Fermion dynamical symmetry and strongly-correlated electrons: A comprehensive model of high-temperature superconductivity
Mike Guidry1(), Yang Sun2(), Lian-Ao Wu3(), Cheng-Li Wu4()
1. Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA 2. School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China 3. IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain, and Department of Theoretical Physics and History of Science, Basque Country University (EHU/UPV), Post Office Box 644, 48080 Bilbao, Spain 4. Department of Physics, Chung-Yuan Christian University, Chungli, Taiwan 320, China
We review application of the SU(4) model of strongly-correlated electrons to cuprate and iron-based superconductors. A minimal self-consistent generalization of BCS theory to incorporate antiferromagnetism on an equal footing with pairing and strong Coulomb repulsion is found to account systematically for the major features of high-temperature superconductivity, with microscopic details of the parent compounds entering only parametrically. This provides a systematic procedure to separate essential from peripheral, suggesting that many features exhibited by the high-Tc data set are of interest in their own right but are not central to the superconducting mechanism. More generally, we propose that the surprisingly broad range of conventional and unconventional superconducting and superfluid behavior observed across many fields of physics results from the systematic appearance of similar algebraic structures for the emergent effective Hamiltonians, even though the microscopic Hamiltonians of the corresponding parent states may differ radically from each other.
Corresponding Author(s):
Mike Guidry,Yang Sun,Lian-Ao Wu,Cheng-Li Wu
引用本文:
. [J]. Frontiers of Physics, 2020, 15(4): 43301.
Mike Guidry, Yang Sun, Lian-Ao Wu, Cheng-Li Wu. Fermion dynamical symmetry and strongly-correlated electrons: A comprehensive model of high-temperature superconductivity. Front. Phys. , 2020, 15(4): 43301.
J. G. Bednorz and K. A. Muller, Possible high Tcsuperconductivity in the Ba–La–Cu–O system, Z. Phys. B 64(2), 189 (1986) https://doi.org/10.1007/BF01303701
2
See: D. A. Bonn, Are high-temperature superconductors exotic? Nat. Phys. 2, 159 (2006); M. R. Norman and C. Pépin, The electronic nature of high temperature cuprate superconductors, Rep. Prog. Phys. 66(10), 1547 (2003) (and Refs. therein) https://doi.org/10.1088/0034-4885/66/10/R01
3
Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, Iron-based layered superconductor LaO1–xFxFeAs (x= 0.05–0.12) with Tc= 26 K, J. Am. Chem. Soc. 130(11), 3296 (2008) https://doi.org/10.1021/ja800073m
4
Y. Sun, M. W. Guidry, and C. L. Wu, A new family of high Tccompounds—Stepping stones toward understanding unconventional superconductivity, Chin. Sci. Bull. 53, 1617 (2009) https://doi.org/10.1007/s11434-008-0250-4
5
J. Guo, S. Jin, G. Wang, S. Wang, K. Zhu, T. Zhou, M. He, and X. Chen, Superconductivity in the iron selenide KxFe2Se2, Phys. Rev. B 82(18), 180520 (2010) https://doi.org/10.1103/PhysRevB.82.180520
6
D. C. Johnston, The puzzle of high temperature superconductivity in layered iron pnictides and chalcogenides, Adv. Phys. 59(6), 803 (2010) https://doi.org/10.1080/00018732.2010.513480
7
J. Paglione and R. L. Greene, High-temperature superconductivity in iron-based materials, Nat. Phys. 6(9), 645 (2010) https://doi.org/10.1038/nphys1759
8
D. Mou, L. Zhao, and X. Zhou, Structural, magnetic and electronic properties of the iron-chalcogenide AxFe2–ySe2 (A=K, Cs, Rb, and Tl, etc.) superconductors, Front. Phys. 6(4), 410 (2011) https://doi.org/10.1007/s11467-011-0229-5
9
