Pseudogap phenomena in ultracold atomic Fermi gases

Abstract

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 T_csuperconductors. 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 T_csuperconductivity. One obstacle to the ultimate understanding of high T_csuperconductivity, 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 T_csuperconductivity.

Graphical abstract

Keywords

pseudogap / pairing fluctuation theory / atomic Fermi gases / BCS–BEC crossover / high Tcsuperconductivity

References

[1] 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

[2] I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys., 2008, 80: 885

[3] 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

[6] M. Čubrović, J. Zaanen, and K. Schalm, String theory, quantum phase transitions, and the emergent Fermi liquid, Science, 2009, 325(5939): 439

[7] T. Timusk and B. Statt, The pseudogap in high-temperature superconductors: An experimental survey, Rep. Prog. Phys., 1999, 62(1): 61

[8] J. R. Schrieffer, Theory of Supercondutivity, 3rd Ed., Perseus Books, Reading, MA, 1983

[9] S. N. Bose, Plancks gesetz und lichtquantenhypothese, Z. Phys., 1924, 26(1): 178

[10] 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

[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

[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

[16] R. Friedberg and T. D. Lee, Boson–Fermion model of superconductivity, Phys. Lett. A, 1989, 138(8): 423

[17] 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

[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

[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

[20] B. Jankó, J. Maly, and K. Levin, Pseudogap effects induced by resonant pair scattering, Phys. Rev. B, 1997, 56(18): R11407(R)

[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

[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

[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

[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

[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)

[26] R. Micnas, J. Ranninger, and S. Robaszkiewicz, Superconductivity in narrow-band systems with local nonretarded attractive interactions, Rev. Mod. Phys., 1990, 62: 113

[27] R. Micnas and S. Robaszkiewicz, Superconductivity in systems with local attractive interactions, Cond. Matt. Phys. (Lviv), 1998, 13: 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

[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)

[30] M. Drechsler and W. Zwerger, Crossover from BCSsuperconductivity to Bose-condensation, Ann. Phys., 1992, 1: 15

[31] R. Haussmann, Crossover from BCS superconductivity to Bose–Einstein condensation: A self-consistent theory, Z. Phys. B, 1993, 91(3): 291

[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

[33] O. Tchernyshyov, Noninteracting Cooper pairs inside a pseudogap, Phys. Rev. B, 1997, 56: 3372

[34] 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

[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

[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

[37] B. DeMarco and D. S. Jin, Onset of Fermi degeneracy in a trapped atomic gas, Science, 1999, 285(5434): 1703

[38] 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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[51] M. W. Zwierlein, A. Schirotzek, C. H. Schunck, and W. Ketterle, Fermionic superfluidity with imbalanced spin populations, Science, 2006, 311(5760): 492

[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

[53] P. Fulde and R. A. Ferrell, Superconductivity in a strong spin-exchange field, Phys. Rev., 1964, 135(3A): A550

[54] 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

[58] S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of ultracold atomic Fermi gases, Rev. Mod. Phys., 2008, 80(4): 1215

[59] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Feshbach resonances in ultracold gases, Rev. Mod. Phys., 2010, 82(2): 1225

[60] 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

[62] Ch. Renner, B. Revaz, K. Kadowaki, I. Maggio-Aprile, and O. Fischer, Observation of the low temperature pseudogap in the vortex cores of Bi₂Sr₂CaCu₂O_8+δ, Phys. Rev. Lett., 1998, 80(16): 3606

[63] Ch. Renner, B. Revaz, J. Y. Genoud, K. Kadowaki, and O. Fischer, Pseudogap precursor of the superconducting gap in under- and overdoped Bi₂Sr₂CaCu₂O_8+δ, Phys. Rev. Lett., 1998, 80(1): 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

[65] M. Kugler, O. Fischer, Ch. Renner, S. Ono, and Y. Ando, Scanning tunneling spectroscopy of Bi₂Sr₂CuO_6+δ: New evidence for the common origin of the pseudogap and superconductivity, Phys. Rev. Lett., 2001, 86(21): 4911

[66] J. W. Loram, K. Mirza, J. Cooper, W. Liang, and J. Wade, Electronic specific heat of YBa₂Cu₃O_6+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

[68] D. Walker, A. P. Mackenzie, and J. R. Cooper, Transport properties of zinc-doped YBa₂Cu₃O_7−δ thin films, Phys. Rev. B, 1995, 51: 15653(R)

[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

[70] Y. F. Yan, P. Matl, J. M. Harris, and N. P. Ong, Negative magnetoresistance in the c-axis resistivity of Bi₂Sr₂CaCu₂O_8+δ and YBa₂Cu₃O_6+x, Phys. Rev. B, 1995, 52: R751(R)

[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

[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

[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(Lu_0.7Ca_0.3)Cu₂O_y: ⁶³Cu and ²⁰⁵Tl NMR studies, Phys. Rev. B, 1996, 54: 3070

[74] A. Goto, H. Yasuoka, K. Otzschi, and Y. Ueda, Phase diagram for the spin pseudogap in LaBa₂Cu₃O_y studied by NMR, Phys. Rev. B, 1997, 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 HgBa₂CuO_4+δ, Phys. Rev. Lett., 1997, 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 Bi₂Sr₂CaCu₂O_8+δ probed by Cu NMR, Phys. Rev. B, 1998, 58: R5960(R)

[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

[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

[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

[80] J. M. Tranquada, P. M. Gehring, G. Shirane, S. Shamoto, and M. Sato, Neutron-scattering study of the dynamical spin susceptibility in YBa₂Cu₃O_6.6, Phys. Rev. B, 1992, 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 YBa₂Cu₃O_6.6, Nature, 2000, 406: 965

