Please wait a minute...
Shopping Cart Email List Chinese
Advanced Search Fig/Tab
Frontiers of Physics

ISSN 2095-0462

ISSN 2095-0470(Online)

CN 11-5994/O4

Postal Subscription Code 80-965

2018 Impact Factor: 2.483

Featured Articles  |  Most Downloaded  |  Most Reading
Online Submission Subscribe to Our RSS Content Feed
Front. Phys.    2019, Vol. 14 Issue (4) : 43402    https://doi.org/10.1007/s11467-019-0896-1
Review article
Topological gapless matters in three-dimensional ultracold atomic gases
Yong Xu()
Center for Quantum Information, IIIS, Tsinghua University, Beijing 100084, China
 Download: PDF(10266 KB)  
 Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks
Abstract

Three-dimensional topological gapless matters with gapless degeneracies protected by a topological invariant defined over a closed manifold in momentum space have attracted considerable interest in various fields ranging from condensed matter materials to ultracold atomic gases. As a highly controllable and disorder free system, ultracold atomic gases provide a versatile platform to simulate topological gapless matters. Here, the current progress in studies of topological gapless phenomena in three-dimensional cold atom systems is summarized in the review. It is mainly focused on Weyl points, structured (type-II) Weyl points, Dirac points, nodal rings and Weyl exceptional rings in cold atoms. Since interactions in cold atoms can be controlled via Feshbach resonances, the progress in both superfluids for attractive interactions and non-interacting cold atom gases is reviewed.
Keywords ultracold atomic gases      Weyl points      Dirac points      nodal rings      Weyl exceptional rings     
Corresponding Author(s): Yong Xu   
Issue Date: 17 April 2019
 Cite this article:   
Yong Xu. Topological gapless matters in three-dimensional ultracold atomic gases[J]. Front. Phys. , 2019, 14(4): 43402.
 URL:  
https://academic.hep.com.cn/fop/EN/10.1007/s11467-019-0896-1
https://academic.hep.com.cn/fop/EN/Y2019/V14/I4/43402
1 P. A. M. Dirac, The quantum theory of the electron, Proc. R. Soc. Lond. A117(778), 610 (1928)
https://doi.org/10.1098/rspa.1928.0023
2 H. Weyl, Elektron und gravitation (I), Z. Phys.56(5–6), 330 (1929)
https://doi.org/10.1007/BF01339504
3 A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Rev. Mod. Phys.81(1), 109 (2009)
https://doi.org/10.1103/RevModPhys.81.109
4 S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, Electronic transport in two-dimensional graphene, Rev. Mod. Phys.83(2), 407 (2011)
https://doi.org/10.1103/RevModPhys.83.407
5 C. Herring, Accidental degeneracy in the energy bands of crystals, Phys. Rev.52(4), 365 (1937)
https://doi.org/10.1103/PhysRev.52.365
6 G. Volovik, Zeros in the fermion spectrum in superfluid systems as diabolical points, JETP Lett.46(2), 98 (1987)
7 G. E. Volovik, The Universe in a Helium Droplet, Clarendon Press, Oxford, 2003
8 X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates, Phys. Rev. B83(20), 205101 (2011)
https://doi.org/10.1103/PhysRevB.83.205101
9 K. Y. Yang, Y. M. Lu, and Y. Ran, Quantum Hall effects in a Weyl semimetal: Possible application in pyrochlore iridates, Phys. Rev. B84(7), 075129 (2011)
https://doi.org/10.1103/PhysRevB.84.075129
10 A. A. Burkov and L. Balents, Weyl semimetal in a topological insulator multilayer, Phys. Rev. Lett.107(12), 127205 (2011)
https://doi.org/10.1103/PhysRevLett.107.127205
11 G. Xu, H. Weng, Z. Wang, X. Dai, and Z. Fang, Chern semimetal and the quantized anomalous Hall effect in HgCr2Se4, Phys. Rev. Lett.107(18), 186806 (2011)
https://doi.org/10.1103/PhysRevLett.107.186806
12 C. Fang, M. J. Gilbert, X. Dai, and B. A. Bernevig, Multi-Weyl topological semimetals stabilized by point group symmetry, Phys. Rev. Lett.108(26), 266802 (2012)
https://doi.org/10.1103/PhysRevLett.108.266802
13 S. A. Yang, H. Pan, and F. Zhang, Dirac and Weyl superconductors in three dimensions, Phys. Rev. Lett.113(4), 046401 (2014)
https://doi.org/10.1103/PhysRevLett.113.046401
14 H. Weng, C. Fang, Z. Fang, B. A. Bernevig, and X. Dai, Weyl semimetal phase in noncentrosymmetric transitionmetal monophosphides, Phys. Rev. X5(1), 011029 (2015)
https://doi.org/10.1103/PhysRevX.5.011029
15 H. Ishizuka, T. Hayata, M. Ueda, and N. Nagaosa, Emergent electromagnetic induction and adiabatic charge pumping in noncentrosymmetric Weyl Semimetals, Phys. Rev. Lett.117(21), 216601 (2016)
https://doi.org/10.1103/PhysRevLett.117.216601
16 C. Fang, L. Lu, J. Liu, and L. Fu, Topological semimetals with helicoid surface states, Nat. Phys.12(10), 936 (2016)
17 M. Gong, S. Tewari, and C. Zhang, BCS–BEC crossover and topological phase transition in 3D spin–orbit coupled degenerate Fermi gases, Phys. Rev. Lett.107(19), 195303 (2011)
https://doi.org/10.1103/PhysRevLett.107.195303
18 J. H. Jiang, Tunable topological Weyl semimetal from simple-cubic lattices with staggered fluxes, Phys. Rev. A85(3), 033640 (2012)
https://doi.org/10.1103/PhysRevA.85.033640
19 B. M. Anderson, G. Juzeliunas, V. M. Galitski, and I. B. Spielman, Synthetic 3D spin–orbit coupling, Phys. Rev. Lett.108(23), 235301 (2012)
https://doi.org/10.1103/PhysRevLett.108.235301
20 Y. Xu, R. L. Chu, and C. Zhang, Anisotropic Weyl fermions from the quasiparticle excitation spectrum of a 3D Fulde–Ferrell superfluid, Phys. Rev. Lett.112(13), 136402 (2014)
https://doi.org/10.1103/PhysRevLett.112.136402
21 Y. Xu and C. Zhang, Topological Fulde–Ferrell superfluids of a spin–orbit coupled Fermi gas, Int. J. Mod. Phys. B29(01), 1530001 (2015)
https://doi.org/10.1142/S0217979215300017
