The pairing and superfluid phenomena in a two-component ultracold atomic Fermi gas is an analogue of Cooper pairing and superconductivity in an electron system, in particular, the high Tcsuperconductors. Owing to the various tunable parameters that have been made accessible experimentally in recent years, atomic Fermi gases can be explored as a prototype or quantum simulator of superconductors. It is hoped that, utilizing such an analogy, the study of atomic Fermi gases may shed light to the mysteries of high Tcsuperconductivity. One obstacle to the ultimate understanding of high Tcsuperconductivity, from day one of its discovery, is the anomalous yet widespread pseudogap phenomena, for which a consensus is yet to be reached within the physics community, after over 27 years of intensive research efforts. In this article, we shall review the progress in the study of pseudogap phenomena in atomic Fermi gases in terms of both theoretical understanding and experimental observations. We show that there is strong, unambiguous evidence for the existence of a pseudogap in strongly interacting Fermi gases. In this context, we shall present a pairing fluctuation theory of the pseudogap physics and show that it is indeed a strong candidate theory for high Tcsuperconductivity.
Q. J. Chen, J. Stajic, S. N. Tan, and K. Levin, BCS–BEC crossover: From high temperature superconductors to ultracold superfluids, Phys. Rep., 2005, 412(1): 1 https://doi.org/10.1016/j.physrep.2005.02.005
J. M. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys., 1998, 2: 231, see also: arXiv: hep-th/9711200v3
4
E. Witten, Anti De Sitter space and holography, Adv. Theor. Math. Phys., 1998, 2: 253
5
O. Aharony, S. S. Gubser, J. Maldacena, H. Ooguri, and Y. Oz, Large N field theories, string theory and gravity, Phys. Rep., 2000, 323(3−4): 183 https://doi.org/10.1016/S0370-1573(99)00083-6
6
M. Čubrović, J. Zaanen, and K. Schalm, String theory, quantum phase transitions, and the emergent Fermi liquid, Science, 2009, 325(5939): 439 https://doi.org/10.1126/science.1174962
7
T. Timusk and B. Statt, The pseudogap in high-temperature superconductors: An experimental survey, Rep. Prog. Phys., 1999, 62(1): 61 https://doi.org/10.1088/0034-4885/62/1/002
8
J. R. Schrieffer, Theory of Supercondutivity, 3rd Ed., Perseus Books, Reading, MA, 1983
A. Einstein, Quantentheorie des einatomigen idealen gases (II), Sitzungsberichte der Preussischen Akademie der Wissenschaften, 1925, 1: 3
11
L. Pitaevskii and S. Stringari, Bose–Einstein Condensation, New York: Oxford, 2003
12
C. J. Pethik and H. Smith, Bose–Einstein Condensation in Dilute Gases, Cambridge: Cambridge University Press, 2002
13
D. M. Eagles, Possible pairing without superconductivity at low carrier concentrations in bulk and thin-film superconducting semiconductors, Phys. Rev., 1969, 186(2): 456 https://doi.org/10.1103/PhysRev.186.456
14
A. J. Leggett, Diatomic molecules and cooper pairs, in: Modern Trends in the Theory of Condensed Matter, Berlin: Springer-Verlag, 1980, pp. 13−27 https://doi.org/10.1007/BFb0120125
15
P. Nozières and S. Schmitt-Rink, Bose condensation in an attractive fermion gas: From weak to strong coupling superconductivity, J. Low Temp. Phys., 1985, 59(3−4): 195 https://doi.org/10.1007/BF00683774
T. Friedberg and T. D. Lee, Gap energy and long-range order in the boson–fermion model of superconductivity, Phys. Rev. B, 1989, 40: 6745 https://doi.org/10.1103/PhysRevB.40.6745
18
C. A. R. Sá de Melo, M. Randeria, and J. R. Engelbrecht, Crossover from BCS to Bose superconductivity: Transition temperature and time-dependent Ginzburg–Landau theory, Phys. Rev. Lett., 1993, 71: 3202 https://doi.org/10.1103/PhysRevLett.71.3202
19
M. Randeria, Crossover from BCS theory to Bose-Einstein Condensation, in: Bose–Einstein Condensation, edited by A. Griffin, D. Snoke, and S. Stringari, Cambridge: Cambridge University Press, 1995, pp. 355−392 https://doi.org/10.1017/CBO9780511524240.017
20
B. Jankó, J. Maly, and K. Levin, Pseudogap effects induced by resonant pair scattering, Phys. Rev. B, 1997, 56(18): R11407(R) https://doi.org/10.1103/PhysRevB.56.R11407
21
J. Maly, B. Jankó, and K. Levin, Numerical studies of the s-wave pseudogap state and related Tc: The “pairing approximation” theory, Physica C, 1999, 321(1−2): 113 https://doi.org/10.1016/S0921-4534(99)00326-3
22
J. Maly, B. Jankó, and K. Levin, Superconductivity from a pseudogapped normal state: A mode-coupling approach to precursor superconductivity, Phys. Rev. B, 1999, 59: 1354 https://doi.org/10.1103/PhysRevB.59.1354
23
Q. J. Chen, I. Kosztin, B. Jankó, and K. Levin, Pairing fluctuation theory of superconducting properties in underdoped to overdoped cuprates, Phys. Rev. Lett., 1998, 81: 4708 https://doi.org/10.1103/PhysRevLett.81.4708
24
Q. J. Chen, I. Kosztin, B. Jankó, and K. Levin, Superconducting transitions from the pseudogap state: d-wave symmetry, lattice, and low-dimensional effects, Phys. Rev. B, 1999, 59: 7083 https://doi.org/10.1103/PhysRevB.59.7083
25
I. Kosztin, Q. J. Chen, B. Jankó, and K. Levin, Relationship between the pseudo- and superconducting gaps: Effects of residual pairing correlations below Tc, Phys. Rev. B, 1998, 58: R5936(R) https://doi.org/10.1103/PhysRevB.58.R5936
26
R. Micnas, J. Ranninger, and S. Robaszkiewicz, Superconductivity in narrow-band systems with local nonretarded attractive interactions, Rev. Mod. Phys., 1990, 62: 113 https://doi.org/10.1103/RevModPhys.62.113