H. Oh, J. Moon, D. Shin, C. Moon, and H. J. Choi, Brief review on iron-based superconductors: Are there clues for unconventional superconductivity? Progress in Superconductivity 13, 65 (2011)
For example, see M. R. Norman, The challenge of unconventional superconductivity, Science 332(6026), 196 (2011) (and references cited there) https://doi.org/10.1126/science.1200181
13
D. Jérome, Organic superconductors: When correlations and magnetism walk in, arXiv: 1201.5796 (2012)
14
See, for example, P. Ring, and P. Schuck, The Nuclear Many-Body Problem, Springer–Verlag, 1980
15
D. Page, M. Prakash, J. M. Lattimer, and A. W. Steiner, Superfluid Neutrons in the core of the neutron star in cassiopeia A, arXiv: 1110.5116 (2011) https://doi.org/10.22323/1.142.0005
16
T. Noda, M. Hashimoto, N. Yasutake, T. Maruyama, T. Tatsumi, and M. Fujimoto, Cooling of compact stars with color superconducting phase in quark hadron mixed phase, Astrophys. J. 765(1), 1 (2012) https://doi.org/10.1088/0004-637X/765/1/1
17
M. G. Alford, A. Schmitt, K. Rajagopal, and T. Schäfer, Color superconductivity in dense quark matter, Rev. Mod. Phys. 80(4), 1455 (2008) https://doi.org/10.1103/RevModPhys.80.1455
R. Bijker, F. Iachello, and A. Leviatan, Algebraic models of hadron structure (I): Nonstrange baryons, Ann. Phys. 236(1), 69 (1994) https://doi.org/10.1006/aphy.1994.1108
21
F. Iachello and R. D. Levine, Algebraic Theory of Molecules, Oxford University Press, Oxford, 1995
For a review, see, Ref. [6] and P. J. Hirschfeld, M. M. Korshunov, and I. I. Mazin, Gap symmetry and structure of Fe-based superconductors, Rep. Prog. Phys. 74, 124508 (2011) https://doi.org/10.1088/0034-4885/74/12/124508
26
T. Timusk and B. Statt, The pseudogap in hightemperature superconductors: An experimental survey, Rep. Prog. Phys. 62(1), 61 (1999) https://doi.org/10.1088/0034-4885/62/1/002
G. Sheet, M. Mehta, D. A. Dikin, S. Lee, C. W. Bark, J. Jiang, J. D. Weiss, E. E. Hellstrom, M. S. Rzchowski, C. B. Eom, and V. Chandrasekhar, Phaseincoherent superconducting pairs in the normal state of Ba(Fe1–xCox)2As2, Phys. Rev. Lett. 105(16), 167003 (2010) https://doi.org/10.1103/PhysRevLett.105.167003
29
T. Mertelj, V. V. Kabanov, C. Gadermaier, N. D. Zhigadlo, S. Katrych, J. Karpinski, and D. Mihailovic, Distinct pseudogap and quasiparticle relaxation dynamics in the superconducting state of nearly optimally doped Sm- FeAsO0.8F0.2 single crystals, Phys. Rev. Lett. 102(11), 117002 (2009) https://doi.org/10.1103/PhysRevLett.102.117002
30
K. Ahilan, F. L. Ning, T. Imai, A. S. Sefat, R. Jin, M. A. McGuire, B. C. Sales, and D. Mandrus, 19F NMR investigation of the iron pnictide superconductor LaFeAsO0.89F0.11, Phys. Rev. B 78, 100501(R) (2008) https://doi.org/10.1103/PhysRevB.78.100501
31
T. Sato, K. Nakayama, Y. Sekiba, T. Arakane, K. Terashima, S. Souma, T. Takahashi, Y. Kamihara, M. Hirano, and H. Hosono, Doping dependence of pseudogap in LaFeAsO1–xFx, J. Phys. Soc. Jpn. C 77(Suppl.), 65 (2008) https://doi.org/10.1143/JPSJS.77SC.65
32
F. Ning, K. Ahilan, T. Imai, A. S. Sefat, R. Jin, M. A. McGuire, B. C. Sales, and D. Mandrus, 59Co and 75As NMR investigation of electron-doped high Tc superconductor BaFe1.8Co0.2As2 (Tc= 22 K), J. Phys. Soc. Jpn. 77(10), 103705 (2008) https://doi.org/10.1143/JPSJ.77.103705
33
S. M. Hayden, H. A. Mook, P. Dai, T. G. Perring, and F. Doğan, The structure of the high-energy spin excitations in a high-transition-temperature superconductor, Nature 429(6991), 531 (2004) https://doi.org/10.1038/nature02576
34
J. M. Tranquada, H. Woo, T. G. Perring, H. Goka, G. D. Gu, G. Xu, M. Fujita, and K. Yamada, Quantum magnetic excitations from stripes in copper oxide superconductors, Nature 429(6991), 534 (2004) https://doi.org/10.1038/nature02574