22 B. Liu, X. Li, L. Yin, and W. V. Liu, Weyl superfluidity in a three-dimensional dipolar Fermi gas, Phys. Rev. Lett.114(4), 045302 (2015)
https://doi.org/10.1103/PhysRevLett.114.045302
23 S. Ganeshan, and S. Das Sarma, Constructing a Weyl semimetal by stacking one-dimensional topological phases, Phys. Rev. B91(12), 125438 (2015)
https://doi.org/10.1103/PhysRevB.91.125438
24 X. Li and S. D. Sarma, Exotic topological density waves in cold atomic Rydberg-dressed fermions, Nat. Commun.6(1), 7137 (2015)
https://doi.org/10.1038/ncomms8137
25 T. Dubček, C. J. Kennedy, L. Lu, W. Ketterle, M. Soljačić, and H. Buljan, Weyl points in three-dimensional optical lattices: Synthetic magnetic monopoles in momentum space, Phys. Rev. Lett.114(22), 225301 (2015)
https://doi.org/10.1103/PhysRevLett.114.225301
26 Y. Xu, F. Zhang, and C. Zhang, Structured Weyl points in spin–orbit coupled fermionic superfluids, Phys. Rev. Lett.115(26), 265304 (2015)
https://doi.org/10.1103/PhysRevLett.115.265304
27 W. Y. He, S. Zhang, and K. T. Law, Realization and detection of Weyl semimetals and the chiral anomaly in cold atomic systems, Phys. Rev. A94(1), 013606 (2016)
https://doi.org/10.1103/PhysRevA.94.013606
28 Y.-Q. Wang and X.-J. Liu, Predicted scaling behavior of Bloch oscillation in Weyl semimetals, Phys. Rev. A94, 031603(R) (2016)
29 Y. Xu and L. M. Duan, Type-II Weyl points in threedimensional cold-atom optical lattices, Phys. Rev. A94(5), 053619 (2016)
https://doi.org/10.1103/PhysRevA.94.053619
30 S. V. Syzranov, M. L. Wall, B. Zhu, V. Gurarie, and A. M. Rey, Emergent Weyl excitations in systems of polar particles, Nat. Commun.7(1), 13543 (2016)
https://doi.org/10.1038/ncomms13543
31 L. Lepori, I. C. Fulga, A. Trombettoni, and M. Burrello, PT-invariant Weyl semimetals in gauge-symmetric systems, Phys. Rev. B94(8), 085107 (2016)
https://doi.org/10.1103/PhysRevB.94.085107
32 L. Lepori, I. C. Fulga, A.Trombettoni, and M. Burrello, Double Weyl points and Fermi arcs of topological semimetals in non-Abelian gauge potentials, Phys. Rev. A94(5), 053633 (2016)
https://doi.org/10.1103/PhysRevA.94.053633
33 X. Y. Mai, D. W. Zhang, Z. Li, and S. L. Zhu, Exploring topological double-Weyl semimetals with cold atoms in optical lattices, Phys. Rev. A95(6), 063616 (2017)
https://doi.org/10.1103/PhysRevA.95.063616
34 L. J. Lang, S. L. Zhang, K. T. Law, and Q. Zhou, Weyl points and topological nodal superfluids in a facecentered-cubic optical lattice, Phys. Rev. B96(3), 035145 (2017)
https://doi.org/10.1103/PhysRevB.96.035145
35 B.-Z. Wang, Y.-H. Lu, W. Sun, S. Chen, Y. J. Deng, and X.-J. Liu, Dirac-, Rashba-, and Weyl-type spin–orbit couplings: Toward experimental realization in ultracold atoms, Phys. Rev. A97, 011605(R) (2018)
36 Y. B. Yang, L. M. Duan, and Y. Xu, Continuously tunable topological pump in high-dimensional cold atomic gases, Phys. Rev. B98(16), 165128 (2018)
https://doi.org/10.1103/PhysRevB.98.165128
37 X. F. Zhou, X. W. Luo, G. Chen, S. T. Jia, and C. Zhang, Rashba and Weyl spin–orbit coupling in an optical lattice clock, arXiv: 1804.09282 (2018)
38 Y. Xu and Y. Hu, Scheme to equilibrate the quantized Hall response of topological systems from coherent dynamics, arXiv: 1807.09732 (2018)
39 L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, Weyl points and line nodes in gyroid photonic crystals, Nat. Photonics7(4), 294 (2013)
https://doi.org/10.1038/nphoton.2013.42
40 M. Xiao, W. J. Chen, W. Y. He, and C. T. Chan, Synthetic gauge flux and Weyl points in acoustic systems, Nat. Phys.11(11), 920 (2015)
41 D. Z. Rocklin, B. G. Chen, M. Falk, V. Vitelli, and T. C. Lubensky, Mechanical Weyl modes in topological Maxwell lattices, Phys. Rev. Lett.116(13), 135503 (2016)
https://doi.org/10.1103/PhysRevLett.116.135503
42 B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao, J. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, Z. Fang, X. Dai, T. Qian, and H. Ding, Experimental discovery of Weyl semimetal TaAs, Phys. Rev. X5(3), 031013 (2015)
https://doi.org/10.1103/PhysRevX.5.031013
43 S. Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C. C. Lee, S. M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia, and M. Z. Hasan, Discovery of a Weyl fermion semimetal and topological Fermi arcs, Science349(6248), 613 (2015)
https://doi.org/10.1126/science.aaa9297
44 L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M. Soljačić, Experimental observation of Weyl points, Science349(6248), 622 (2015)
https://doi.org/10.1126/science.aaa9273
45 W. J. Chen, M. Xiao, and C. T. Chan, Photonic crystals possessing multiple Weyl points and the experimental observation of robust surface states, Nat. Commun.7(1), 13038 (2016)
https://doi.org/10.1038/ncomms13038
46 B. Yang, Q. H. Guo, B. Tremain, R. J.Liu, L. E. Barr, Q. H. Yan, W. L. Gao, H. C. Liu, Y. J. Xiang, J. Chen, C. Fang, A. Hibbins, L. Lu, and S. Zhang, Ideal Weyl points and helicoid surface states in artificial photonic crystal structures, Science359(6379), 1013 (2018)
https://doi.org/10.1126/science.aaq1221
47 A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu, M. Troyer, X. Dai, and B. A. Bernevig, Type-II Weyl semimetals, Nature527(7579), 495 (2015)
https://doi.org/10.1038/nature15768
48 E. J. Bergholtz, Z. Liu, M. Trescher, R. Moessner, and M. Udagawa, Topology and interactions in a frustrated slab: Tuning from Weyl semimetals to C>1 fractional Chern insulators, Phys. Rev. Lett.114(1), 016806 (2015)
https://doi.org/10.1103/PhysRevLett.114.016806
49 Z. Yu, Y. Yao, and S. A. Yang, Predicted unusual magnetoresponse in type-II Weyl semimetals, Phys. Rev. Lett.117(7), 077202 (2016)
https://doi.org/10.1103/PhysRevLett.117.077202