27
R. Micnas and S. Robaszkiewicz, Superconductivity in systems with local attractive interactions, Cond. Matt. Phys. (Lviv), 1998, 13: 89 https://doi.org/10.5488/CMP.1.1.89
28
R. Micnas, M. H. Pedersen, S. Schafroth, T. Schneider, J. J. Rodríguez-Núñez, and H. Beck, Excitation spectrum of the attractive Hubbard model, Phys. Rev. B, 1995, 52: 16223 https://doi.org/10.1103/PhysRevB.52.16223
29
J. Ranninger and J. M. Robin, Manifestations of the pseudogap in the boson–fermion model for Bose–Einsteincondensation-driven superconductivity, Phys. Rev. B, 1996, 53: R11961(R) https://doi.org/10.1103/PhysRevB.53.R11961
R. Haussmann, Crossover from BCS superconductivity to Bose–Einstein condensation: A self-consistent theory, Z. Phys. B, 1993, 91(3): 291 https://doi.org/10.1007/BF01344058
32
R. Haussmann, Properties of a Fermi liquid at the superfluid transition in the crossover region between BCS superconductivity and Bose–Einstein condensation, Phys. Rev. B, 1994, 49: 12975 https://doi.org/10.1103/PhysRevB.49.12975
E. V. Gorbar, V. M. Loktev, and S. G. Sharapov, Crossover from BCS to composite-boson (local-pair) superconductivity in quasi-2D systems, Physica C, 1996, 257(3−4): 355 https://doi.org/10.1016/0921-4534(95)00773-3
35
V. P. Gusynin, V. M. Loktev, and S. G. Sharapov, Phase diagram of a 2D metal system with a variable number of carriers, JETP Lett., 1997, 65(2): 182 https://doi.org/10.1134/1.567308
36
M. Marini, F. Pistolesi, and G. C. Strinati, Evolution from BCS superconductivity to Bose condensation: Analytic results for the crossover in three dimensions, Eur. Phys. J. B, 1998, 1(2): 151 https://doi.org/10.1007/s100510050165
M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Observation of Bose–Einstein condensation in a dilute atomic vapor, Science, 1995, 269(5221): 198 https://doi.org/10.1126/science.269.5221.198
39
C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Evidence of Bose–Einstein condensation in an atomic gas with attractive interactions, Phys. Rev. Lett., 1995, 75: 1687 https://doi.org/10.1103/PhysRevLett.75.1687
40
C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Evidence of Bose–Einstein condensation in an atomic gas with attractive interactions [Phys. Rev. Lett. 75, 1687 (1995)], Phys. Rev. Lett., 1997, 79: 1170 https://doi.org/10.1103/PhysRevLett.79.1170
41
K. B. Davis, M. Mewes, M. R. Andrews, D. S. Durfee, D. M. Kurn, W. Ketterle, and W. Ketterle, Bose–Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett., 1995, 75(22): 3969 https://doi.org/10.1103/PhysRevLett.75.3969
42
M. Greiner, C. A. Regal, and D. S. Jin, Emergence of a molecular Bose–Einstein condensate from a Fermi gas, Nature, 2003, 426(6966): 537 https://doi.org/10.1038/nature02199
43
S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, S. Riedl, C. Chin, J. H. Denschlag, and R. Grimm, Bose–Einstein condensation of molecules, Science, 2003, 302(5653): 2101 https://doi.org/10.1126/science.1093280
44
M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. Raupach, S. Gupta, Z. Hadzibabic, and W. Ketterle, Observation of Bose–Einstein condensation of molecules, Phys. Rev. Lett., 2003, 91(25): 250401 https://doi.org/10.1103/PhysRevLett.91.250401
45
C. A. Regal, M. Greiner, and D. S. Jin, Observation of resonance condensation of fermionic atom pairs, Phys. Rev. Lett., 2004, 92(4): 040403 https://doi.org/10.1103/PhysRevLett.92.040403
46
M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J. H. Denschlag, and R. Grimm, Crossover from a molecular Bose–Einstein condensate to a degenerate Fermi gas, Phys. Rev. Lett., 2004, 92(12): 120401 https://doi.org/10.1103/PhysRevLett.92.120401
47
C. Chin, M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, J. H. Denschlag, and R. Grimm, Observation of the pairing gap in a strongly interacting Fermi gas, Science, 2004, 305(5687): 1128 https://doi.org/10.1126/science.1100818
48
M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. Raupach, A. J. Kerman, and W. Ketterle, Condensation of pairs of fermionic atoms near a Feshbach resonance, Phys. Rev. Lett., 2004, 92(12): 120403 https://doi.org/10.1103/PhysRevLett.92.120403
49
J. Kinast, A. Turlapov, J. E. Thomas, Q. J. Chen, J. Stajic, and K. Levin, Heat capacity of a strongly interacting Fermi gas, Science, 2005, 307(5713): 1296 https://doi.org/10.1126/science.1109220
50
M. W. Zwierlein, J. R. Abo-Shaeer, and A. Schirotzek, and W. Ketterle, Vortices and superfluidity in a strongly interacting Fermi gas, Nature, 2005, 435: 1047 https://doi.org/10.1038/nature03858
51
M. W. Zwierlein, A. Schirotzek, C. H. Schunck, and W. Ketterle, Fermionic superfluidity with imbalanced spin populations, Science, 2006, 311(5760): 492 https://doi.org/10.1126/science.1122318
52
G. B. Partridge, W. Li, R. I. Kamar, Y. A. Liao, and R. G. Hulet, Pairing and phase separation in a polarized Fermi gas, Science, 2006, 311(5760): 503 https://doi.org/10.1126/science.1122876
A. I. Larkin and Yu. N. Ovchinnikov, Neodnorodnoe sostoyanie sverkhprovodnikov, Zh. Eksp. Teor. Fiz., 1964, 47: 1136
55
I. Larkin and Yu. N. Ovhinnikov, Nonuniform state of superconductors, Sov. Phys. JETP, 1965, 20: 762
56
Q. J. Chen, Generalization of BCS theory to short coherence length superconductors: A BCS-Bose-Einstein crossover scenario, Ph.D. thesis, University of Chiago, 2000, available in the ProQuest Dissertations & Theses Database online.
