Document Type : Review Article
Author
Department of physics, Sharif University of Technology, Tehran, Iran
Abstract
By using concepts of topology in mathematics, quantum mechanics, and their synergetic development during the past few decades, condensed matter physicists have discovered new phases of matter and introduced general frameworks to classify them. The research includes a vast gamut from chemistry of atomic orbitals to material science, promising new applications in the growing technologies. This review article aims to provide a better understanding of these unprecedented electron systems and their underlying topological principles. The article consist of two parts. First, there is a historical review of using topological concepts in condensed matter systems. Then, in the second part, we elaborate on some basics of topology in quantum mechanics and the concept of topological invariants.
Keywords
- M Tinkham, “Introduction to Superconductivity”, NY: McGraw-Hill, (1996).
- N Goldenfeld, “Lectures On Phase Transitions and the Renormalization Group”, Addison-Wesley (1992).
- L D L A V L Ginzburg, Eksp. Teor. Fiz. 20 (1950) 1064.
- H Wagner and N D Mermin, Physical Review Letters 17 (1966) 1133.
- H E Stanley, and T A Kaplan, Phys. Rev. Lett. 17 (1966) 913.
- F Wegner, Zeitschrift für Physik 206 (1967) 465.
- V L Berezinskii, Sov. Phys. JETP 32 (1970) 493.
- V L Berezinskii, Sov. Phys. JETP 34 (1972) 610.
- J M Kosterlitz and D J Thouless, Journal of Physics C: Solid State Physics 6 (1973) 1181.
- N D Mermin, Rev. Mod. Phys. 51 (1979) 591.
- J M Kosterlitz, Rev. Mod. Phys. 89 (2017) 40501.
- K V Klitzing, G Dorda, and M Pepper, Phys. Rev. Lett. 45 (1980) 494.
- D J Thouless, M Kohmoto, M P Nightingale, and M D Nijs, Phys. Rev. Lett. 49 (982) 405.
- D J Thouless, “Topological Quantum Numbers in Nonrelativistic Physics”, World Scientific, Singapore (1998).
- M V Berry, Proceedings of the Royal Society A 392 (1984) 45.
- F D M Haldane, Phys. Rev. Lett. 61 (1988) 2015.
- Y Hatsugai, Phys. Rev. Lett. 71 (1993) 3697.
- G Jotzu, M Messer, R Desbuquois, M Lebrat, T Uehlinger, D Greif, and T Esslinger, Nature 515 (2014) 237.
- C Kane and E Mele, Phys. Rev. Lett. 95 (2005) 226801.
- C Kane and E Mele, Phys. Rev. Lett. 95 (2005) 146802.
- B Bernevig, T Hughes, and S C. Zhang, Science 314 (2006) 1757.
- [M König, S Wiedmann, C Brüne, A Roth, H Buhmann, L Molenkamp, X Qi and S Zhang, Science 318 (2007) 766.
- M Z Hasan and C L Kane, Rev. Mod. Phys. 82 (2010) 3045.
- A Kitaev, AIP Conference Proceedings 1134 (2009) 22.
- X L. Qi and S C. Zhang, Rev. Mod. Phys. 83 (2011) 1057.
- A Kitaev, Physics-Uspekhi 44 (2001) 131.
- C Nayak, S H Simon, A Stern, M Freedman, and S D Sarma, Rev. Mod. Phys. 80 (2008) 1083.
- [C K Chiu, J C Teo, A P Schnyder, and S Ryu, Rev. Mod. Phys. 88 (2016) 035005.
- [L Fu, Phys. Rev. Lett. 106 (2011) 106802.
- M Kargarian and G A Fiete, Phys. Rev. Lett. 110 (2013) 156403.
- F Schindler, A M Cook, M G Vergniory, Z Wang, S S P Parkin, B A Bernevig and T Neupert, Science Advances 4 (2018) eaat0346.
- N Armitage, E Mele and A Vishwanath, Rev. Mod. Phys. 90 (2018) 015001.
- S Y Xu and et al., Science 347 (2015) 294.