35
J. E. Hoffman, E. W. Hudson, K. M. Lang, V. Madhavan, H. Eisaki, S. Uchida, and J. C. Davis, A four unit cell periodic pattern of quasi-particle states surrounding vortex cores in Bi2Sr2CaCu2O8+δ, Science 295(5554), 466 (2002) https://doi.org/10.1126/science.1066974
36
M. Vershinin, S. Misra, S. Ono, Y. Abe, Y. Ando, and A. Yazdani, Local ordering in the pseudogap state of the high-Tcsuperconductor Bi2Sr2CaCu2O8+δ, Science 303(5666), 1995 (2004) https://doi.org/10.1126/science.1093384
37
T. Hanaguri, C. Lupien, Y. Kohsaka, D. H. Lee, M. Azuma, M. Takano, H. Takagi, and J. C. Davis, A “checkerboard” electronic crystal state in lightly holedoped Ca2–xNaxCuO2Cl2, Nature 430(7003), 1001 (2004) https://doi.org/10.1038/nature02861
38
K. McElroy, J. Lee, J. A. Slezak, D. -H. Lee, H. Eisaki, S. Uchida, and J. C. Davis, Atomic-scale sources and mechanism of nanoscale electronic disorder in Bi2Sr2CaCu2O8+δ, Science 309(5737), 1048 (2005) https://doi.org/10.1126/science.1113095
39
V. J. Emery and S. A. Kivelson, Importance of phase fluctuations in superconductors with small superfluid density, Nature 374(6521), 434(1995) https://doi.org/10.1038/374434a0
A. Bohr and B. R. Mottelson, Nuclear Structure, Vols. I and II, W. A. Benjamin, 1969 and 1975
45
However the standard methodologies in the elementary particle physics case differ from the ones used here, partially because a relativistic quantum field theory is required there but non-relativistic fields are adequate for the present discussion. An accessible introduction to non- Abelian gauge fields may be found in Gauge Field Theories, An Introduction with Applications, Mike Guidry, Wiley, 1992.
46
M. W. Guidry, L. A. Wu, Y. Sun, and C. L. Wu, SU(4) model of high-temperature superconductivity and antiferromagnetism, Phys. Rev. B 63(13), 134516(2001) https://doi.org/10.1103/PhysRevB.63.134516
47
M. W. Guidry, A fermion dynamical symmetry model of high-temperature superconductivity and antiferromagnetic order, Rev. Mex. Fís. 45(S2), 132 (1999)
48
L. A. Wu, M. W. Guidry, Y. Sun, and C. L. Wu, SO(5) as a critical dynamical symmetry in the SU(4) model of high-temperature superconductivity, Phys. Rev. B 67(1), 014515 (2003) https://doi.org/10.1103/PhysRevB.67.014515
49
M. W. Guidry, Y. Sun, and C. L. Wu, Mott insulators, no-double-occupancy, and non-Abelian superconductivity, Phys. Rev. B 70(18), 184501 (2004) https://doi.org/10.1103/PhysRevB.70.184501
50
Y. Sun, M. W. Guidry, and C. L. Wu, Temperaturedependent gap equations and their solutions in the SU(4) model of high-temperature superconductivity, Phys. Rev. B 73(13), 134519 (2006) https://doi.org/10.1103/PhysRevB.73.134519
51
Y. Sun, M. W. Guidry, and C. L. Wu, Pairing Gaps, Pseudogaps, and Phase Diagrams for Cuprate Superconductors, Phys. Rev. B 75(13), 134511 (2007) https://doi.org/10.1103/PhysRevB.75.134511
52
Y. Sun, M. W. Guidry, and C. L. Wu, k-dependent SU(4) model of high-temperature superconductivity and its coherent state solutions, Phys. Rev. B 78(17), 174524 (2008) https://doi.org/10.1103/PhysRevB.78.174524
53
M. W. Guidry, Y. Sun, and C. L. Wu, Instabilities of doped Mott insulators and the properties of hightemperature superconductors, published in Nuclei and Mesoscopic Physics, p. 160, P. Danielewicz, P. Piecuch, and V. Zelevinsky (Eds.), AIP Conference Proceedings, 2008. https://doi.org/10.1063/1.2915590
54
M. W. Guidry, Y. Sun, and C. L. Wu, A unified description of cuprate and iron arsenide superconductors, Front. Phys. China 4(4), 433 (2009) https://doi.org/10.1007/s11467-009-0068-9
55