57
Q. J. Chen, J. Stajic, and K. Levin, Applying BCS–BEC crossover theory to high temperature superconductors and ultracold atomic Fermi gases, Low Temp. Phys., 2006, 32(4): 406; Fiz. Nizk. Temp., 2006, 32: 538
M. Inguscio, W. Ketterle, and C. Salomon (Eds.), Ultracold Fermi gases, Proceedings of the International School of Physics “Enrico Fermi”, Vol. CLXIV, Varenna, 2006, Società Italiana di Fisca Bologna, Italy (ISO press, Amsterdam, 2008)
61
H. Ding, T. Yokoya, J. C. Campuzano, T. Takahashi, M. Randeria, M. R. Norman, T. Mochiku, K. Hadowaki, and J. Giapintzakis, Spectroscopic evidence for a pseudogap in the normal state of underdoped high-Tc superconductors, Nature, 1996, 382(6586): 51 https://doi.org/10.1038/382051a0
62
Ch. Renner, B. Revaz, K. Kadowaki, I. Maggio-Aprile, and O. Fischer, Observation of the low temperature pseudogap in the vortex cores of Bi2Sr2CaCu2O8+δ, Phys. Rev. Lett., 1998, 80(16): 3606 https://doi.org/10.1103/PhysRevLett.80.3606
63
Ch. Renner, B. Revaz, J. Y. Genoud, K. Kadowaki, and O. Fischer, Pseudogap precursor of the superconducting gap in under- and overdoped Bi2Sr2CaCu2O8+δ, Phys. Rev. Lett., 1998, 80(1): 149 https://doi.org/10.1103/PhysRevLett.80.149
64
V. M. Krasnov, A. Yurgens, D. Winkler, P. Delsing, and T. Claeson, Evidence for coexistence of the superconducting gap and the pseudogap in Bi-2212 from intrinsic tunneling spectroscopy, Phys. Rev. Lett., 2000, 84: 5860 https://doi.org/10.1103/PhysRevLett.84.5860
65
M. Kugler, O. Fischer, Ch. Renner, S. Ono, and Y. Ando, Scanning tunneling spectroscopy of Bi2Sr2CuO6+δ: New evidence for the common origin of the pseudogap and superconductivity, Phys. Rev. Lett., 2001, 86(21): 4911 https://doi.org/10.1103/PhysRevLett.86.4911
66
J. W. Loram, K. Mirza, J. Cooper, W. Liang, and J. Wade, Electronic specific heat of YBa2Cu3O6+x from 1.8 to 300 K, J. Superondutivity, 1994, 7(1): 243
67
G. V. M. Williams, E. M. Haines, and J. L. Tallon, Pair breaking in the presence of a normal-state pseudogap in high-Tc cuprates, Phys. Rev. B, 1998, 57: 146 https://doi.org/10.1103/PhysRevB.57.146
68
D. Walker, A. P. Mackenzie, and J. R. Cooper, Transport properties of zinc-doped YBa2Cu3O7−δ thin films, Phys. Rev. B, 1995, 51: 15653(R) https://doi.org/10.1103/PhysRevB.51.15653
69
T. Graf, J. M. Lawrene, M. F. Hundley, J. D. Thompson, A. Lacerda, E. Haanappel, M. S. Torikahili, Z. Fisk, and P. C. Canfield, Resistivity, magnetization, and specific heat of YbAgCu4 in high magnetic fields, Phys. Rev. B, 1995, 51: 15053 https://doi.org/10.1103/PhysRevB.51.15053
70
Y. F. Yan, P. Matl, J. M. Harris, and N. P. Ong, Negative magnetoresistance in the c-axis resistivity of Bi2Sr2CaCu2O8+δ and YBa2Cu3O6+x, Phys. Rev. B, 1995, 52: R751(R) https://doi.org/10.1103/PhysRevB.52.R751
71
G. Williams, J. L. Tallon, R. Dupree, and R. Michalak, Transport and NMR studies of the effect of Ni substitution on superconductivity and the normal-state pseudogap in YBa2Cu4O8, Phys. Rev. B, 1996, 54: 9532 https://doi.org/10.1103/PhysRevB.54.9532
72
G. Williams, J. L. Tallon, E. M. Haines, R. Michalak, and R. Dupree, NMR evidence for a d-wave normal-state pseudogap, Phys. Rev. Lett., 1997, 78: 721 https://doi.org/10.1103/PhysRevLett.78.721
73
K. Magishi, Y. Kituoka, G.-Q. Zheng, K. Asayama, T. Kondo, Y. Shimakawa, T. Manako, and Y. Kubo, Spin-gap behavior in underdoped TlSr2(Lu0.7Ca0.3)Cu2Oy: 63Cu and 205Tl NMR studies, Phys. Rev. B, 1996, 54: 3070 https://doi.org/10.1103/PhysRevB.54.3070
74
A. Goto, H. Yasuoka, K. Otzschi, and Y. Ueda, Phase diagram for the spin pseudogap in LaBa2Cu3Oy studied by NMR, Phys. Rev. B, 1997, 55: 12736 https://doi.org/10.1103/PhysRevB.55.12736
75
J. Bobroff, H. Alloul, P. Mendels, V. Viallet, J.-F. Marucco, and D. Colson, 17O NMR evidence for a pseudogap in the monolayer HgBa2CuO4+δ, Phys. Rev. Lett., 1997, 78: 3757 https://doi.org/10.1103/PhysRevLett.78.3757
76