50 M. Udagawa and E. J. Bergholtz, Field-selective anomaly and chiral mode reversal in type-II Weyl materials, Phys. Rev. Lett.117(8), 086401 (2016)
https://doi.org/10.1103/PhysRevLett.117.086401
51 T. E. O’Brien, M. Diez, and C. W. J. Beenakker, Magnetic breakdown and Klein tunneling in a type-II Weyl semimetal, Phys. Rev. Lett.116(23), 236401 (2016)
https://doi.org/10.1103/PhysRevLett.116.236401
52 A. A. Zyuzin and R. P. Tiwari, Intrinsic anomalous Hall effect in type-II Weyl semimetals, JETP Lett.103(11), 717 (2016)
https://doi.org/10.1134/S002136401611014X
53 K. Deng, G. Wan, P. Deng, K. Zhang, S. Ding, E. Wang, M. Yan, H. Huang, H. Zhang, Z. Xu, J. Denlinger, A. Fedorov, H. Yang, W. Duan, H. Yao, Y. Wu, S. Fan, H. Zhang, X. Chen, and S. Zhou, Experimental observation of topological Fermi arcs in type-II Weyl semimetal MoTe2, Nat. Phys.12(12), 1105 (2016)
54 L. Huang, T. M. McCormick, M. Ochi, Z. Zhao, M. T. Suzuki, R. Arita, Y. Wu, D. Mou, H. Cao, J. Yan, N. Trivedi, and A. Kaminski, Spectroscopic evidence for a type II Weyl semimetallic state in MoTe2, Nat. Mater.15(11), 1155 (2016)
https://doi.org/10.1038/nmat4685
55 A. Liang, J. W. Huang, S. M. Nie, Y. Ding, Q. Gao, et al., Electronic evidence for type II Weyl semimetal state in MoTe2, arXiv: 1604.01706 (2016)
56 A. Tamai, Q. S. Wu, I. Cucchi, F. Y. Bruno, S. Riccò, T. K. Kim, M. Hoesch, C. Barreteau, E. Giannini, C. Besnard, A. A. Soluyanov, and F. Baumberger, Fermi arcs and their topological character in the candidate type-II Weyl semimetal MoTe2, Phys. Rev. X6(3), 031021 (2016)
https://doi.org/10.1103/PhysRevX.6.031021
57 S. Y. Xu, N. Alidoust, G. Q. Chang, H. Lu, B Singh, I. Belopolski, et al., Discovery of Lorentz-violating type II Weyl fermions in LaAlGe, Sci. Adv.3(6), e1603266 (2017)
https://doi.org/10.1126/sciadv.1603266
58 J. Jiang, Z. K. Liu, Y. Sun, H. F. Yang, C. R. Rajamathi, Y. P. Qi, L. X. Yang, C. Chen, H. Peng, C. C. Hwang, S. Z. Sun, S. K. Mo, I. Vobornik, J. Fujii, S. S. P. Parkin, C. Felser, B. H. Yan, and Y. L. Chen, Signature of type-II Weyl semimetal phase in MoTe2, Nat. Commun.8, 13973 (2017)
https://doi.org/10.1038/ncomms13973
59 A. Liang, J. W. Huang, S. M. Nie, Y. Ding, Q. Gao, et al., Electronic evidence for type II Weyl semimetal state in MoTe2, arXiv: 1604.01706 (2016)
60 N. Xu, Z. J. Wang, A. P. Weber, A. Magrez, P. Bugnon, et al., Discovery of Weyl semimetal state violating Lorentz invariance in MoTe2, arXiv: 1604.02116 (2016)
61 F. Y. Bruno, A. Tamai, Q. S. Wu, I. Cucchi, C. Barreteau, A. de la Torre, S. McKeown Walker, S. Riccò, Z. Wang, T. K. Kim, M. Hoesch, M. Shi, N. C. Plumb, E. Giannini, A. A. Soluyanov, and F. Baumberger, Observation of large topologically trivial Fermi arcs in the candidate type-II Weyl semimetal WTe2, Phys. Rev. B94, 121112(R) (2016)
62 J. Noh, S. Huang, D. Leykam, Y. D. Chong, K. P. Chen, and M. C. Rechtsman, Experimental observation of optical Weyl points and Fermi arc-like surface states, Nat. Phys.13(6), 611 (2017)
63 B. Yang, Q. H. Guo, B. Tremain, L. E. Barr, W. L. Gao, H. C. Liu, B. Béri, Y. J. Xiang, D. Y. Fan, A. P. Hibbins, and S. Zhang, Direct observation of topological surfacestate arcs in photonic metamaterials, Nat. Commun.8(1), 97 (2017)
https://doi.org/10.1038/s41467-017-00134-1
64 A. A. Burkov, M. D. Hook, and L. Balents, Topological nodal semimetals, Phys. Rev. B84(23), 235126 (2011)
https://doi.org/10.1103/PhysRevB.84.235126
65 C. Fang, H. Weng, X. Dai, and Z. Fang, Topological nodal line semimetals, Chin. Phys. B25(11), 117106 (2016)
https://doi.org/10.1088/1674-1056/25/11/117106
66 Y. K. Kim, B. J. Wieder, C. L. Kane, and A. M. Rappe, Dirac line nodes in inversion-symmetric crystals, Phys. Rev. Lett.115(3), 036806 (2015)
https://doi.org/10.1103/PhysRevLett.115.036806
67 L. K. Lim and R. Moessner, Pseudospin vortex ring with a nodal line in three dimensions, Phys. Rev. Lett.118(1), 016401 (2017)
https://doi.org/10.1103/PhysRevLett.118.016401
68 W. Chen, H. Z. Lu, and J. M. Hou, Topological semimetals with a double-helix nodal link, Phys. Rev. B96(4), 041102 (2017)
https://doi.org/10.1103/PhysRevB.96.041102
69 Z. B. Yan, R. Bi, H. T. Shen, L. Lu, S.-C. Zhang, and Z. Wang, Nodal-link semimetals, Phys. Rev. B96, 041103(R) (2017)
70 P. Y. Chang and C. H. Yee, Weyl-link semimetals, Phys. Rev. B96(8), 081114 (2017)
https://doi.org/10.1103/PhysRevB.96.081114
71 R. Bi, Z. Yan, L. Lu, and Z. Wang, Nodal-knot semimetals, Phys. Rev. B96, 201305(R) (2017)
72 C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Observation of parity–time symmetry in optics, Nat. Phys.6(3), 192 (2010)
73 B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, Parity–time-symmetric whispering-gallery microcavities, Nat. Phys.10(5), 394 (2014)
74 L. Feng, Z. J. Wong, R. M. Ma, Y. Wang, and X. Zhang, Single-mode laser by parity-time symmetry breaking, Science346(6212), 972 (2014)
https://doi.org/10.1126/science.1258479
75 H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, Parity–timesymmetric microring lasers, Science346(6212), 975 (2014)
https://doi.org/10.1126/science.1258480
76 J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, Observation of a topological transition in the bulk of a non-Hermitian system, Phys. Rev. Lett.115(4), 040402 (2015)
https://doi.org/10.1103/PhysRevLett.115.040402
77 B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.L. Chua, J. D. Joannopoulos, and M. Soljačić, Spawning rings of exceptional points out of Dirac cones, Nature525(7569), 354 (2015)
https://doi.org/10.1038/nature14889
78 T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. Höfling, Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R. G. Dall, and E. A. Ostrovskaya, Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard, Nature526(7574), 554 (2015)