M. W. Guidry, Y. Sun, and C. L. Wu, Strong anisotropy of cupratepseudogap correlations: Implications for Fermi arcs and Fermi pockets, New J. Phys. 11(12), 123023 (2009) https://doi.org/10.1088/1367-2630/11/12/123023
56
M. Guidry, Y. Sun, and C. L. Wu, Generalizing the Cooper pair instability to doped Mott insulators, Front. Phys. China 5(2), 171 (2010) https://doi.org/10.1007/s11467-010-0006-x
57
M. W. Guidry, Y. Sun, and C. L. Wu, Inhomogeneity, dynamical symmetry, and complexity in high-temperature superconductors: Reconciling a universal phase diagram with rich local disorder, Chin. Sci. Bull. 56(4–5), 367 (2011) https://doi.org/10.1007/s11434-010-4282-1
58
The particle–hole symmetry intrinsic to these models does not mean that hole-doped and electron-doped compounds are expected to behave in the same manner. Although the operators and basis states of the model are particle–hole symmetric, the interactions entering the effective Hamiltonian would not be expected to be the same for holedoped and particle-doped compounds. Thus, the physical properties of hole-doped and electron-doped compounds could differ substantially.
59
We employ an isomorphism between the groups SU(4) and SO(6) to label irreducible representations using SO(6) quantum numbers. The representation structure and relationship of SU(4) and SO(6) is discussed in: J. N. Ginocchio, Ann. Phys. 126, 234 (1980)
60
Groups generally may have more than one Casimir invariant. We shall use the term “Casimir”to refer loosely to the lowest-order such invariants (which are generally quadratic in the group generators). In the context of the present discussion, quadratic Casimirs are associated with 2-body interactions at the microscopic level. Higher-order Casimirs are then generally associated with 3-body and higher interactions. The restriction of our Hamiltonians to polynomials of order 2 in the Casimirs is then a physical restriction to consideration of only 1-body and 2-body interactions.
W. M. Zhang, C. L. Wu, D. H. Feng, J. N. Ginocchio, and M. W. Guidry, Geometrical structure and critical phenomena in the fermion dynamical symmetry model: Sp(6), Phys. Rev. C 38(3), 1475 (1988) https://doi.org/10.1103/PhysRevC.38.1475
63
W. M. Zhang, D. H. Feng, C. L. Wu, H. Wu, and J. N. Ginocchio, Symmetry constrained Hartree–Fock– Bogoliubov theory with applications to the fermion dynamical symmetry model, Nucl. Phys. A 505(1), 7 (1989) https://doi.org/10.1016/0375-9474(89)90414-4
64
W. M. Zhang, D. H. Feng, and J. N. Ginocchio, Geometrical interpretation of SO(7): A critical dynamical symmetry, Phys. Rev. Lett. 59(18), 2032(1987) https://doi.org/10.1103/PhysRevLett.59.2032
65
W. M. Zhang, D. H. Feng, and J. N. Ginocchio, Geometrical structure and critical phenomena in the fermion dynamical symmetry model: SO(8), Phys. Rev. C 37(3), 1281 (1988) https://doi.org/10.1103/PhysRevC.37.1281
J. R. Klauder, Continuous-representation theory: Postulates of continuous-representation theory, J. Math. Phys. 4, 1055 (1963); Continuous‐representation theory (II): Postulates of continuous‐representation theory, J. Math. Phys. 4, 1058 (1963) https://doi.org/10.1063/1.1704034
70
Thus the most general SU(4) coherent state depends on eight real variables. The reduction of the coherent state parameters to only two in Eq. (28) follows from requiring time reversal symmetry and assuming conservation of spin projection Szfor the wave function.
71
B. R. Judd, Operator Techniques in Atomic Spectroscopy, McGraw-Hill, 1963
72
The competition between dynamical symmetries governing the transition between spherical and deformed nuclei is discussed in §4.5 (in particular, §4.5.4) of Ref. [18].
