نوع مقاله : مقاله پژوهشی
نویسندگان
1 1. گروه علوم مهندسی و فیزیک، مرکز آموزش عالی فنی و مهندسی بویینزهرا، بویینزهرا
2 2. دانشکدة فیزیک، دانشگاه تحصیلات تکمیلی علوم پایة زنجان، زنجان
چکیده
در این مقاله مدل تعمیم یافتة زنجیرۀ کیتائف یک بعدی، یعنی زنجیرۀ ابررسانایی موج p بلند برد را مورد بررسی قرار میدهیم. در مدل بلند برد کیتائف، پارامترهای مربوط به جفتشدگی و تونلزنی فرمیونها در هامیلتونی بلند برد هستند و شدت آنها، مستقل از یکدیگر، با عکس فاصلة بین نقاط شبکه به توان یک نما- با افزایش فاصلۀ بین نقاط شبکه- کاهش مییابد. در این مدل، وجود مدهای لبهای بدون جرم مایورانا و همچنین مدهای لبهای جرمدار دیراک را با روش قطریسازی دقیق و نیز با محاسبات تحلیلی بررسی میکنیم. قطریسازی دقیق حاکی از وجود هر دو نوع مد لبهای بدون جرم و جرمدار در طیف انرژی است. همچنین با محاسبۀ عدد پیچش، که ناوردای توپولوژیکی است، نمودار فاز و گذار فازهای توپولوژیکی این مدل را به دست میآوریم.
کلیدواژهها
عنوان مقاله [English]
One-dimensional p-wave superconductivity with long-range hopping and pairing
نویسندگان [English]
- S Ansari 1
- R Jafari 2
1 1. Department of Engineering Sciences and Physics, Buein Zahra Technical University, Buein Zahra, Iran
2 2. Department of Physics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, Iran
چکیده [English]
In this paper, we have considered the extended version of the Kitaev model in one dimension, i.e., a long-range p-wave superconducting wire. In the long-range Kitaev chain, the superconducting hopping and pairing terms in the Hamiltonian decay, independently, in a power-law fashion , where l is the distance between the two sites and x is some positive constant. We have studied the appearance of Majorana zero-energy edge modes and also, massive Dirac edge modes by exact diagonalization, as well as analytical computations. Exact diagonalization indicates the existence of both kinds of massless and massive edge modes in the energy spectrum. Furthermore, we obtain the phase diagram and the topological phase transitions by calculating the winding number, which is the topological invariant.
کلیدواژهها [English]
- p-wave superconductivity
- Kitaev chain
- topological phase transition
- Majorana modes
- long-range interaction
- X G Wen, “Quantum Field Theory and Many Body Systems”, Oxford University Press, Oxford (2007).
- T Kitagawa, E Berg, M Rudner, and E Demler, Phys. Rev. B 82 (2010) 235114.
- N H Lindner, G Refael, and V Galitski, Nature Phys. 7 (2011) 490.
- L Jiang, T Kitagawa, J Alicea, A R Akhmerov, D Pekker, G Refael, J I Cirac, E Demler, M D Lukin, and P Zoller, Phys. Rev. Lett. 106 (2011) 220402.
- Z Gu, H A Fertig, D P Arovas, and A Auerbach, Phys. Rev. Lett. 107 (2011) 216601.
- T Kitagawa, T Oka, A. Brataas, L Fu, and E Demler, Phys. Rev. B 84 (2011) 235108.
- N H Lindner, D L Bergman, G Refael, and V Galitski, Phys. Rev. B 87 (2013) 235131.
- M Trif and Y Tserkovnyak, Phys. Rev. Lett. 109 (2012) 257002.
- A Russomanno, A Silva, and G E Santoro, Phys. Rev. Lett. 109 (2012) 257201.