K. Ishida, K. Yoshida, T. Mito, Y. Tokumaga, Y. Kitaoka, K. Asayama, Y. Nakayama, J. Shimoyama, and K. Kishio, Pseudogap behavior in single-crystal Bi2Sr2CaCu2O8+δ probed by Cu NMR, Phys. Rev. B, 1998, 58: R5960(R) https://doi.org/10.1103/PhysRevB.58.R5960
77
A. V. Puchkov, D. N. Basov, and T. Timusk, The pseudogap state in high-Tc superconductors: An infrared study, J. Phys.: Condens. Matter, 1996, 8(48): 10049 https://doi.org/10.1088/0953-8984/8/48/023
78
D. N. Basov, R. Liang, B. Dabrowski, D. A. Bonn, W. N. Hardy, and T. Timusk, Pseudogap and charge dynamics in CuO2 planes in YBCO, Phys. Rev. Lett., 1996, 77: 4090 https://doi.org/10.1103/PhysRevLett.77.4090
79
D. Basov, C. Homes, E. Singley, M. Strongin, T. Timusk, G. Blumberg, and D. van der Marel, Unconventional energetics of the pseudogap state and superconducting state in high-Tc cuprates, Phys. Rev. B, 2001, 63: 134514 https://doi.org/10.1103/PhysRevB.63.134514
80
J. M. Tranquada, P. M. Gehring, G. Shirane, S. Shamoto, and M. Sato, Neutron-scattering study of the dynamical spin susceptibility in YBa2Cu3O6.6, Phys. Rev. B, 1992, 46: 5561 https://doi.org/10.1103/PhysRevB.46.5561
81
P. C. Dai, H. A. Mook, S. M. Hayden, and F. Dogan, Resonance as a measure of pairing correlations in the high-Tc superconductor YBa2Cu3O6.6, Nature, 2000, 406: 965 https://doi.org/10.1038/35023094
82
B. Lake, G. Aeppli, T. E. Mason, A. Schroeder, D. F. Mc-Morrow, K. Lefmann, M. Isshiki, M. Nohara, H. Takagi, and S. M. Hayden, Spin gap and magnetic coherence in a clean high-temperature superconductor, Nature, 1999, 400: 43 https://doi.org/10.1038/21840
83
G. Ruani and P. Ricci, Transitions at T>Tc in underdoped crystals of YBa2Cu3O7−x observed by resonant Raman scattering, Phys. Rev. B, 1997, 55: 93 https://doi.org/10.1103/PhysRevB.55.93
84
X. K. Chen, J. G. Nacini, K. C. Hewitt, J. C. Irwin, R. Liang, and W. N. Hardy, Electronic Raman scattering in underdoped YBa2Cu3O6.5, Phys. Rev. B, 1997, 56: R513(R) https://doi.org/10.1103/PhysRevB.56.R513
85
R. Nemetschek, M. Opel, C. Hoffmann, P. F. Muller, R. Hackl, H. Berger, L. Forro, A. Er, and E. Walker, Pseudogap and superconducting gap in the electronic Raman spectra of underdoped cuprates, Phys. Rev. Lett., 1997, 78: 4837 https://doi.org/10.1103/PhysRevLett.78.4837
86
J. W. Quilty, H. J. Trodahl, and D. M. Pooke, Electronic Raman scattering from Bi2Sr2CaCu2O8+δ: Doping dependence of the pseudogap and anomalous 600 cm−1 peak, Phys. Rev. B, 1998, 57: R11097 https://doi.org/10.1103/PhysRevB.57.R11097
87
Z. A. Xu, N. Ong, Y. Want, T. Kakeshita, and S. Uchida, Vortex-like excitations and the onset of superconducting phase fluctuation in underdoped La2−xSrxCuO4, Nature, 2000, 406: 486 https://doi.org/10.1038/35020016
88
Y. Wang, Z. A. Xu, T. Kakeshita, S. Uchida, and N. P. Ong, Onset of the vortexlike Nernst signal above Tc in La2−xSrxCuO4 and Bi2Sr2−yLayCuO6, Phys. Rev. B, 2001, 64: 224519 https://doi.org/10.1103/PhysRevB.64.224519
89
Y. Y. Wang, N. P. Ong, Z. A. Xu, T. Kakeshita, S. Uchida, D. Bonn, R. Liang, and W. Hardy, High field phase diagram of cuprates derived from the Nernst effect, Phys. Rev. Lett., 2002, 88: 257003 https://doi.org/10.1103/PhysRevLett.88.257003
90
S. Tan and K. Levin, Nernst effect and anomalous transport in cuprates: A preformed-pair alternative to the vortex scenario, Phys. Rev. B, 2004, 69(6): 064510 https://doi.org/10.1103/PhysRevB.69.064510
91
A. G. Loeser, Z. X. Shen, D. S. Dessau, D. S. Marshall, C. H. Park, P. Fournier, and A. Kapitulnik, Excitation gap in the normal state of underdoped Bi2Sr2CaCu2O8+δ, Science, 1996, 273(5273): 325 https://doi.org/10.1126/science.273.5273.325
92
A. Kanigel, U. Chatterjee, M. Randeria, M. R. Norman, G. Koren, K. Kadowaki, and J. C. Campuzano, Evidence for pairing above the transition temperature of cuprate superconductors from the electronic dispersion in the pseudogap phase, Phys. Rev. Lett., 2008, 101(13): 137002 https://doi.org/10.1103/PhysRevLett.101.137002
93
For simplicity, here we do not discuss electron doping, which is rather similar. Further information can be found in Ref. [7].