[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

[83] G. Ruani and P. Ricci, Transitions at T>Tc in underdoped crystals of YBa₂Cu₃O_7−x observed by resonant Raman scattering, Phys. Rev. B, 1997, 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 YBa₂Cu₃O_6.5, Phys. Rev. B, 1997, 56: R513(R)

[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

[86] J. W. Quilty, H. J. Trodahl, and D. M. Pooke, Electronic Raman scattering from Bi₂Sr₂CaCu₂O_8+δ: Doping dependence of the pseudogap and anomalous 600 cm⁻¹ peak, Phys. Rev. B, 1998, 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 La_2−xSr_xCuO₄, Nature, 2000, 406: 486

[88] Y. Wang, Z. A. Xu, T. Kakeshita, S. Uchida, and N. P. Ong, Onset of the vortexlike Nernst signal above Tc in La_2−xSr_xCuO₄ and Bi₂Sr_2−yLa_yCuO₆, Phys. Rev. B, 2001, 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

[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

[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 Bi₂Sr₂CaCu₂O_8+δ, Science, 1996, 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

[93] For simplicity, here we do not discuss electron doping, which is rather similar. Further information can be found in Ref. [7].

[94] S. Chakravarty, R. B. Laughlin, D. K. Morr, and C. Nayak, Hidden order in the cuprates, Phys. Rev. B, 2001, 63(9): 094503

[95] P. A. Lee, High Tc superconductors as doped Mott insulators: Fluctuating current and spin chirality, Physica C, 2000, 341−348: 63

[96] P. A. Lee and X.-G. Wen, Vortex structure in underdoped cuprates, Phys. Rev. B, 2001, 63(22): 224517

[97] 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

[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

[99] C. M. Varma, Theory of the pseudogap state of the cuprates, Phys. Rev. B, 2006, 73(15): 155113

[100] 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

[101] J. L. Tallon and J. W. Loram, The doping dependence of T^∗ – What is the real high-Tc phase diagram? Physica C, 2001, 349(1−2): 53

[102] 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

[103] P. W. Anderson, The resonating valence bond state in La₂CuO₄ and superconductivity, Science, 1987, 235(4793): 1196

[104] 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

[105] N. Nagaosa and P. A. Lee, Ginzburg–Landau theory of the spin-charge-separated system, Phys. Rev. B, 1992, 45: 966

[106] For a review of spin-harge separation, see: P. A. Lee, Pseudogaps in underdoped cuprates, Physica C, 1999, 317−318: 194

[107] Y. J. Uemura, G. M. Luke, B. J. Sternlieb, J. H. Brewer, J. F. Carolan, , Universal Correlations between Tc and ns/m^∗ (carrier density over effective mass) in high-Tc cuprate superconductors, Phys. Rev. Lett., 1989, 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

[110] V. J. Emery and S. A. Kivelson, Importance of phase fluctuations in superconductors with small superfluid density, Nature, 1995, 374: 434

[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

[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

[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

[114] Y. Ohashi and A. Griffin, BCS–BEC crossover in a gas of Fermi atoms with a Feshbach resonance, Phys. Rev. Lett., 2002, 89(13): 130402

[115] 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

[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

[117] H. Hu, P. D. Drummond, and X. J. Liu, Universal thermodynamics of strongly interacting Fermi gases, Nat. Phys., 2007, 3(7): 469

[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

[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

[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

[121] R. Haussmann, W. Rantner, S. Cerrito, and W. Zwerger, Thermodynamics of the BCS–BEC crossover, Phys. Rev. A, 2007, 75(2): 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

[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

[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)

[125] L. P. Kadanoff and P. C. Martin, Theory of many-particle systems (II): Superconductivity, Phys. Rev., 1961, 124(3): 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

[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

[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)

[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

[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

[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

[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)

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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 Bi₂Sr₂CaCu₂O_8+δ from tunneling spectroscopy, Phys. Rev. Lett., 1999, 83(5): 1018

[148] T.-L. Ho, Universal thermodynamics of degenerate quantum gases in the unitarity limit, Phys. Rev. Lett., 2004, 92(9): 090402

[149] 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

[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.)

[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.

[153] Q. J. Chen, J. Stajic, and K. Levin, Thermodynamics of interacting fermions in atomic traps, Phys. Rev. Lett., 2005, 95(26): 260405

[154] 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)

[155] Note that the experimental data cannot be measuring N_c = N as shown in Fig. 18, since at 1/(kF a) = −1,N_c = N is far below the experimental threshold of detection.

[156] J. Stajic, Q. J. Chen, and K. Levin, Density profiles of strongly interacting trapped Fermi gases, Phys. Rev. Lett., 2005, 94: 060401

[157] 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)

[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

[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

[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

[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

[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

[164] M. Punk and W. Zwerger, Theory of RF-spectroscopy of strongly interacting fermions, Phys. Rev. Lett., 2007, 99(17): 170404

[165] 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

[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

[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

[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

[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

[171] O. Goulko and M. Wingate, Thermodynamics of balanced and slightly spin-imbalanced Fermi gases at unitarity, Phys. Rev. A, 2010, 82(5): 053621

[172] J. Kinnunen, M. Rodríguez, and P. Törmä, Pairing gap and in-gap excitations in trapped fermionic superfluids, Science, 2004, 305(5687): 1131

[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)

[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)

[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

[176] Q. J. Chen and K. Levin, Momentum resolved radio frequency spectroscopy in trapped fermi gases, Phys. Rev. Lett., 2009, 102(19): 190402

[177] 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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

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