10. V M Bastidas, C Emary, G Schaller, and T Brandes, Phys. Rev. A 86 (2012) 063627.
11. V M Bastidas, C Emary, B Regler, and T Brandes, Phys. Rev. Lett. 108 (2012) 043003.
12. M Tomka, A Polkovnikov, and V Gritsev, Phys. Rev. Lett. 108 (2012) 080404.
13. A Gomez-Leon and G Platero, Phys. Rev. Lett. 110 (2013) 200403.
14. B Dora, J Cayssol, F Simon, and R Moessner, Phys. Rev. Lett. 108 (2012) 056602.
15. D E Liu, A Levchenko, and H U Baranger, Phys. Rev. Lett. 111 (2013) 047002.
16. Q J Tong, J H An, J Gong, H G Luo, and C H Oh, Phys. Rev. B 87 (2013) 201109(R).
17. Y T Katan and D Podolsky, Phys. Rev. Lett. 110 (2013) 016802.
18. A Kundu and B Seradjeh, Phys. Rev. Lett. 111 (2013) 136402.
19. V M Bastidas, C Emary, G Schaller, A Gomez-Leon, G Platero, and T Brandes, arXiv:1302.0781v2.
20. T L Schmidt, A Nunnenkamp, and C Bruder, New J. Phys. 15 (2013) 025043.
21. A A Reynoso and D Frustaglia, Phys. Rev. B 87 (2013) 115420.
22. A Y Kitaev, Phys. Usp. 44 (2001) 131.
23. I C Fulga, F. Hassler, A. R. Akhmerov, and C. W. J. Beenakker, Phys. Rev. B 83 (2011) 155429.
24. J D Sau and S. Das Sarma, Nature Comm. 3 (2012) 964.
25. R M Lutchyn and M P A. Fisher, Phys. Rev. B 84 (2011) 214528.
26. W DeGottardi, D Sen, and S Vishveshwara, New J. Phys. 13 (2011) 065028.
27. W DeGottardi, D Sen, and S Vishveshwara, Phys. Rev. Lett. 110 (2013) 146404.
28. M Thakurathi, A A Patel, D Sen, and A Dutta, Phys. Rev. B 88 (2013) 155133.
29. W DeGottardi, M Thakurathi, S Vishveshwara, and D Sen, Phys. Rev. B 88 (2013) 165111.
30. L J Lang and S Chen, Phys. Rev. B 86 (2012) 205135.
31. Y Niu, S B Chung, C H Hsu, I Mandal, S Raghu, and S Chakravarty, Phys. Rev. B 85 (2012) 035110.
- A A Zvyagin, Phys. Rev. B 90 (2014) 014507.
33. S Nadj-Perge, I K Drozdov, J Li, H Chen, S Jeon, J Seo, A H MacDonald, B A Bernevig, and A Yazdani, Science 346 (2014) 602.
34. V Mourik, K Zuo, S M Frolov, S R Plissard, E P A M Bakkers, and L P Kouwenhoven, Science 336 (2012) 1003.
35. M T Deng, C L Yu, G Y Huang, M Larsson, P Caro, and H Q Xu, Nano Lett. 12 (2012) 6414.
36. A Das, Y Ronen, Y Most, Y Oreg, M Heiblum, and H Shtrikman, Nat. Phys.8 (2012) 887.
37. W Chang, V Manucharyan, T Jespersen, J Nygard, and C Marcus, Phys. Rev. Lett. 110 (2013) 217005.
38. L P Rokhinson, X Liu, and J K Furdyna, Nat. Phys. 8 (2012) 795.
39. H O H. Churchill, V Fatemi, K Grove-Rasmussen, M T Deng, P Caroff, H Q Xu, and C M Marcus, Phys. Rev. B 87 (2013) 241401(R).
40. A D K Finck, D J Van Harlingen, P K Mohseni, K Jung, and X Li, Phys. Rev. Lett. 110 (2013) 126406.
41. J Alicea, Rep. Prog. Phys. 75 (2012) 076501.
42. [42] M. Leijnse and K. Flensberg, Semicond. Sci. Technol. 27 (2012) 124003.
43. C W J Beenakker, Annu. Rev. Condens. Matter Phys. 4 (2013) 113.
44. T D Stanescu and S Tewari, J. Phys.: Condens. Matter 25 (2013) 233201.
45. J Alicea, Y Oreg, G Refael, F Von Oppen, and M P A Fisher, Nat. Phys. 7 (2011) 412.
46. A R Akhmerov, Phys. Rev. B 82 (2010) 020509(R).
47. P Pfeuty, Ann. Phys. 57 (1970) 79.
48. E Lieb, T Schultz, and D Mattis, Ann. Phys. 16 (1961) 407.
49. A Ghazaryan and T Chakraborty, Phys. Rev. B 92 (2015) 115138.
50. D Vodola, L Lepori, E Ercolessi, A V Gorshkov, and G Pupillo, Phys, Rev. Lett. 113 (2014) 156402.
51. D Vodola, L Lepori, E Ercolessi, and G Pupillo, New J. Phys. 18 (2016) 015001.
52. O Viyuela, D Vodola, G Pupillo, and M A Delgado, Phys. Rev. B 94 (2016) 125121.
53. U Bhattacharya and A Dutta, Phys. Rev. B 97 (2018) 214505.
54. P Uhrich, N Defenu, R Jafari, and J C Halimeh, Phys. Rev. B 101 (2020) 245148.
55. L Lepori, L Dell’Anna, New J. Phys. 19 (2017) 103030.
56. Z C Shi, X Q Shao, and X X Yi, arXiv:1507.03657v3.
57. F Pientka, L I Glazman, and F Von Oppen, Phys. Rev. B 88 (2013) 155420.
58. F Pientka, Y Peng, L Glazman, and F Von Oppen, Phys. Scr. T164 (2015) 014008.
59. K Patrick, T Neupert, and J K Pachos, Phys. Rev. Lett. 118 (2017).
60. Z Huang and D P Arovas, Phys. Rev. Lett. 113 (2014) 076407.
61. O Viyuela, A Rivas and M A Martin-Delgado, 2D Mater. 2 (2015) 034006.
62. M Van Regemortel, D Sels and M Wouters Phys. Rev. A 93 (2016) 032311.
63. D Giuliano, S Paganelli and L Lepori Phys. Rev. B 97 (2018) 155113.
64. P Cats, A Quelle, O Viyuela, M A Martin-Delgado, and C Morais Smith, Phys. Rev. B 97 (2018) 121106 (R).
65. S Tewari and J D Sau, Phys. Rev. Lett. 109 (2012) 150408.
66. A Altland and M R Zirnbauer, Phys. Rev. B 55 (1997) 1142.
67. L Santos, Y Nishida, C Chamon, and C Mudry, Phys. Rev. B 83 (2011) 104522.
68. L Li, C Yang, and S Chen, Europhys. Lett. 112 1 (2015) 10004.
69. P Ghosh, J D Sau, S Tewari, and S Das Sarma, Phys. Rev. B 82 (2010) 184525.