73
M. W. Guidry and Y. Sun, Superconductivity and superfluidity as universal emergent phenomena in diverse fermionic systems, Front. Phys. 10(4), 1 (2015) https://doi.org/10.1007/s11467-015-0502-0
74
M. W. Guidry, Universality of emergent states in diverse physical systems, AIP Conf. Proc. 1912, 020005 (2017) https://doi.org/10.1063/1.5016130
75
L. A. Wu and M. W. Guidry, The ground state of monolayer graphene in a strong magnetic field, Sci. Rep. 6(1), 22423 (2016) https://doi.org/10.1038/srep22423
76
L. A. Wu, M. Murphy, and M. W. Guidry, SO(8) fermion dynamical symmetry and strongly correlated quantum Hall states in monolayer graphene, Phys. Rev. B 95(11), 115117 (2017) https://doi.org/10.1103/PhysRevB.95.115117
77
M. W. Guidry, SO(8) fermion dynamical symmetry and quantum hall states for graphene in a strong magnetic field, Fortschr. Phys. 65(6–8), 1600057 (2017) https://doi.org/10.1002/prop.201600057
78
J. L. Tallon, J. W. Loram, J. R. Cooper, C. Panagopoulos, and C. Bernhard, Superfluid density in cuprate high-Tcsuperconductors: A new paradigm, Phys. Rev. B 68(18), 180501 (2003) https://doi.org/10.1103/PhysRevB.68.180501
79
P. Dai, H. A. Mook, S. M. Hayden, G. Aeppli, T. G. Perring, R. D. Hunt, and F. Doğan, The magnetic excitation spectrum and thermodynamics of high-Tcsuperconductors, Science 284(5418), 1344 (1999) https://doi.org/10.1126/science.284.5418.1344
80
J. C. Campuzano, H. Ding, M. R. Norman, H. M. Fretwell, M. Randeria, A. Kaminski, J. Mesot, T. Takeuchi, T. Sato, T. Yokoya, T. Takahashi, T. Mochiku, K. Kadowaki, P. Guptasarma, D. G. Hinks, Z. Konstantinovic, Z. Z. Li, and H. Raffy, Electronic spectra and their relation to the (π, π) collective mode in high-Tcsuperconductors, Phys. Rev. Lett. 83(18), 3709 (1999) https://doi.org/10.1103/PhysRevLett.83.3709
81
Z. A. Xu, N. P. Ong, Y. Wang, T. Kakeshita, and S. Uchida, Vortex-like excitations and the onset of superconducting phase fluctuation in underdoped La2–xSrxCuO4, Nature 406(6795), 486 (2000) https://doi.org/10.1038/35020016
82
N. P. Ong, Y. Wang, S. Ono, Y. Ando, and S. Uchida, Vorticity and the Nernst effect in cuprate superconductors, Ann. Phys. 13(12), 9 (2004) https://doi.org/10.1002/andp.200310034
D. Vaknin, S. K. Sinha, D. E. Moncton, D. C. Johnston, J. M. Newsam, C. R. Safinya, and H. E. King, Antiferromagnetism in La2CuO4–y, Phys. Rev. Lett. 58(26), 2802 (1987) https://doi.org/10.1103/PhysRevLett.58.2802
85
T. S. Nunner, B. M. Anderson, A. Melikyan, and P. J. Hirschfeld, Dopant-modulated pair interaction in cuprate superconductors, Phys. Rev. Lett. 95(17), 177003 (2005) https://doi.org/10.1103/PhysRevLett.95.177003
86
A. C. Fang, L. Capriotti, D. J. Scalapino, S. A. Kivelson, N. Kaneko, M. Greven, and A. Kapitulnik, Gapinhomogeneity-induced electronic states in superconducting Bi2Sr2CaCu2O8+δ, Phys. Rev. Lett. 96(1), 017007 (2006) https://doi.org/10.1103/PhysRevLett.96.017007
87
Y. He, T. S. Nunner, P. J. Hirschfeld, and H. P. Cheng, Local electronic structure of Bi2Sr2CaCu2O8 near oxygen dopants: A window on the high-Tcpairing mechanism, Phys. Rev. Lett. 96(19), 197002 (2006) https://doi.org/10.1103/PhysRevLett.96.197002
88