C. Honerkamp and P. A. Lee, Staggered flux fluctuations and the quasiparticle scattering rate in the SU(2) gauge theory of the t–J model, Phys. Rev. Lett., 2003, 90(24): 246402 https://doi.org/10.1103/PhysRevLett.90.246402
98
C. M. Varma, Non-Fermi-liquid states and pairing instability of a general model of copper oxide metals, Phys. Rev. B, 1997, 55(21): 14554 https://doi.org/10.1103/PhysRevB.55.14554
J. W. Loram, K. A. Mirza, J. R. Cooper, and J. L. Tallon, Specific heat evidence on the normal state pseudogap, J. Phys. Chem. Solids, 1998, 59(10−12): 2091 https://doi.org/10.1016/S0022-3697(98)00180-2
Q. J. Chen, K. Levin, and I. Kosztin, Superconducting phase coherence in the presence of a pseudogap: Relation to specific heat, tunneling, and vortex core spectroscopies, Phys. Rev. B, 2001, 63(18): 184519 https://doi.org/10.1103/PhysRevB.63.184519
P. W. Anderson, P. A. Lee, M. Randeria, T. M. Rie, N. Trivedi, and F. C. Zhang, The physics behind hightemperature superconducting cuprates: The “plain vanilla” version of RVB, J. Phys.: Condens. Matter, 2004, 16(24): R755 https://doi.org/10.1088/0953-8984/16/24/R02
Y. J. Uemura, G. M. Luke, B. J. Sternlieb, J. H. Brewer, J. F. Carolan, et al., Universal Correlations between Tc and ns/m∗ (carrier density over effective mass) in high-Tc cuprate superconductors, Phys. Rev. Lett., 1989, 62: 2317 https://doi.org/10.1103/PhysRevLett.62.2317
108
Y. J. Uemura, Bose–Einstein to BCS crossover picture for high-Tc cuprates, Physica C, 1997, 282−287: 194
109
V. Mishra, U. Chatterjee, J. C. Campuzano, and M. R. Norman, Effect of the pseudogap on the transition temperature in the cuprates and implications for its origin, Nat. Phys., 2014, 10(5): 357 https://doi.org/10.1038/nphys2926
110
V. J. Emery and S. A. Kivelson, Importance of phase fluctuations in superconductors with small superfluid density, Nature, 1995, 374: 434 https://doi.org/10.1038/374434a0
111
M. Franz, Z. B. Tesanovic, and O. Vafek, QED3 theory of pairing pseudogap in cuprates: From d-wave superconductor to antiferromagnet via “algebraic” Fermi liquid, Phys. Rev. B, 2002, 66: 054535 https://doi.org/10.1103/PhysRevB.66.054535
112
I. Ussishkin, S. L. Sondhi, and D. A. Huse, Gaussian superconducting fluctuations, thermal transport, and the Nernst effect, Phys. Rev. Lett., 2002, 89(28): 287001 https://doi.org/10.1103/PhysRevLett.89.287001
113
J. N. Milstein, S. J. J. M. F. Kokkelmans, and M. J. Holland, Resonance theory of the crossover from Bardeen–Cooper-Schrieffer superfluidity to Bose–Einstein condensation in a dilute Fermi gas, Phys. Rev. A, 2002, 66(4): 043604 https://doi.org/10.1103/PhysRevA.66.043604
N. Andrenacci, P. Pieri, and G. C. Strinati, Evolution from BCS superconductivity to Bose–Einstein condensation: Current correlation function in the broken-symmetry phase, Phys. Rev. B, 2003, 68: 144507 https://doi.org/10.1103/PhysRevB.68.144507
116
A. Perali, P. Pieri, L. Pisani, and G. C. Strinati, BCS–BEC crossover at finite temperature for superfluid trapped Fermi atoms, Phys. Rev. Lett., 2004, 92(22): 220404 https://doi.org/10.1103/PhysRevLett.92.220404
117
H. Hu, P. D. Drummond, and X. J. Liu, Universal thermodynamics of strongly interacting Fermi gases, Nat. Phys., 2007, 3(7): 469 https://doi.org/10.1038/nphys598
118
K. Levin, Q. J. Chen, Y. He, and C.-C. Chien, Comparison of different pairing fluctuation approaches to BCS–BEC crossover, Ann. Phys., 2010, 325(2): 233 https://doi.org/10.1016/j.aop.2009.09.011
119
N. E. Bickers, D. J. Scalapino, and S. R. White, Conserving approximations for strongly correlated electron systems: Bethe–Salpeter equation and dynamics for the twodimensional hubbard model, Phys. Rev. Lett., 1989, 62: 961 https://doi.org/10.1103/PhysRevLett.62.961
120
N. E. Bickers and D. J. Scalapino, Conserving approximations for strongly fluctuating electron systems (I): Formalism and calculational approach, Ann. Phys., 1989, 193: 206 https://doi.org/10.1016/0003-4916(89)90359-X
121
R. Haussmann, W. Rantner, S. Cerrito, and W. Zwerger, Thermodynamics of the BCS–BEC crossover, Phys. Rev. A, 2007, 75(2): 023610 https://doi.org/10.1103/PhysRevA.75.023610
122
Y. O. R. Watanabe and S. Tsuchiya, Superfluid density of states and pseudogap phenomenon in the BCS–BEC crossover regime of a superfluid Fermi gas, Phys. Rev. A, 2010, 82: 043630 https://doi.org/10.1103/PhysRevA.82.043630
123
P. Magierski, G. Wlazöwski, A. Bulgac, and J. E. Drut, Finite-temperature pairing gap of a unitary Fermi gas by quantum Monte Carlo calculations, Phys. Rev. Lett., 2009, 103(21): 210403 https://doi.org/10.1103/PhysRevLett.103.210403
124
P. Pieri, A. Perali, G. C. Strinati, S. Riedl, M. J. Wright, A. Altmeyer, C. Kohstall, E. R. S. Guajardo, J. H. Denschlag, and R. Grimm, Pairing-gap, pseudogap, and no-gap phases in the radio-frequency spectra of a trapped unitary 6Li gas, Phys. Rev. A, 2011, 84: 011608(R) https://doi.org/10.1103/PhysRevA.84.011608
125
L. P. Kadanoff and P. C. Martin, Theory of many-particle systems (II): Superconductivity, Phys. Rev., 1961, 124(3): 670 https://doi.org/10.1103/PhysRev.124.670
126
J. Stajic, J. N. Milstein, Q. J. Chen, M. L. Chiofalo, M. J. Holland, and K. Levin, Nature of superfluidity in ultracold Fermi gases near Feshbach resonances, Phys. Rev. A, 2004, 69(6): 063610 https://doi.org/10.1103/PhysRevA.69.063610
127
While a general interaction V(k−k′) may not be separable, it can however be de composed into different channels as V(k−k′)=∑lϕklϕk′l, where ϕkl represents s-, p-, d-wave channels, etc. In most cases, only one channel dominates the superfluid order so that we may neglect other channels. In this way, the use of a separable potential is justified.