https://doi.org/10.1038/nature15522
79 H. Xu, D. Mason, L. Jiang, and J. G. E. Harris, Topological energy transfer in an optomechanical system with exceptional points, Nature537(7618), 80 (2016)
https://doi.org/10.1038/nature18604
80 J. Li, A. K. Harter, J. Liu, L. de Melo, Y. N. Joglekar, and L. Luo, Observation of parity–time symmetry breaking transitions in a dissipative Floquet system of ultracold atoms, arXiv: 1608.05061 (2016)
81 S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, Topologically protected bound states in photonic parity–time-symmetric crystals, Nat. Mater.16(4), 433 (2017)
https://doi.org/10.1038/nmat4811
82 W. J. Chen, S. K. Özdemir, G. M. Zhao, J. Wiersig, and L. Yang, Exceptional points enhance sensing in an optical microcavity, Nature548(7666), 192 (2017)
https://doi.org/10.1038/nature23281
83 L. Xiao, X. Zhan, Z. H. Bian, K. K. Wang, X. Zhang, X. P. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami, W. Yi, H. Obuse, B. C. Sanders, and P. Xue, Observation of topological edge states in parity–time-symmetric quantum walks, Nat. Phys.13(11), 1117 (2017)
84 H. Zhou, C. Peng, Y. Yoon, C. W. Hsu, K. A. Nelson, L. Fu, J. D. Joannopoulos, M. Soljačić, and B. Zhen, Observation of bulk Fermi arc and polarization half charge from paired exceptional points, Science359(6379), 1009 (2018)
https://doi.org/10.1126/science.aap9859
85 A. Cerjan, S. Huang, K. P. Chen, Y. Chong, and M. C. Rechtsman, Experimental realization of a Weyl exceptional ring, arXiv: 1808.09541 (2018)
86 M. S. Rudner and L. S. Levitov, Topological transition in a non-Hermitian quantum walk, Phys. Rev. Lett.102(6), 065703 (2009)
https://doi.org/10.1103/PhysRevLett.102.065703
87 K. Esaki, M. Sato, K. Hasebe, and M. Kohmoto, Edge states and topological phases in non-Hermitian systems, Phys. Rev. B84(20), 205128 (2011)
https://doi.org/10.1103/PhysRevB.84.205128
88 C.E. Bardyn, M. A. Baranov, C. V. Kraus, E. Rico, A. İmamoǧlu, P. Zoller, and S. Diehl, Topology by dissipation, New J. Phys.15(8), 085001 (2013)
https://doi.org/10.1088/1367-2630/15/8/085001
89 S. Malzard, C. Poli, and H. Schomerus, Topologically protected defect states in open photonic systems with non-Hermitian charge-conjugation and parity–time symmetry, Phys. Rev. Lett.115(20), 200402 (2015)
https://doi.org/10.1103/PhysRevLett.115.200402
90 T. E. Lee, Anomalous edge state in a non-Hermitian lattice, Phys. Rev. Lett.116(13), 133903 (2016)
https://doi.org/10.1103/PhysRevLett.116.133903
91 Q. B. Zeng, B. Zhu, S. Chen, L. You, and R. Lü, Non-Hermitian Kitaev chain with complex on-site potentials, Phys. Rev. A94(2), 022119 (2016)
https://doi.org/10.1103/PhysRevA.94.022119
92 D. Leykam, K. Y. Bliokh, C. Huang, Y. Chong, and F. Nori, Edge modes, degeneracies, and topological numbers in non-Hermitian systems, Phys. Rev. Lett.118(4), 040401 (2017)
https://doi.org/10.1103/PhysRevLett.118.040401
93 Y. Xu, S. T. Wang, and L. M. Duan, Weyl exceptional rings in a three-dimensional dissipative cold atomic gas, Phys. Rev. Lett.118(4), 045701 (2017)
https://doi.org/10.1103/PhysRevLett.118.045701
94 H. Menke and M. M. Hirschmann, Topological quantum wires with balanced gain and loss, Phys. Rev. B95(17), 174506 (2017)
https://doi.org/10.1103/PhysRevB.95.174506
95 H. Shen, B. Zhen, and L. Fu, Topological band theory for Non-Hermitian Hamiltonians, Phys. Rev. Lett.120(14), 146402 (2018)
https://doi.org/10.1103/PhysRevLett.120.146402
96 S. Lieu, Topological phases in the non-Hermitian Su-Schrieffer-Heeger model, Phys. Rev. B97(4), 045106 (2018)
https://doi.org/10.1103/PhysRevB.97.045106
97 A. A. Zyuzin and A. Yu. Zyuzin, Flat band in disorderdriven non-Hermitian Weyl semimetals, Phys. Rev. B97, 041203(R) (2018)
98 C. Yin, H. Jiang, L. Li, R. Lü, and S. Chen, Geometrical meaning of winding number and its characterization of topological phases in one-dimensional chiral non-Hermitian systems, Phys. Rev. A97(5), 052115 (2018)
https://doi.org/10.1103/PhysRevA.97.052115
99 S. Yao and Z. Wang, Edge states and topological invariants of Non-Hermitian systems, Phys. Rev. Lett.121(8), 086803 (2018)
https://doi.org/10.1103/PhysRevLett.121.086803
100 S. Yao and Z. Wang, Non-Hermitian Chern bands, Phys. Rev. Lett.121(13), 136802 (2018)
https://doi.org/10.1103/PhysRevLett.121.136802
101 Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, and M. Ueda, Topological phases of non-Hermitian systems, Phys. Rev. X8(3), 031079 (2018)
https://doi.org/10.1103/PhysRevX.8.031079
102 A. Cerjan, M. Xiao, L. Yuan, and S. Fan, Effects of non-Hermitian perturbations on Weyl Hamiltonians with arbitrary topological charges, Phys. Rev. B97(7), 075128 (2018)
https://doi.org/10.1103/PhysRevB.97.075128
103 J. Carlström and E. J. Bergholtz, Exceptional links and twisted Fermi ribbons in non-Hermitian systems, Phys. Rev. A98(4), 042114 (2018)
https://doi.org/10.1103/PhysRevA.98.042114
104 M. S. Rudner, M. Levin, and L. S. Levitov, Survival, decay, and topological protection in non-Hermitian quantum transport, arXiv: 1605.07652 (2016)
105 K. Kawabata, S. Higashikawa, Z. Gong, Y. Ashida, and M. Ueda, arXiv: 1804.04676 (2018)
106 K. Kawabata, K. Shiozaki, and M. Ueda, Anomalous helical edge states in a non-Hermitian Chern insulator, Phys. Rev. B98, 165148 (2018)
https://doi.org/10.1103/PhysRevB.98.165148
107 Y. Chen and H. Zhai, Hall conductance of a non-Hermitian Chern insulator, Phys. Rev. B98, 245130 (2018)
https://doi.org/10.1103/PhysRevB.98.245130
108 H. Wang, J. Ruan, and H. Zhang, Non-Hermitian nodalline semimetals with an anomalous bulk-boundary correspondence, Phys. Rev. B99, 075130 (2019)
