, Scalapino D J, Schrieffer J R 1995 Phys. Rev. B 52 1271
Google Scholar
[5] Bergeret F S, Volkov A F, Efetov K B 2005 Rev. Mod. Phys. 77 1321
Google Scholar
[6] Tanaka Y, Golubov A A, Kashiwaya S, Ueda M 2007 Phys. Rev. Lett. 99 037005
Google Scholar
[7] Keizer R S, Goennenwein S T B, Klapwijk T M, Miao G, Xiao G, Gupta A 2006 Nature 439 525
[8] Eschrig M, Löfwander T 2008 Nat. Phys. 4 138
Google Scholar
[9] Robinson J W A, Witt J D S, Blamire M G 2010 Science 329 59
Google Scholar
[10] Wen L, Jin L J, Chen Y, Zha G Q, Zhou S P 2014 Europhys. Lett. 105 27007
Google Scholar
[11] Annica M, Black S, Alexander V B 2013 Phys. Rev. B 88 104514
Google Scholar
[12] Stewart G R 2011 Rev. Mod. Phys. 83 1589
Google Scholar
[13] Rourke P M C, Tanatar M A, Turel C S, Berdeklis J, Petrovic C, Wei J Y T 2005 Phys. Rev. Lett. 94 107005
Google Scholar
[14] Klitzing K V, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494
Google Scholar
[15] Tsui D C, Stormer H L, Gossard A C 1982 Phys. Rev. Lett. 48 1559
Google Scholar
[16] Volovik G E, Yakovenko V M 1989 Condensed Matter Physics 1 5263
Google Scholar
[17] Bernevig B A, Zhang S C 2006 Phys. Rev. Lett. 96 106802
Google Scholar
[18] Babaev E, Speight M 2005 Phys. Rev. B 72 180502
Google Scholar
[19] Moshchalkov V, Menghini M, Nishio T, Chen Q H, Silhanek A V, Dao V H, Chibotaru L F, Zhigadlo N D, Karpinski J 2009 Phys. Rev. Lett. 102 117001
Google Scholar
[20] Nishio T, Dao V H, Chen Q H, Chibotaru L F, Kadowaki K, Moshchalkov V 2010 Phys. Rev. B 81 020506
Google Scholar
[21] Kogan V G, Schmalian J 2011 Phys. Rev. B 83 054515
Google Scholar
[22] Geyer J, Fernandes R M, Kogan V G, Schmalian J 2010 Phys. Rev. B 82 104521
Google Scholar
[23] Babaev E, Silaev M 2012 Phys. Rev. B 86 016501
[24] Šimánek E 2006 Phys. Rev. B 74 052501
[25] Iavarone M, Karapetrov G, Koshelev A E, Kwok W K, Crabtree G W, Hinks D G, Kang W N, Choi E M, Kim H J, Kim H J, Lee S I 2002 Phys. Rev. Lett. 89 187002
Google Scholar
[26] Kong Y, Dolgov O V, Jepsen O, Andersen O K 2001 Phys. Rev. B 64 020501
Google Scholar
[27] Zhitomirsky M E, Dao V H 2004 Phys. Rev. B 69 054508
Google Scholar
[28] Silaev M, Babaev E 2012 Phys. Rev. B 85 134514
Google Scholar
[29] Garaud J, Babaev E 2014 Phys. Rev. Lett. 112 017003
Google Scholar
[30] Garaud J, Silaev M, Babaev E 2016 Phys. Rev. Lett. 116 097002
Google Scholar
[31] Garaud J, Silaev M, Babaev E 2017 Phys. Rev. B 96 140503
Google Scholar
[32] Lin S Z, Maiti S, Chubukov A 2016 Phys. Rev. B 94 064519
Google Scholar
[33] Grinenko V, Materne P, Sarkar R, Luetkens H, Kihou K, Lee C H, Akhmadaliev S, Efremov D V, Drechsler S L, Klauss H H 2017 Phys. Rev. 95 214511
Google Scholar
[34] Zhang L F, Covaci L, Milošević M V 2017 Phys. Rev. B 96 224512
Google Scholar
[35] Zyuzin A A, Garaud J, Babaev E 2017 Phys. Rev. Lett. 119 167001
Google Scholar
[36] Yu X Z, Onose Y, Kanazawa N, Park J H, Han J H, Matsui Y, Nagaosa N, Tokura Y 2010 Nature 465 901
Google Scholar
[37] Choi J Y, Kwon W J, Shin Y I 2012 Phys. Rev. Lett. 108 035301
Google Scholar
[38] Sato M, Ando Y 2017 Rep. Prog. Phys. 80 076501
Google Scholar
[39] Linder J, Yokoyama T, Sudb A E, Schrig M 2009 Phys. Rev. Lett. 102 107008
Google Scholar
[40] Tanaka Y, Tanuma Y 2007 Phys. Rev. B 76 054522
Google Scholar
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