- Z K Liu and et al., Science 343 (2014) 864.
- Z Wang, H Weng, Q Wu, X Dai and Z Fang, Phys. Rev. B, 88 (2013) 125427.
- S Jeon, B B Zhou, A Gyenis, B E Feldman, I Kimchi, A C Potter, Q D Gibson, R J Cava, A Vishwanath and A Yazdani, Nature Materials 13 (2014) 851.
- S M. Huang, S Y. Xu, I Belopolski, C C. Lee, G Chang, B K. Wang, N Alidoust, G Bian, M Neupane, C Zhang, S Jia, A Bansil, H Lin and M Z Hasan, Nature Communications 6 (2015) 7373.
- H Weng, C Fang, Z Fang, B A Bernevig and X Dai, Phys. Rev. X 5 (2015) 011029.
- S Huang, J Kim, W A Shelton, E W Plummer and R Jin, PNAS (2017) 1706657114.
- C L Zhang and et al., Nature Communications 7 (2016) 10735.
- Q Li, D E Kharzeev, C Zhang, Y Huang, I Pletikosić, A V Fedorov, R D Zhong, J A Schneeloch, G D Gu and T Valla, Nature Physics 12 (2016) 550.
- B Bradlyn and et al., Science 353 (2016) aaf5037.
- Z Rao and et al., Nature 567 (2019) 496.
- G Chang, S Y Xu, B J Wieder, D S Sanchez, S M Huang, I Belopolski, T R. Chang, S Zhang, A Bansil, H Lin and M Z Hasan, Phys. Rev. Lett. 119 (2017) 206401.
- [T H Hsieh, H Lin, J Liu, W Duan, A Bansil and L Fu, Nature Communications 3 (2012) 982.
- P Dziawa, B J Kowalski, K Dybko, R Buczko, A Szczerbakow, M Szot, E Łusakowska, T Balasubramanian, B M Wojek, M H Berntsen, O Tjernberg and T Story, Nature Materials 11 (2012) 1023.
- A P Mackenzie and Y. Maeno, Rev. Mod. Phys. 75 (2003) 657.
- X Gong, M Kargarian, A Stern, D Yue, H Zhou, X Jin, V M Galitski, V M Yakovenko and J Xia, Science Advances 3 (2017) e1602579.
- A Das, Y Ronen, Y Most, Y Oreg, M Heiblum and H Shtrikman, Nature Physics 8 (2012) 887.
- M X Wang and e. al, Science 336 (2012) 6077.
- L Jiao, S Howard, S Ran, Z Wang, J O Rodriguez, M Sigrist, Z Wang, N P Butch and V Madhavan, Nature 579 (2020) 523.
- Y Wu, N H Jo, L L Wang, C A Schmidt, K M.Neilson, B Schrunk, P Swatek, A Eaton, S L Bud'ko, P C Canfield and A Kaminski, Phys. Rev. B 99 (2019) 161113(R).
- C Le, X Wu, S Qin, Y Li, R Thomale, F C. Zhang and J Hu, PNAS 115 (2018) 8311.
- N B M Schröter and e. al, Nature Physics 15 (2019) 759.
- M Dzero, K Sun, V Galitski and P Coleman, Phys. Rev. Lett. 104 (2010) 106408.
- K Hagiwara and e. al, Nature Communications 7 (2016) 12690.
- [T Itou, A Oyamada, S Maegawa, M Tamura and R Kato, Phys. Rev. B 77 (2008) 104413.
- J S Helton, K Matan, M P Shores, E A Nytko, B M Bartlett, Y Yoshida, Y Takano, A Suslov, Y Qiu, J H. Chung, D G Nocera and Y S Lee, Phys. Rev. Lett. 98 (2007) 107204.
- H L Stormer, D C Tsui and A. C Gossard, Rev. Mod. Phys. 71 (1999) S298.
- F. Wilczek, Phys. Rev. Lett. 49 (1982) 957.
- X G Wen, “Quantum Field Theory of Many-Body Systems:From the Origin of Sound to an Origin of Light and Electrons”, Oxford: OUP (2004).