https://doi.org/10.1103/PhysRevB.99.075130
109 K. Luo, J. Feng, Y. X. Zhao, and R. Yu, Nodal manifolds bounded by exceptional points on non-Hermitian honeycomb lattices and electrical-circuit realizations, arXiv: 1810.09231 (2018)
110 Q. B. Zeng, Y. B. Yang, and Y. Xu, Topological non-Hermitian quasicrystals, arXiv: 1901.08060 (2019)
111 N. Goldman, J. C. Budich, and P. Zoller, Topological quantum matter with ultracold gases in optical lattices, Nat. Phys.12(7), 639 (2016)
112 D. W. Zhang, Y. Q. Zhu, Y. X. Zhao, H. Yan, and S. L. Zhu, Topological quantum matter with cold atoms, arXiv: 1810.09228 (2018)
113 M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, Direct measurement of the Zak phase in topological Bloch bands, Nat. Phys.9, 795 (2013)
114 M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices, Phys. Rev.Lett.111(18), 185301 (2013)
https://doi.org/10.1103/PhysRevLett.111.185301
115 H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Burton, and W. Ketterle, Realizing the harper Hamiltonian with laser-assisted tunneling in optical lattices, Phys. Rev. Lett.111(18), 185302 (2013)
https://doi.org/10.1103/PhysRevLett.111.185302
116 G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, Experimental realization of the topological Haldane model with ultracold fermions, Nature515(7526), 237 (2014)
https://doi.org/10.1038/nature13915
117 M. Aidelsburger, M. Lohse, C. Schweizer, M. Atala, J. T. Barreiro, S. Nascimbène, N. R. Cooper, I. Bloch, and N. Goldman, Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms, Nat. Phys.11(2), 162 (2015)
118 N. Fläschner, B. Rem, M. Tarnowski, D. Vogel, D. S. Lühmann, K. Sengstock, and C. Weitenberg, Experimental reconstruction of the Berry curvature in a Floquet Bloch band, Science352(6289), 1091 (2016)
https://doi.org/10.1126/science.aad4568
119 Z. Wu, L. Zhang, W. Sun, X. T. Xu, B. Z. Wang, S. C. Ji, Y. J. Deng, S. Chen, X. J. Liu, and J. W. Pan, Realization of two-dimensional spin–orbit coupling for Bose–Einstein condensates, Science354(6308), 83 (2016)
https://doi.org/10.1126/science.aaf6689
120 W. Sun, B. Z. Wang, X. T. Xu, C. R. Yi, L. Zhang, Z. Wu, Y. J. Deng, X. J. Liu, S. Chen, and J. W. Pan, Highly controllable and robust 2D spin–orbit coupling for quantum gases, Phys. Rev. Lett.121(15), 150401 (2018)
https://doi.org/10.1103/PhysRevLett.121.150401
121 S. Nakajima, T. Tomita, S. Taie, T. Ichinose, H. Ozawa, L. Wang, M. Troyer, and Y. Takahashi, Topological Thouless pumping of ultracold fermions, Nat. Phys.12(4), 296 (2016)
122 M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger, and I. Bloch, A Thouless quantum pump with ultracold bosonic atoms in an optical superlattice, Nat. Phys.12(4), 350 (2016)
123 M. Lohse, C. Schweizer, H. M. Price, O. Zilberberg, and I. Bloch, Exploring 4D quantum Hall physics with a 2D topological charge pump, Nature553(7686), 55 (2018)
https://doi.org/10.1038/nature25000
124 P. Soltan-Panahi, J. Struck, P. Hauke, A. Bick, W. Plenkers, G. Meineke, C. Becker, P. Windpassinger, M. Lewenstein, and K. Sengstock, Multi-component quantum gases in spin-dependent hexagonal lattices, Nat. Phys.7(5), 434 (2011)
125 L. Tarruell, D. Greif, T. Uehlinger, G. Jotzu, and T. Esslinger, Creating, moving and merging Dirac points with a Fermi gas in a tunable honeycomb lattice, Nature483(7389), 302 (2012)
https://doi.org/10.1038/nature10871
126 L. Huang, Z. Meng, P. Wang, P. Peng, S. L. Zhang, L. Chen, D. Li, Q. Zhou, and J. Zhang, Experimental realization of two-dimensional synthetic spin–orbit coupling in ultracold Fermi gases, Nat. Phys.12(6), 540 (2016)
127 Z. Meng, L. Huang, P. Peng, D. Li, L. C. Chen, Y. Xu, C. Zhang, P. Wang, and J. Zhang, Experimental observation of a topological band gap opening in ultracold Fermi gases with two-dimensional spin–orbit coupling, Phys. Rev. Lett.117(23), 235304 (2016)
https://doi.org/10.1103/PhysRevLett.117.235304
128 X. T. Xu, C. R. Yi, B. Z. Wang, W. Sun, Y. Deng, X. J. Liu, S. Chen, and J. W. Pan, Precision mapping the topological bands of 2D spin–orbit coupling with microwave spin-injection spectroscopy, Sci. Bull.63(22), 1464 (2018)
https://doi.org/10.1016/j.scib.2018.11.002
129 S. Sugawa, F. S. Carcoba, A. R. Perry, Y. C. Yue, and I. B. Spielman, Second Chern number of a quantum-simulated non-Abelian Yang monopole, Science360(6396), 1429 (2018)
https://doi.org/10.1126/science.aam9031
130 B. Song, C. He, S. Niu, L. Zhang, Z. J. Ren, X.J. Liu, and G. B. Jo, Observation of nodal-line semimetal with ultracold fermions in an optical lattice, arXiv: 1808.07428 (2018)
131 H. Hu, L. Dong, Y. Cao, H. Pu, and X. J. Liu, Gapless topological Fulde–Ferrell superfluidity induced by an inplane Zeeman field, Phys. Rev. A90(3), 033624 (2014)
https://doi.org/10.1103/PhysRevA.90.033624
132 K. J. Seo, L. Han, and C. A. R. Sá de Melo, Emergence of Majorana and Dirac particles in ultracold fermions via tunable interactions, spin–orbit effects, and Zeeman fields, Phys. Rev. Lett.109(10), 105303 (2012)
https://doi.org/10.1103/PhysRevLett.109.105303
133 Y. J. Lin, K. J. Garcia, and I. B. Spielman, Spin–orbitcoupled Bose–Einstein condensates, Nature471(7336), 83 (2011)
https://doi.org/10.1038/nature09887
134 J. Y. Zhang, S. C. Ji, Z. Chen, L. Zhang, Z. D. Du, B. Yan, G. S. Pan, B. Zhao, Y. J. Deng, H. Zhai, S. Chen, and J. W. Pan, Collective dipole oscillations of a spin–orbit coupled Bose–Einstein condensate, Phys. Rev. Lett.109(11), 115301 (2012)
https://doi.org/10.1103/PhysRevLett.109.115301
135 P. Wang, Z. Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H. Zhai, and J. Zhang, Spin–orbit coupled degenerate Fermi gases, Phys. Rev. Lett.109(9), 095301 (2012)