128
S. J. J. M. F. Kokkelmans, J. N. Milstein, M. L. Chiofalo, R. Walser, and M. J. Holland, Resonance superfluidity: Renormalization of resonance scattering theory, Phys. Rev. A, 2002, 65(5): 053617 https://doi.org/10.1103/PhysRevA.65.053617
129
Here we will mainly discuss s-wave short range contact potential for atomic Fermi gases. At present, p-wave superfluids are not yet available experimentally in atomic Fermi gases.
130
H. Guo, C.-C. Chien, Q. J. Chen, Y. He, and K. Levin, Finite-temperature behavior of an interspecies fermionic superfluid with population imbalance, Phys. Rev. A, 2009, 80: 011601(R) https://doi.org/10.1103/PhysRevA.80.011601
131
J. B. Wang, Y. M. Che, L. F. Zhang, and Q. J. Chen, Searching for the elusive exotic Fulde–Ferrell–Larkin–Ovchinnikov states in Fermi–Fermi mixtures of ultracold quantum gases, arXiv: 1404.5696, 2014
132
C.-C. Chien, Q. J. Chen, Y. He, and K. Levin, Intermediatetemperature superfluidity in an atomic Fermi gas with population imbalance, Phys. Rev. Lett., 2006, 97(9): 090402 https://doi.org/10.1103/PhysRevLett.97.090402
133
C.-C. Chien, Q. J. Chen, Y. He, and K. Levin, Superfluid phase diagrams of trapped Fermi gases with population imbalance, Phys. Rev. Lett., 2007, 98(11): 110404 https://doi.org/10.1103/PhysRevLett.98.110404
134
Q. J. Chen, Y. He, C.-C. Chien, and K. Levin, Theory of superfluids with population imbalance: Finite-temperature and BCS–BEC crossover effects, Phys. Rev. B, 2007, 75(1): 014521 https://doi.org/10.1103/PhysRevB.75.014521
135
J. B. Wang, H. Guo, and Q. J. Chen, Exotic phase separation and phase diagrams of a Fermi–Fermi mixture in a trap at finite temperature, Phys. Rev. A, 2013, 87: 041601(R) https://doi.org/10.1103/PhysRevA.87.041601
136
K. M. O’Hara, S. L. Hemmer, M. E. Gehm, S. R. Granade, and J. E. Thomas, Observation of a strongly interacting degenerate Fermi gas of atoms, Science, 2002, 298(5601): 2179 https://doi.org/10.1126/science.1079107
137
T. Bourdel, L. Khaykovich, J. Cubizolles, J. Zhang, F. Chevy, M. Teichmann, L. Tarruell, S. J. Kokkelmans, and C. Salomon, Experimental study of the BEC–BCS crossover region in lithium 6, Phys. Rev. Lett., 2004, 93(5): 050401 https://doi.org/10.1103/PhysRevLett.93.050401
138
J. Carlson, S. Y. Chang, V. R. Pandharipande, and K. E. Schmidt, Superfluid Fermi gases with large scattering length, Phys. Rev. Lett., 2003, 91(5): 050401 https://doi.org/10.1103/PhysRevLett.91.050401
139
I. Kosztin, Q. J. Chen, Y.-J. Kao, and K. Levin, Pair excitations, collective modes, and gauge invariance in the BCS-Bose–Einstein crossover scenario, Phys. Rev. B, 2000, 61(17): 11662 https://doi.org/10.1103/PhysRevB.61.11662
140
Q. J. Chen, Y. He, C.-C. Chien, and K. Levin, Stability conditions and phase diagrams for two-component Fermi gases with population imbalance, Phys. Rev. A, 2006, 74(6): 063603 https://doi.org/10.1103/PhysRevA.74.063603
141
In fact, the parameter γ can be taken from experiment, as has been done in Ref. [102], where one can find more details.