- Y Kasahara, T Ohnishi, Y Mizukami, O Tanaka, S Ma, K Sugii, N Kurita, H Tanaka, J Nasu, Y Motome, T Shibauchi and Y Matsuda, Nature 559 (2018) 227.
- A Kitaev, Annals of Physics 321 (2006) 2.
- A Kitaev, Annals of Physics 303 (2003) 2.
- H Bombin and M A Martin-Delgado, Phys. Rev. Lett. 97 (2006) 180501.
- M Kargarian, Phys. Rev. A 78 (2008) 062312.
- F D M Haldane, Phys. Rev. Lett. 50 (1983) 1153.
- F D M Haldane, Rev. Mod. Phys. 89 (2017) 40502.
- V Mourik, K Zuo, S M Frolov, S R Plissard, E P A M. Bakkers and L P Kouwenhoven, Science 336 (2012) 1003.
- S Trebst, "Kitaev Materials," arXiv, (2017).
- M Kargarian, M Randeria and Y M Lu, PNAS 113 (2016) 8648.
- B Bradlyn, L Elcoro, J Cano, M G Vergniory, Z Wang, C Felser, M I Aroyo and B A Bernevig, Nature 547 (2017) 298.
- [M G Vergniory, L Elcoro, C Felser, N Regnault, B A Bernevig and Z Wang, Nature 566 (2019) 480.
- R Bistritzer and A H. MacDonald, PNAS 108 (2011) 12233.
- Y Cao, V Fatemi, A Demir, S Fang, S L Tomarken, J Y Luo, J D Sanchez-Yamagishi, K Watanabe, T Taniguchi, E Kaxiras, R C Ashoori and P Jarillo-Herrero, Nature 556 (2018) 80.
- Y Cao, V Fatemi, S Fang, K Watanabe, T Taniguchi, E Kaxiras and P Jarillo-Herrero, Nature 556 (2018) 43.
- [H C Po, L Zou, A Vishwanath and T Senthil, Phys. Rev. X 8 (2018) 031089.
- B Lian, Z Wang and B A Bernevig, Phys. Rev. Lett. 122 (2019) 257002.
- F Wu, A MacDonald and I Martin, Phys. Rev. Lett. 121 (2018) 257001.
- A L Sharpe, E J Fox, A W Barnard, J Finney, K Watanabe, T Taniguchi, M A Kastner and D Goldhaber-Gordon, Science 365 (2019) 605.
- Z Song, Z Wang, G L Wujun Shi, C Fang and B A Bernevig, Phys. Rev. Lett. 123 (2019) 036401.
- S Ran, C Eckberg, Q P Ding, Y Furukawa, T Metz, S R Saha, L Liu, M Zic, H Kim, J Paglione and N P Butch, Science365 (2019) 684.
- S M Bhattacharjee, “Topology and Condensed Matter Physics, Texts and Readings in Physical Sciences”, Springer, Singapore, 19 (2017) 171.
- A Altland and B D Simons, “Condensed Matter Field Theory”, Cambridge: Cambridge University Press, (2010).
- D Xiao, M C. Chang and Q Niu, Rev. Mod. Phys. 82 (2010) 1959.
- J E Moore and L Balents, Phys. Rev. B 75 (2007). 121306(R.
- R Roy, Phys. Rev. B 79 (2009) 195322.
- B A Bernevig and T L Hughes, “Topological Insulators and Topological Superconductors”, Princeton: Princeton University Press (2013).
- Editorial, "Topology on top," Nature Physics 12 (2016) 615.
- G A Fiete, Nature 547 (2017) 287.
- A B Khanikaev, S H Mousavi, W K Tse, M Kargarian, A H MacDonald and G Shvets, Nature Materials 12 (2013) 233.
- M Hafezi, et al, Nature Photonics 7 (2013) 1001.
- C L Kane and T C Lubensky, Nature Physics 10 (2014) 39.
E Cohen, H Larocque, FBouchard, F Nejadsattari, Y Gefen and E Karimi, Nature Physics Reviews 1 (2019) 437.
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