M. M. Maśka, Z. Sledz, K. Czajka, and M. Mierzejewski, Inhomogeneity-induced enhancement of the pairing interaction in cuprate superconductors, Phys. Rev. Lett. 99(14), 147006 (2007) https://doi.org/10.1103/PhysRevLett.99.147006
89
S. Petit and M. B. Lepetit, Real-space fluctuations of effective exchange integrals in high-Tccuprates, Europhys. Lett. 87(6), 67005 (2009) https://doi.org/10.1209/0295-5075/87/67005
90
K. Foyevtsova, R. Valentı, and P. J. Hirschfeld, Effect of dopant atoms on local superexchange in cuprate superconductors: A perturbative treatment, Phys. Rev. B 79(14), 144424 (2009) https://doi.org/10.1103/PhysRevB.79.144424
91
S. Johnston, F. Vernay, and T. P. Devereaux, Impact of an oxygen dopant in Bi2Sr2CaCu2O8+δ, Europhys. Lett. 86(3), 37007 (2009) https://doi.org/10.1209/0295-5075/86/37007
92
S. Okamoto and T. A. Maier, Microscopic inhomogeneity and superconducting properties of a two-dimensional Hubbard model for high-Tccuprates, Phys. Rev. B 81(21), 214525 (2010) https://doi.org/10.1103/PhysRevB.81.214525
93
G. Khaliullin, M. Mori, T. Tohyama, and S. Maekawa, Enhanced pairing correlations near oxygen dopants in cuprate superconductors, Phys. Rev. Lett. 105(25), 257005 (2010) https://doi.org/10.1103/PhysRevLett.105.257005
94
J. W. Loram, J. L. Tallon, and W. Y. Liang, Absence of gross static inhomogeneity in cuprate superconductors, Phys. Rev. B 69, 060502(R) (2004) https://doi.org/10.1103/PhysRevB.69.060502
95
J. Bobroff, H. Alloul, S. Ouazi, P. Mendels, A. Mahajan, N. Blanchard, G. Collin, V. Guillen, and J. F. Marucco, Absence of static phase separation in the high Tccuprate YBa2Cu3O6+y, Phys. Rev. Lett. 89(15), 157002 (2002) https://doi.org/10.1103/PhysRevLett.89.157002
96
I. Bozovic, G. Logvenov, M. A. J. Verhoeven, P. Caputo, E. Goldobin, and M. R. Beasley, Giant proximity effect in cuprate superconductors, Phys. Rev. Lett. 93(15), 157002 (2004) https://doi.org/10.1103/PhysRevLett.93.157002
97
G. Alvarez, M. Mayr, A. Moreo, and E. Dagotto, Areas of superconductivity and giant proximity effects in underdoped cuprates, Phys. Rev. B 71(1), 014514 (2005) https://doi.org/10.1103/PhysRevB.71.014514
98
I. Bozovic, G. Logvenov, M. A. J. Verhoeven, P. Caputo, E. Goldobin, and T. H. Geballe, No mixing of superconductivity and antiferromagnetism in a high-temperature superconductor, Nature 422(6934), 873 (2003) https://doi.org/10.1038/nature01544
99
E. Demler, A. J. Berlinsky, C. Kallin, G. B. Arnold, and M. R. Beasley, Proximity effect and Josephson coupling in the SO(5) theory of high-Tcsuperconductivity, Phys. Rev. Lett. 80(13), 2917 (1998) https://doi.org/10.1103/PhysRevLett.80.2917
M. R. Norman and C. Pépin, The electronic nature of high temperature cuprate superconductors, Rep. Prog. Phys. 66(10), 1547 (2003) https://doi.org/10.1088/0034-4885/66/10/R01
102
A. Damascelli, A. Hussain, and Z. X. Shen, Angleresolved photoemission studies of the cuprate superconductors, Rev. Mod. Phys. 75(2), 473 (2003) https://doi.org/10.1103/RevModPhys.75.473
M. R. Norman, H. Ding, M. Randeria, J. C. Campuzano, T. Yokoya, T. Takeuchi, T. Takahashi, T. Mochiku, K. Kadowaki, P. Guptasarma, and D. G. Hinks, Destruction of the Fermi surface in underdoped high-Tcsuperconductors, Nature 392(6672), 157 (1998) https://doi.org/10.1038/32366