https://doi.org/10.1103/PhysRevLett.109.095301
136 L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah, W. S. Bakr, and M. W. Zwierlein, Spin-injection spectroscopy of a spin–orbit coupled Fermi Gas, Phys. Rev. Lett.109(9), 095302 (2012)
https://doi.org/10.1103/PhysRevLett.109.095302
137 C. L. Qu, C. Hamner, M. Gong, C. W. Zhang, and P. Engels, Observation of Zitterbewegung in a spin–orbitcoupled Bose–Einstein condensate, Phys. Rev. A88, 021604(R) (2013)
138 A. J. Olson, S.J. Wang, R. J. Niffenegger, C.H. Li, C. H. Greene, and Y. P. Chen, Tunable Landau–Zener transitions in a spin–orbit-coupled Bose–Einstein condensate, Phys. Rev. A90(1), 013616 (2014)
https://doi.org/10.1103/PhysRevA.90.013616
139 X. Luo, L. Wu, J. Chen, Q. Guan, K. Gao, Z.F. Xu, L. You, and R. Wang, Tunable atomic spin–orbit coupling synthesized with a modulating gradient magnetic field, Sci. Rep.6(1), 18983 (2016)
https://doi.org/10.1038/srep18983
140 N. Q. Burdick, Y. J. Tang, and B. L. Lev, Long-lived spin–orbit-coupled degenerate dipolar Fermi gas, Phys. Rev. X6(3), 031022 (2016)
https://doi.org/10.1103/PhysRevX.6.031022
141 B. Song, C. He, S. Zhang, E. Hajiyev, W. Huang, X.-J. Liu, and G.-B. Jo, Spin–orbit-coupled two-electron Fermi gases of ytterbium atoms, Phys. Rev. A 94, 061604(R) (2016)
142 J. Li, W. Huang, B. Shteynas, S. Burchesky, F. Ç. Top, E. Su, J. Lee, A. O. Jamison, and W. Ketterle, Spin–orbit coupling and spin textures in optical superlattices, Phys. Rev. Lett.117(18), 185301 (2016)
https://doi.org/10.1103/PhysRevLett.117.185301
143 S. Kolkowitz, S. L. Bromley, T. Bothwell, M. L. Wall, G. E. Marti, A. P. Koller, X. Zhang, A. M. Rey, and J. Ye, Spin–orbit-coupled fermions in an optical lattice clock, Nature542(7639), 66 (2017)
https://doi.org/10.1038/nature20811
144 J. R. Li, J. Lee, W. Huang, S. Burchesky, B. Shteynas, F. Ç. Top, A. O. Jamison, and W. Ketterle, A stripe phase with supersolid properties in spin–orbit-coupled Bose–Einstein condensates, Nature543(7643), 91 (2017)
https://doi.org/10.1038/nature21431
145 B. Song, L. Zhang, C. He, T. F. J. Poon, E. Hajiyev, S. Zhang, X. J. Liu, and G. B. Jo, Observation of symmetryprotected topological band with ultracold fermions, Sci. Adv.4(2), 4748 (2018)
https://doi.org/10.1126/sciadv.aao4748
146 V. Galitski and I. B. Spielman, Spin–orbit coupling in quantum gases, Nature494(7435), 49 (2013)
https://doi.org/10.1038/nature11841
147 H. Zhai, Degenerate quantum gases with spin–orbit coupling: A review, Rep. Prog. Phys.78(2), 026001 (2015)
https://doi.org/10.1088/0034-4885/78/2/026001
148 L. Zhang and X.J. Liu, Spin–orbit coupling and topological phases for ultracold atoms, arXiv: 1806.05628 (2018)
149 Y. J. Wu, W. Y. Zhou, and S. P. Kou, Bogoliubov excitations in the Bose–Hubbard extension of a Weyl semimetal, Phys. Rev. A95(2), 023620 (2017)
https://doi.org/10.1103/PhysRevA.95.023620
150 B. Huang and X. Yang, Mott insulator–superfluid phase transition in two-band Bose–Hubbard models with gapless nodal lines, J. Phys. At. Mol. Opt. Phys.51(1), 015302 (2018)
https://doi.org/10.1088/1361-6455/aa9787
151 W. Y. Zhou, Y. J. Wu, and S. P. Kou, Bogoliubov excitations in a Bose–Hubbard model on a hyperhoneycomb lattice, Chin. Phys. B27(5), 050302 (2018)
https://doi.org/10.1088/1674-1056/27/5/050302
152 C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Feshbach resonances in ultracold gases, Rev. Mod. Phys.82(2), 1225 (2010)
https://doi.org/10.1103/RevModPhys.82.1225
153 B. M. Anderson, G. Juzeliunas, V. M. Galitski, and I. B. Spielman, Synthetic 3D spin–orbit coupling, Phys. Rev. Lett.108(23), 235301 (2012)
https://doi.org/10.1103/PhysRevLett.108.235301
154 T. Dubček, C. J. Kennedy, L. Lu, W. Ketterle, M. Soljačić, and H. Buljan, Weyl points in three-dimensional optical lattices: Synthetic magnetic monopoles in momentum space, Phys. Rev. Lett.114(22), 225301 (2015)
https://doi.org/10.1103/PhysRevLett.114.225301
155 Y. Xu and C. Zhang, Dirac and Weyl rings in threedimensional cold-atom optical lattices, Phys. Rev. A93(6), 063606 (2016)
https://doi.org/10.1103/PhysRevA.93.063606
156 D. W. Zhang, Y. X. Zhao, R. B. Liu, Z. Y. Xue, S. L. Zhu, and Z. D. Wang, Quantum simulation of exotic PT-invariant topological nodal loop bands with ultracold atoms in an optical lattice, Phys. Rev. A93(4), 043617 (2016)
https://doi.org/10.1103/PhysRevA.93.043617
157 A. M. Turner and A. Vishwanath, Beyond band insulators: Topology of semimetals and interacting phases, Topol. Insul.6, 293 (2013)
https://doi.org/10.1016/B978-0-444-63314-9.00011-1
158 P. Hosur and X. Qi, Recent developments in transport phenomena in Weyl semimetals, C. R. Phys.14(9–10), 857 (2013)
https://doi.org/10.1016/j.crhy.2013.10.010
159 O. Vafek and A. Vishwanath, Dirac fermions in solids: From high-Tc cuprates and graphene to topological insulators and Weyl semimetals, Annu. Rev. Condens. Matter Phys.5(1), 83 (2014)
https://doi.org/10.1146/annurev-conmatphys-031113-133841
160 D. E. Kharzeev, The Chiral Magnetic Effect and anomaly-induced transport, Prog. Part. Nucl. Phys.75, 133 (2014)
https://doi.org/10.1016/j.ppnp.2014.01.002
161 T. O. Wehling, A. M. Black-Schaffer, and A. V. Balatsky, Dirac materials, Adv. Phys.63(1), 1 (2014)
https://doi.org/10.1080/00018732.2014.927109
162 W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Balents, Correlated quantum phenomena in the strong spin–orbit regime, Annu. Rev. Condens. Matter Phys.5(1), 57 (2014)
https://doi.org/10.1146/annurev-conmatphys-020911-125138
163 A. Burkov, Chiral anomaly and transport in Weyl metals, J. Phys.: Condens. Matter27(11), 113201 (2015)