142
P. Pieri, L. Pisani, and G. C. Strinati, BCS–BEC crossover at finite temperature in the broken-symmetry phase, Phys. Rev. B, 2004, 70(9): 094508 https://doi.org/10.1103/PhysRevB.70.094508
143
N. Fukushima, Y. Ohashi, E. Taylor, and A. Griffin, Superfluid density and condensate fraction in the BCS–BEC crossover regime at finite temperatures, Phys. Rev. A, 2007, 75(3): 033609 https://doi.org/10.1103/PhysRevA.75.033609
144
I. Kosztin and A. J. Leggett, Nonlocal effects on the magnetic penetration depth in d-wave superconductors, Phys. Rev. Lett., 1997, 79(1): 135 https://doi.org/10.1103/PhysRevLett.79.135
145
S. Hufner, M. A. Hossain, A. Damaselli, and G. Sawatzky, Two gaps make a high-temperature superconductor? Rep. Prog. Phys., 2008, 71(6): 062501 https://doi.org/10.1088/0034-4885/71/6/062501
146
G. Baskaran, Z. Zou, and P. W. Anderson, The resonating valence bond state and high-Tc superconductivity —A mean field theory, Solid State Commun., 1987, 63(11): 973 https://doi.org/10.1016/0038-1098(87)90642-9
147
N. Miyakawa, J. Zasadzinski, L. Ozyuzer, P. Guptasarma, D. Hinks, C. Kendziora, and K. Gray, Predominantly superconducting origin of large energy gaps in underdoped Bi2Sr2CaCu2O8+δ from tunneling spectroscopy, Phys. Rev. Lett., 1999, 83(5): 1018 https://doi.org/10.1103/PhysRevLett.83.1018
M. L. Chiofalo, S. J. J. M. F. Kokkelmans, J. N. Milstein, and M. J. Holland, Signatures of resonance superfluidity in a quantum Fermi gas, Phys. Rev. Lett., 2002, 88(9): 090402 https://doi.org/10.1103/PhysRevLett.88.090402
150
Note here that the definition for nc and np differ from that in Ref. [156] by a factor of 2.
151
G. E. Astrakharchik, J. Boronat, J. Casulleras, and S. Giorgini, Momentum distribution and condensate fraction of a fermion gas in the BCS–BEC Crossover, Phys. Rev. Lett., 2005, 95: 230405 (Their result seems to suggest a tendency of decrease in the condensate fraction with an increasing particle number used for simulation.) https://doi.org/10.1103/PhysRevLett.95.230405
152
The curves in Fig. 19 were calculated using a two-channel model. Nevertheless, for wide Feshbach resonances such as in 6Li and 40K, the closed-channel fraction is very small [191, 192] so that the quantitative difference in the entropy s(r) between the two-channel and one-channel model is negligible.
Q. J. Chen, C. A. Regal, M. Greiner, D. S. Jin, and K. Levin, Understanding the superfluid phase diagram in trapped Fermi gases, Phys. Rev. A, 2006, 73: 041601(R) https://doi.org/10.1103/PhysRevA.73.041601
155
Note that the experimental data cannot be measuring Nc = N as shown in Fig. 18, since at 1/(kF a) = −1,Nc = N is far below the experimental threshold of detection.
While one may argue that the kink, if it exists, may be smoothed out by the ∫ dydz integration, we note that as of the time of this writing, no kink behavior has ever been reported in 3D density profiles obtained via an inverse Abel transformation of experimental data.
158
Q. J. Chen, C. A. Regal, D. S. Jin, and K. Levin, Finitetemperature momentum distribution of a trapped Fermi gas, Phys. Rev. A, 2006, 74: 011601(R) https://doi.org/10.1103/PhysRevA.74.011601
159
Q. J. Chen, Y. He, C.-C. Chien, and K. Levin, Theory of radio frequency spectroscopy experiments in ultracold Fermi gases and their relation to photoemission in the cuprates, Rep. Prog. Phys., 2009, 72(12): 122501 https://doi.org/10.1088/0034-4885/72/12/122501
160
C. H. Shunk, Y. Shin, A. Schirotzek, M. W. Zwierlein, and W. Ketterle, Determination of the fermion pair size in a resonantly interacting superfluid, Nature, 2008, 454(7205): 739 https://doi.org/10.1038/nature07176
161
C. H. Schunk, Y. Shin, A. Schirotzek, M. W. Zwierlein, and W. Ketterle, Pairing without superfluidity: The ground state of an imbalanced Fermi mixture, Science, 2007, 316(5826): 867 https://doi.org/10.1126/science.1140749
162
Z. Yu and G. Baym, Spin-correlation functions in ultracold paired atomic-fermion systems: Sum rules, self-consistent approximations, and mean fields, Phys. Rev. A, 2006, 73(6): 063601 https://doi.org/10.1103/PhysRevA.73.063601
163
G. Baym, C. J. Pethick, Z. H. Yu, and M. W. Zwierlein, Coherence and clock shifts in ultracold Fermi gases with resonant interactions, Phys. Rev. Lett., 2007, 99(19): 190407 https://doi.org/10.1103/PhysRevLett.99.190407
A. Perali, P. Pieri, and G. C. Strinati, Competition between final-state and pairing-gap effects in the radio-frequency spectra of ultracold Fermi atoms, Phys. Rev. Lett., 2008, 100(1): 010402 https://doi.org/10.1103/PhysRevLett.100.010402
166
S. Basu and E. J. Müller, Final-state effects in the radio frequency spectrum of strongly interacting fermions, Phys. Rev. Lett., 2008, 101(6): 060405 https://doi.org/10.1103/PhysRevLett.101.060405
167
Y. He, C. C. Chien, Q. J. Chen, and K. Levin, Temperature and final state effects in radio frequency spectroscopy experiments on atomic Fermi gases, Phys. Rev. Lett., 2009, 102(2): 020402 https://doi.org/10.1103/PhysRevLett.102.020402
168
M. J. H. Ku, A. T. Sommer, L. W. Cheuk, and M. W. Zwierlein, Revealing the superfluid lambda transition in the universal thermodynamics of a unitary Fermi gas, arXiv: 1110.3309, 2011
169
E. Burovski, N. Prokof’ev, B. Svistunov, and M. Troyer, Critical temperature and thermodynamics of attractive fermions at unitarity, Phys. Rev. Lett., 2006, 96(16): 160402 https://doi.org/10.1103/PhysRevLett.96.160402
170
E. Burovski, E. Kozik, N. Prokof’ev, B. Svistunov, and M. Troyer, Critical temperature curve in BEC–BCS crossover, Phys. Rev. Lett., 2008, 101(9): 090402 https://doi.org/10.1103/PhysRevLett.101.090402
171
O. Goulko and M. Wingate, Thermodynamics of balanced and slightly spin-imbalanced Fermi gases at unitarity, Phys. Rev. A, 2010, 82(5): 053621 https://doi.org/10.1103/PhysRevA.82.053621
172
J. Kinnunen, M. Rodríguez, and P. Törmä, Pairing gap and in-gap excitations in trapped fermionic superfluids, Science, 2004, 305(5687): 1131 https://doi.org/10.1126/science.1100782
173
Y. He, Q. J. Chen, and K. Levin, Radio-frequency spectroscopy and the pairing gap in trapped Fermi gases, Phys. Rev. A, 2005, 72: 011602(R) https://doi.org/10.1103/PhysRevA.72.011602
174
P. Massignan, G. M. Bruun, and H. T. C. Stoof, Twin peaks in RF spectra of Fermi gases at unitarity, Phys. Rev. A, 2008, 77: 031601(R) https://doi.org/10.1103/PhysRevA.77.031601
175
J. T. Stewart, J. P. Gaebler, and D. S. Jin, Using photoemission spectroscopy to probe a strongly interacting Fermi gas, Nature, 2008, 454(7205): 744 https://doi.org/10.1038/nature07172
D. S. Jin, Private communications; D.S. Jin, American Physical Society March Meeting Talk B8.00002, 2009
178
J. P. Gaebler, J. T. Stewart, T. E. Drake, D. S. Jin, A. Perali, P. Pieri, and G. C. Strinati, Observation of pseudogap behaviour in a strongly interacting Fermi gas, Nat. Phys., 2010, 6(8): 569 https://doi.org/10.1038/nphys1709
179
A. Perali, F. Palestini, P. Pieri, G. C. Strinati, J. T. Stewart, J. P. Gaebler, T. E. Drake, and D. S. Jin, Evolution of the normal state of a strongly interacting Fermi gas from a pseudogap phase to a molecular Bose gas, Phys. Rev. Lett., 2011, 106(6): 060402 https://doi.org/10.1103/PhysRevLett.106.060402
180
A. Perali, P. Pieri, G. C. Strinati, and C. Castellani, Pseudogap and spectral function from superconducting fluctuations to the bosonic limit, Phys. Rev. B, 2002, 66(2): 024510 https://doi.org/10.1103/PhysRevB.66.024510
181
P. Pieri, L. Pisani, and G. C. Strinati, Pairing fluctuation effects on the single-particle spectra for the superconducting state, Phys. Rev. Lett., 2004, 92(11): 110401 https://doi.org/10.1103/PhysRevLett.92.110401
182
Y. Shin, M. W. Zwierlein, C. H. Schunck, A. Schirotzek, and W. Ketterle, Observation of phase separation in a strongly interacting imbalanced Fermi gas, Phys. Rev. Lett., 2006, 97(3): 030401 https://doi.org/10.1103/PhysRevLett.97.030401
183
S. Nascimbène, N. Navon, K. Jiang, F. Chevy, and C. Salomon, Exploring the thermodynamics of a universal Fermi gas, Nature, 2010, 463(7284): 1057 https://doi.org/10.1038/nature08814
184
S. Nascimbène, N. Navon, S. Pilati, F. Chevy, S. Giorgini, A. Georges, and C. Salomon, Fermi-liquid behavior of the normal phase of a strongly interacting gas of cold atoms, Phys. Rev. Lett., 2011, 106(21): 215303 https://doi.org/10.1103/PhysRevLett.106.215303
185
L. P. Gor’kov and T. K. Melik-Barkhudarov, Contribution to the theory of superfluidity in an imperfect fermi gas, Sov. Phys. JETP, 1961, 13: 1018
186
H. Heiselberg, C. J. Pethick, H. Smith, and L. Viverit, Influence of induced interactions on the superfluid transition in dilute Fermi gases, Phys. Rev. Lett., 2000, 85(12): 2418 https://doi.org/10.1103/PhysRevLett.85.2418
187
D. H. Kim, P. Törmä, and J.-P. Martikainen, Induced interactions for ultracold Fermi gases in optical lattices, Phys. Rev. Lett., 2009, 102(24): 245301 https://doi.org/10.1103/PhysRevLett.102.245301
188
J. P. Martikainen, J. J. Kinnunen, P. Törmä, and C. J. Pethick, Induced interactions and the superfluid transition temperature in a three-component Fermi gas, Phys. Rev. Lett., 2009, 103(26): 260403 https://doi.org/10.1103/PhysRevLett.103.260403
189
Z. Q. Yu, K. Huang, and L. Yin, Induced interaction in a Fermi gas with a BEC–BCS crossover, Phys. Rev. A, 2009, 79(5): 053636 https://doi.org/10.1103/PhysRevA.79.053636
190
Q. J. Chen, Effect of the particle-hole channel on BCS–Bose–Einstein condensation crossover in atomic Fermi gases, arXiv: 1109.2307, 2011
191
Q. J. Chen and K. Levin, Population of closed-channel molecules in trapped Fermi gases with broad Feshbach resonances, Phys. Rev. Lett., 2005, 95(26): 260406 https://doi.org/10.1103/PhysRevLett.95.260406
192
G. B. Partridge, K. E. Strecker, R. I. Kamar, M. W. Jack, and R. G. Hulet, Molecular probe of pairing in the BEC–BCS crossover, Phys. Rev. Lett., 2005, 95(2): 020404 https://doi.org/10.1103/PhysRevLett.95.020404
193
H. Guo, C.-C. Chien, and K. Levin, Establishing the presence of coherence in atomic Fermi superfluids: Spin-flip and spin-preserving Bragg scattering at finite temperatures, Phys. Rev. Lett., 2010, 105(12): 120401 https://doi.org/10.1103/PhysRevLett.105.120401
194
M. G. Lingham, K. Fenech, S. Hoinka, and C. J. Vale, Local observation of pair condensation in a Fermi gas at unitarity, Phys. Rev. Lett., 2014, 112(10): 100404 https://doi.org/10.1103/PhysRevLett.112.100404