105
X. J. Zhou, T. Yoshida, D. H. Lee, W. L. Yang, V. Brouet, F. Zhou, W. X. Ti, J. W. Xiong, Z. X. Zhao, T. Sasagawa, T. Kakeshita, H. Eisaki, S. Uchida, A. Fujimori, Z. Hussain, and Z. X. Shen, Dichotomy between nodal and antinodal quasiparticles in underdoped (La2–xSrx)CuO4 superconductor, Phys. Rev. Lett. 92(18), 187001 (2004) https://doi.org/10.1103/PhysRevLett.92.187001
106
A. Kanigel, M. R. Norman, M. Randeria, U. Chatterjee, S. Souma, A. Kaminski, H. M. Fretwell, S. Rosenkranz, M. Shi, T. Sato, T. Takahashi, Z. Z. Li, H. Raffy, K. Kadowaki, D. Hinks, L. Ozyuzer, and J. C. Campuzano, Evolution of the pseudogap from Fermi arcs to the nodal liquid, Nat. Phys. 2(7), 447 (2006) https://doi.org/10.1038/nphys334
107
N. Doiron-Leyraud, C. Proust, D. LeBoeuf, J. Levallois, J. B. Bonnemaison, R. Liang, D. A. Bonn, W. N. Hardy, and L. Taillefer, Quantum oscillations and the Fermi surface in an underdoped high-Tcsuperconductor, Nature 447(7144), 565 (2007) https://doi.org/10.1038/nature05872
108
D. LeBoeuf, N. Doiron-Leyraud, J. Levallois, R. Daou, J. B. Bonnemaison, N. E. Hussey, L. Balicas, B. J. Ramshaw, R. Liang, D. A. Bonn, W. N. Hardy, S. Adachi, C. Proust, and L. Taillefer, Electron pockets in the Fermi surface of hole-doped high-Tcsuperconductors, Nature 450(7169), 533 (2007) https://doi.org/10.1038/nature06332
109
E. A. Yelland, J. Singleton, C. H. Mielke, N. Harrison, F. F. Balakirev, B. Dabrowski, and J. R. Cooper, Quantum oscillations in the underdopedcuprate YBa2Cu4O8, Phys. Rev. Lett. 100(4), 047003 (2008) https://doi.org/10.1103/PhysRevLett.100.047003
110
A. F. Bangura, J. D. Fletcher, A. Carrington, J. Levallois, M. Nardone, B. Vignolle, P. J. Heard, N. Doiron- Leyraud, D. LeBoeuf, L. Taillefer, S. Adachi, C. Proust, and N. E. Hussey, Small Fermi surface pockets in underdopedhigh temperature superconductors: Observation of Shubnikov–de Haas oscillations in YBa2Cu4O8, Phys. Rev. Lett. 100(4), 047004 (2008) https://doi.org/10.1103/PhysRevLett.100.047004
111
C. Jaudet, D. Vignolles, A. Audouard, J. Levallois, D. LeBoeuf, N. Doiron-Leyraud, B. Vignolle, M. Nardone, A. Zitouni, R. Liang, D. A. Bonn, W. N. Hardy, L. Taillefer, and C. Proust, de Haas–van Alphen oscillations in the underdoped high-temperature superconductor YBa2Cu3O6.5, Phys. Rev. Lett. 100(18), 187005 (2008) https://doi.org/10.1103/PhysRevLett.100.187005
G. F. Chen, Z. Li, G. Li, J. Zhou, D. Wu, J. Dong, W. Z. Hu, P. Zheng, Z. J. Chen, H. Q. Yuan, J. Singleton, J. L. Luo, and N. L. Wang, Superconducting properties of the Fe-based layered superconductor LaFeAsO0.9F0.1, Phys. Rev. Lett. 101(5), 057007 (2008) https://doi.org/10.1103/PhysRevLett.101.057007
114
H. H. Wen, G. Mu, L. Fang, H. Yang, and X. Zhu, Superconductivity at 25 K in hole-doped (La1–xSrx)OFeAs, Europhys. Lett. 82(1), 17009 (2008) https://doi.org/10.1209/0295-5075/82/17009
115
X. H. Chen, T. Wu, G. Wu, R. H. Liu, H. Chen, and D. F. Fang, Superconductivity at 43 K in SmFeAsO1–xFx, Nature 453(7196), 761 (2008) https://doi.org/10.1038/nature07045
116
G. F. Chen, Z. Li, D. Wu, G. Li, W. Z. Hu, J. Dong, P. Zheng, J. L. Luo, and N. L. Wang, Superconductivity at 41 K and its competition with spin-density-wave instability in layered CeO1–xFxFeAs, Phys. Rev. Lett. 100(24), 247002 (2008) https://doi.org/10.1103/PhysRevLett.100.247002
117
Z. A. Ren, J. Yang, W. Lu, W. Yi, X. L. Shen, Z. C. Li, G. C. Che, X. L. Dong, L. L. Sun, F. Zhou, and Z. X. Zhao, Superconductivity in the iron-based F-doped layered quaternary compound Nd[O1–xFx]FeAs, Europhys. Lett. 82(5), 57002 (2008) https://doi.org/10.1209/0295-5075/82/57002
118
G. F. Chen, Z. Li, D. Wu, J. Dong, G. Li, W. Z. Hu, P. Zheng, J. L. Luo, and N. L. Wang, Element substitution effect in transition metal oxypnictide Re(O1–xFx)TAs (Re= rare earth, T= transition metal), Chin. Phys. Lett. 25(6), 2235 (2008) https://doi.org/10.1088/0256-307X/25/6/086
119
M. Daghofer, A. Moreo, J. A. Riera, E. Arrigoni, D. J. Scalapino, and E. Dagotto, Model for the magnetic order and pairing channels in Fe pnictide superconductors, Phys. Rev. Lett. 101(23), 237004 (2008) https://doi.org/10.1103/PhysRevLett.101.237004
120
A. Moreo, M. Daghofer, J. A. Riera, and E. Dagotto, Properties of a two-orbital model for oxypnictide superconductors: Magnetic order, B2g spin-singlet pairing channel, and its nodal structure, Phys. Rev. B 79(13), 134502 (2009) https://doi.org/10.1103/PhysRevB.79.134502
121
K. Nakayama, T. Sato, P. Richard, Y. M. Xu, Y. Sekiba, S. Souma, G. F. Chen, J. L. Luo, N. L. Wang, H. Ding, and T. Takahashi, Superconducting gap symmetry of Ba0.6K0.4Fe2As2 studied by angle-resolved photoemission spectroscopy, Europhys. Lett. 85(6), 67002 (2009) https://doi.org/10.1209/0295-5075/85/67002
122
H. Ding, P. Richard, K. Nakayama, K. Sugawara, T. Arakane, Y. Sekiba, A. Takayama, S. Souma, T. Sato, T. Takahashi, Z. Wang, X. Dai, Z. Fang, G. F. Chen, J. L. Luo, and N. L. Wang, Observation of Fermi-surface-dependent nodeless superconducting gaps in Ba0.6K0.4Fe2As2, Europhys. Lett. 83(4), 47001 (2008) https://doi.org/10.1209/0295-5075/83/47001
123
L. Zhao, H. Y. Liu, W. T. Zhang, J. Q. Meng, X. W. Jia, G. D. Liu, X. L. Dong, G. F. Chen, J. L. Luo, N. L. Wang, W. Lu, G. L. Wang, Y. Zhou, Y. Zhu, X. Y. Wang, Z. Y. Xu, C. T. Chen, and X. J. Zhou, Multiple nodeless superconducting gaps in (Ba0.6K0.4)Fe2As2 superconductor from angle resolved photoemission spectroscopy, Chin. Phys. Lett. 25(12), 4402 (2008)
124
K. Umezawa, Y. Li, H. Miao, K. Nakayama, Z. H. Liu, P. Richard, T. Sato, J. B. He, D. M. Wang, G. F. Chen, H. Ding, T. Takahashi, and S. C. Wang, Unconventional anisotropic s-wave superconducting gaps of LiFeAs ironpnictide superconductor, Phys. Rev. Lett. 108(3), 037002 (2012) https://doi.org/10.1103/PhysRevLett.108.037002
125
Z. H. Liu, P. Richard, K. Nakayama, G. F. Chen, S. Dong, J. B. He, D. M. Wang, T. L. Xia, K. Umezawa, T. Kawahara, S. Souma, T. Sato, T. Takahashi, T. Qian, Y. Huang, N. Xu, Y. Shi, H. Ding, and S. C. Wang, Unconventional superconducting gap in NaFe0.95Co0.05As observed by angle-resolved photoemission spectroscopy, Phys. Rev. B 84(6), 064519 (2011) https://doi.org/10.1103/PhysRevB.84.064519
The broken particle number symmetry can be restored by particle-number projection, but in practice this procedure may not be necessary as we are dealing with a system having a very large number of fermions.