https://doi.org/10.1088/0953-8984/27/11/113201
164 A. P. Schnyder and P. M. R. Brydon, Topological surface states in nodal superconductors, J. Phys.: Condens. Matter27(24), 243201 (2015)
https://doi.org/10.1088/0953-8984/27/24/243201
165 M. Z. Hasan, S. Y. Xu, and G. Bian, Topological insulators, topological superconductors and Weyl fermion semimetals: Discoveries, perspectives and outlooks, Phys. Scr.T164, 014001 (2015)
https://doi.org/10.1088/0031-8949/2015/T164/014001
166 E. Witten, Three lectures on topological phases of matter, Riv. Nuovo Cim.39, 313 (2016)
167 H. Weng, X. Dai, and Z. Fang, Topological semimetals predicted from first-principles calculations, J. Phys.: Condens. Matter28(30), 303001 (2016)
https://doi.org/10.1088/0953-8984/28/30/303001
168 A. Bansil, H. Lin, and T. Das, Topological band theory, Rev. Mod. Phys.88(2), 021004 (2016)
https://doi.org/10.1103/RevModPhys.88.021004
169 A. A. Burkov, Topological semimetals, Nat. Mater.15(11), 1145 (2016)
https://doi.org/10.1038/nmat4788
170 S. Jia, S. Y. Xu, and M. Z. Hasan, Weyl semimetals, Fermi arcs and chiral anomalies, Nat. Mater.15(11), 1140 (2016)
https://doi.org/10.1038/nmat4787
171 B. Yan and C. Felser, Topological materials: Weyl semimetals, Annu. Rev. Condens. Matter Phys.8(1), 337 (2017)
https://doi.org/10.1146/annurev-conmatphys-031016-025458
172 M. Z. Hasan, S. Y. Xu, I. Belopolski, and S. M. Huang, Discovery of Weyl fermion semimetals and topological Fermi arc states, Annu. Rev. Condens. Matter Phys.8(1), 289 (2017)
https://doi.org/10.1146/annurev-conmatphys-031016-025225
173 L. Šmejkal, T. Jungwirth, and J. Sinova, Route towards Dirac and Weyl antiferromagnetic spintronics, Phys. Status Solidi Rapid Res. Lett.11(4), 1770317 (2017)
https://doi.org/10.1002/pssr.201770317
174 A. Burkov, Weyl metals, Annu. Rev. Condens. Matter Phys.9(1), 359 (2018)
https://doi.org/10.1146/annurev-conmatphys-033117-054129
175 S. V. Syzranov and L. Radzihovsky, High-dimensional disorder-driven phenomena in Weyl semimetals, semiconductors, and related systems, Ann. Rev. Condens. Matter Phys.9(1), 35 (2018)
https://doi.org/10.1146/annurev-conmatphys-033117-054037
176 N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyl and Dirac semimetals in three-dimensional solids, Rev. Mod. Phys.90(1), 015001 (2018)
https://doi.org/10.1103/RevModPhys.90.015001
177 T. Meng and L. Balents, Weyl superconductors, Phys. Rev. B86(5), 054504 (2012)
https://doi.org/10.1103/PhysRevB.86.054504
178 G. B. Halasz and L. Balents, Time-reversal invariant realization of the Weyl semimetal phase, Phys. Rev. B85(3), 035103 (2012)
https://doi.org/10.1103/PhysRevB.85.035103
179 M. O. Goerbig, J. N. Fuchs, G. Montambaux, and F. Piéchon, Tilted anisotropic Dirac cones in quinoid-type graphene and _-(BEDT-TTF)2I3, Phys. Rev. B78(4), 045415 (2008)
https://doi.org/10.1103/PhysRevB.78.045415
180 J. Yang and N. Nagaosa, Classification of stable threedimensional Dirac semimetals with nontrivial topology, Nat. Commun.5(1), 4898 (2014)
https://doi.org/10.1038/ncomms5898
181 H. Huang, S. Zhou, and W. Duan, Type-II Dirac fermions in the PtSe2 class of transition metal dichalcogenides, Phys. Rev. B94(12), 121117 (2016)
https://doi.org/10.1103/PhysRevB.94.121117
182 M. Yan, H. Huang, K. Zhang, E. Wang, W. Yao, K. Deng, G. Wan, H. Zhang, M. Arita, H. Yang, Z. Sun, H. Yao, Y. Wu, S. Fan, W. Duan, and S. Zhou, Lorentz-violating type-II Dirac fermions in transition metal dichalcogenide PtTe2, Nat. Commun.8(1), 257 (2017)
https://doi.org/10.1038/s41467-017-00280-6
183 G. G. Pyrialakos, N. S. Nye, N. V. Kantartzis, and D. N. Christodoulides, Emergence of type-II Dirac points in graphynelike photonic lattices, Phys. Rev. Lett.119(11), 113901 (2017)
https://doi.org/10.1103/PhysRevLett.119.113901
184 H. X. Wang, Y. Chen, Z. H. Hang, H. Y. Kee, and J. H. Jiang, Type-II Dirac photons, Quantum Materials2(1), 54 (2017)
https://doi.org/10.1038/s41535-017-0058-z
185 H. J. Noh, J. Jeong, E. J. Cho, K. Kim, B. I. Min, and B. G. Park, Experimental realization of type-II Dirac fermions in a PdTe2 superconductor, Phys. Rev. Lett.119(1), 016401 (2017)
https://doi.org/10.1103/PhysRevLett.119.016401
186 F. C. Fei, X. Y. Bo, R. Wang, B. Wu, J. Jiang, D. Z. Fu, M. Gao, H. Zheng, Y. L. Chen, X. F. Wang, H. J. Bu, F. Q. Song, X. G. Wan, B. G. Wang, and G. H. Wang, Nontrivial Berry phase and type-II Dirac transport in the layered material PdTe2, Phys. Rev. B96, 041201(R) (2017)
187 Y. X. Zhao, A. P. Schnyder, and Z. D. Wang, Unified theory of PT and CP invariant topological metals and nodal superconductors, Phys. Rev. Lett.116(15), 156402 (2016)
https://doi.org/10.1103/PhysRevLett.116.156402
188 S. Q. Shen, Topological Insulators: Dirac equation in Condensed Matters, Springer, Berlin, 2012
https://doi.org/10.1007/978-3-642-32858-9
189 N. B. Kopnin, T. T. Heikkilä, and G. E. Volovik, Hightemperature surface superconductivity in topological flatband systems, Phys. Rev. B83, 220503(R) (2011)
190 S. Li, Z.M. Yu, Y. Liu, S. Guan, S.S. Wang, X. Zhang, Y. Yao, and S. A. Yang, Type-II nodal loops: Theory and material realization, Phys. Rev. B96(8), 081106 (2017)
https://doi.org/10.1103/PhysRevB.96.081106
191 J. He, X. Kong, W. Wang, and S. P. Kou, Type II nodal line semimetal, arXiv: 1709.08287 (2017)
192 T. R. Chang, I. Pletikosic, T. Kong, G. Bian, A. Huang, J. Denlinger, S. K. Kushwaha, B. Sinkovic, H.-T. Jeng, T. Valla, W. W. Xie, and R. J. Cava, Realization of a type-II nodal-line semimetal in Mg3Bi2, arXiv: 1711.09167 (2017)
193 N. Moiseyev, Non-Hermitian Quantum Mechanics, Cambridge University Press, Cambridge, 2011
https://doi.org/10.1017/CBO9780511976186
194 M. V. Berry, Physics of non-Hermitian degeneracies, Czech. J. Phys.54(10), 1039 (2004)
https://doi.org/10.1023/B:CJOP.0000044002.05657.04
195 I. Rotter, A non-Hermitian Hamilton operator and the physics of open quantum systems, J. Phys. A Math. Theor.42(15), 153001 (2009)
https://doi.org/10.1088/1751-8113/42/15/153001
196 W. D. Heiss, The physics of exceptional points, J. Phys. A Math. Theor.45(44), 444016 (2012)
https://doi.org/10.1088/1751-8113/45/44/444016
197 C. M. Bender and S. Boettcher, Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett.80(24), 5243 (1998)
https://doi.org/10.1103/PhysRevLett.80.5243
198 M. Kleman and O. D. Lavrentovich, Soft Matter Physics: An Introduction, XXV, 637 pp. Springer, Berlin Heidelberg New York Tokyo 2003, Hardcover EUR 39.95 (excl. VAT)
199 G. Sundaram and Q. Niu, Wave-packet dynamics in slowly perturbed crystals: Gradient corrections and Berry-phase effects, Phys. Rev. B59(23), 14915 (1999)
https://doi.org/10.1103/PhysRevB.59.14915
200 Z. Zheng, M. Gong, X. Zou, C. Zhang, and G. Guo, Route to observable Fulde–Ferrell–Larkin–Ovchinnikov phases in three-dimensional spin–orbit-coupled degenerate Fermi gases, Phys. Rev. A87, 031602(R) (2013)
201 F. Wu, G. C. Guo, W. Zhang, and W. Yi, Unconventional superfluid in a two-dimensional Fermi gas with anisotropic spin–orbit coupling and Zeeman fields, Phys. Rev. Lett.110(11), 110401 (2013)
https://doi.org/10.1103/PhysRevLett.110.110401
202 Y. Xu, C. Qu, M. Gong, and C. Zhang, Competing superfluid orders in spin–orbit-coupled fermionic cold-atom optical lattices, Phys. Rev. A89(1), 013607 (2014)
https://doi.org/10.1103/PhysRevA.89.013607
203 P. Fulde and R. A. Ferrell, Superconductivity in a strong spin-exchange field, Phys. Rev.135(3A), A550 (1964)
https://doi.org/10.1103/PhysRev.135.A550
204 A. I. Larkin and Y. N. Ovchinnikov, Nonuniform state of superconductors, Zh. Eksp. Teor. Fiz.47, 1136 (1964) (Sov. Phys. JETP20, 762 (1965))
205 D. Xiao, M. C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys.82(3), 1959 (2010)
https://doi.org/10.1103/RevModPhys.82.1959
206 Y. Cao, S. H. Zou, X. J. Liu, S. Yi, G. L. Long, and H. Hu, Gapless topological Fulde–Ferrell superfluidity in spin–orbit coupled Fermi gases, Phys. Rev. Lett.113(11), 115302 (2014)
https://doi.org/10.1103/PhysRevLett.113.115302
207 Y. Xu and C. Zhang, Berezinskii–Kosterlitz–Thouless phase transition in 2D spin–orbit-coupled Fulde–Ferrell superfluids, Phys. Rev. Lett.114(11), 110401 (2015)
https://doi.org/10.1103/PhysRevLett.114.110401
208 Y. Cao, X. J. Liu, L. He, G. L. Long, and H. Hu, Superfluid density and Berezinskii–Kosterlitz–Thouless transition of a spin–orbit-coupled Fulde–Ferrell superfluid, Phys. Rev. A91(2), 023609 (2015)
https://doi.org/10.1103/PhysRevA.91.023609
209 X. Z. Ying and A. Kamenev, Symmetry-protected topological metals, Phys. Rev. Lett.121(8), 086810 (2018)
https://doi.org/10.1103/PhysRevLett.121.086810
210 W. Y. He, D. H. Xu, B. T. Zhou, Q. Zhou, and K. T. Law, From nodal-ring topological superfluids to spiral Majorana modes in cold atomic systems, Phys. Rev. A97(4), 043618 (2018)
https://doi.org/10.1103/PhysRevA.97.043618
211 M. Z. Hasan and C. L. Kane, Topological insulators, Rev. Mod. Phys.82(4), 3045 (2010)
https://doi.org/10.1103/RevModPhys.82.3045
212 X. L. Qi and S. C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys.83(4), 1057 (2011)
https://doi.org/10.1103/RevModPhys.83.1057
213 J. Alicea, New directions in the pursuit of Majorana fermions in solid state systems, Rep. Prog. Phys.75(7), 076501 (2012)
https://doi.org/10.1088/0034-4885/75/7/076501
214 X. J. Liu, K. T. Law, and T. K. Ng, Realization of 2D spin–orbit interaction and exotic topological orders in cold atoms, Phys. Rev. Lett.112(8), 086401 (2014)
https://doi.org/10.1103/PhysRevLett.112.086401
215 B. J. Wieder, Y. Kim, A. M. Rappe, and C. L. Kane, Double Dirac semimetals in three dimensions, Phys. Rev. Lett.116(18), 186402 (2016)
https://doi.org/10.1103/PhysRevLett.116.186402
216 B. Bradlyn, J. Cano, Z. Wang, M. G. Vergniory, C. Felser, R. J. Cava, and B. A. Bernevig, Beyond Dirac and Weyl fermions: Unconventional quasiparticles in conventional crystals, Science353(6299), 6299 (2016)
https://doi.org/10.1126/science.aaf5037
217 Y. Q. Zhu, D. W. Zhang, H. Yan, D. Y. Xing, and S. L. Zhu, Emergent pseudospin-1 Maxwell fermions with a threefold degeneracy in optical lattices, Phys. Rev. A96(3), 033634 (2017)
https://doi.org/10.1103/PhysRevA.96.033634
218 H. Hu, J. Hou, F. Zhang, and C. Zhang, Topological triply degenerate points induced by spin-tensormomentum couplings, Phys. Rev. Lett.120(24), 240401 (2018)
https://doi.org/10.1103/PhysRevLett.120.240401
219 I. C. Fulga, L. Fallani, and M. Burrello, Geometrically protected triple-point crossings in an optical lattice, Phys. Rev. B97, 121402(R) (2018)
220 X. Y. Mai, Y. Q. Zhu, Z. Li, D. W. Zhang, and S. L. Zhu, Topological metal bands with double-triple-point fermions in optical lattices, arXiv: 1810.11560 (2018)
221 B. Lian and S. C. Zhang, Five-dimensional generalization of the topological Weyl semimetal, Phys. Rev. B94(4), 041105 (2016)
https://doi.org/10.1103/PhysRevB.94.041105
222 B. Lian and S. C. Zhang, Weyl semimetal and topological phase transition in five dimensions, Phys. Rev. B95(23), 235106 (2017)
https://doi.org/10.1103/PhysRevB.95.235106
223 H. M. Price, O. Zilberberg, T. Ozawa, I. Carusotto, and N. Goldman, Four-dimensional quantum Hall effect with ultracold atoms, Phys. Rev. Lett.115(19), 195303 (2015)
https://doi.org/10.1103/PhysRevLett.115.195303
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed