狄拉克费米子体系中的手征相变
|
摘要
高温超导和石墨烯是凝聚态物理中受到广泛、深入研究的两个强关联体系,其共同特点是低能元激发是无质量的狄拉克费米子,满足相对论性的狄拉克方程,可用(2+1)维量子场论模型来描述。其中高温超导的低能有效理论是(2+1)维量子电动力学,而石墨烯体系中狄拉克费米子之间存在库仑相互作用。QED3中的动力学手征对称破缺(DCSB)已经被研究超过二十年,而在石墨烯中的研究也已经持续了接近十年。这些研究不仅有助于加深对量子色动力学(QCD)中的DCSB的理解,而且也可以用于广泛理解高温超导和石墨烯中的许多重要物理现象。
本论文通过对规范相互作用和库仑相互作用导致的动力学手征对称破缺的讨论,研究了高温超导和石墨烯中的手征量子相变,以及该相变对体系低能物理行为的影响。DCSB可以看作由相互作用导致的费米子-反费米子配对机制。研究表明,当狄拉克费米子之间的相互作用足够强时,狄拉克费米子会通过真空凝聚而获得有限的质量,并同时破坏体系原来的手征对称性。通过对Dyson-Schwinger (DS)方程详细地理论分析和计算,我们确定了在高温超导和石墨烯中DCSB发生的条件,指出规范和库仑相互作用的长程性在DCSB的发生中起关键作用,并将所得结果用于讨论DCSB的物理效应。经过具体计算,我们发现,温度、化学势和杂质散射以及规范场质量等因素都会破坏相互作用的长程性,降低动力学费米子质量生成的可能性,既降低费米子临界味数Nc,也会减小动力学质量m。
此外,为了更好地理解石墨烯中手征相变带来的物理效应,我们具体计算了半金属和绝缘体两个相中的比热和磁化率。我们发现,在半金属相中,库仑相互作用会导致比热和磁化率的非费米液体行为；而在绝缘体相中,由于费米子激发具有有限能隙,比热和磁化率被大大压制。可见,比热和磁化率能够很好地区分半金属和绝缘体两个相,这为实验上寻找手征相变提供重要信息。通过这些物理量与实验结果的比较,可以加深对手征相变的物理效应的理解。
High temperature superconductor (HTSC) and graphene are two strongly cor-related electron systems those have attracted great research interests in recent years. They share one common feature that the low-energy elementary excitations are mass-less Dirac fermions, arising from the special band structures. These Dirac fermions satisfy relativistic Dirac wave equation and can be described by three-dimensional quantum field theory. The dynamical chiral symmetry breaking caused by gauge field and Coulomb interaction, which can describe the low-energy physics of high tempera-ture superconductor and graphene respectively. The theory describing the low-energy physics of of HTSC is (2+1)-dimension QED(QED3), while that is Coulomb inter-action for the Dirac fermions in graphene. The dynamic chiral symmetry breaking (DCSB) in QED3 has been investigated intensively for more than twenty years and about decade for that in graphene, On the one hand, these investigations may help to gain deeper understanding of DCSB in QCD. On the other hand, this non-perturbative phenomenon can be widely used to understand many important physical phenomena in HTSC and graphene.
In this thesis, the chiral quantum phase transition and the consequent effects on low-energy physics of system are studied by discussing on the DCSB driven by gauge field and Coulomb interaction in HTSC and graphene, respectively. The DCSB is realized by forming fermion-anti-fermion pairs mediated by strong gauge (Coulomb) interaction. Research shows that when the interaction between Dirac fermions is strong enough, the Dirac fermions will acquire finite dynamic mass by vacuum condensations and break the original chiral symmetry contemporarily. Through the theoretical anal-ysis and calculation of the Dyson-Schwinger (DS) equation, we have identified the conditions of the occurrence of DCSB in the HTSC and the graphene, pointed out that the long-range nature of gauge (Coulomb) interaction play a key role in DCSB. The results are also used to discuss the physical effects induced by DCSB. After the specific calculations, we found that temperature, chemical potential, impurity scattering even the mass of gauge field will make the interaction become short-ranged and prevent the generation of dynamic mass, then reduce the fermion critical flavor number Nc and dynamical mass m.
Furthermore, in order to get a better understanding of the physical effects induced by chiral phase transition in graphene, we calculate the specific heat and susceptibil-ity of the system and show that they exhibit distinct behaviors in the semimetal and insulator phases. In the semimetal phase, the Coulomb interaction will lead to non-Fermi liquid behavior of specific heat and susceptibility; while in the insulating phase, the specific heat and susceptibility are strongly suppressed because of finite gap of fermion excitations. Apparently, both specific heat and susceptibility manifest quite different behaviors in these two phases and can distinguish them clearly, which pro-vides important information on experiment research on chiral phase transition. These quantities can be compared with experimental results and hence may help to understand the physical consequence of chiral phase transition.
本论文通过对规范相互作用和库仑相互作用导致的动力学手征对称破缺的讨论,研究了高温超导和石墨烯中的手征量子相变,以及该相变对体系低能物理行为的影响。DCSB可以看作由相互作用导致的费米子-反费米子配对机制。研究表明,当狄拉克费米子之间的相互作用足够强时,狄拉克费米子会通过真空凝聚而获得有限的质量,并同时破坏体系原来的手征对称性。通过对Dyson-Schwinger (DS)方程详细地理论分析和计算,我们确定了在高温超导和石墨烯中DCSB发生的条件,指出规范和库仑相互作用的长程性在DCSB的发生中起关键作用,并将所得结果用于讨论DCSB的物理效应。经过具体计算,我们发现,温度、化学势和杂质散射以及规范场质量等因素都会破坏相互作用的长程性,降低动力学费米子质量生成的可能性,既降低费米子临界味数Nc,也会减小动力学质量m。
此外,为了更好地理解石墨烯中手征相变带来的物理效应,我们具体计算了半金属和绝缘体两个相中的比热和磁化率。我们发现,在半金属相中,库仑相互作用会导致比热和磁化率的非费米液体行为；而在绝缘体相中,由于费米子激发具有有限能隙,比热和磁化率被大大压制。可见,比热和磁化率能够很好地区分半金属和绝缘体两个相,这为实验上寻找手征相变提供重要信息。通过这些物理量与实验结果的比较,可以加深对手征相变的物理效应的理解。
High temperature superconductor (HTSC) and graphene are two strongly cor-related electron systems those have attracted great research interests in recent years. They share one common feature that the low-energy elementary excitations are mass-less Dirac fermions, arising from the special band structures. These Dirac fermions satisfy relativistic Dirac wave equation and can be described by three-dimensional quantum field theory. The dynamical chiral symmetry breaking caused by gauge field and Coulomb interaction, which can describe the low-energy physics of high tempera-ture superconductor and graphene respectively. The theory describing the low-energy physics of of HTSC is (2+1)-dimension QED(QED3), while that is Coulomb inter-action for the Dirac fermions in graphene. The dynamic chiral symmetry breaking (DCSB) in QED3 has been investigated intensively for more than twenty years and about decade for that in graphene, On the one hand, these investigations may help to gain deeper understanding of DCSB in QCD. On the other hand, this non-perturbative phenomenon can be widely used to understand many important physical phenomena in HTSC and graphene.
In this thesis, the chiral quantum phase transition and the consequent effects on low-energy physics of system are studied by discussing on the DCSB driven by gauge field and Coulomb interaction in HTSC and graphene, respectively. The DCSB is realized by forming fermion-anti-fermion pairs mediated by strong gauge (Coulomb) interaction. Research shows that when the interaction between Dirac fermions is strong enough, the Dirac fermions will acquire finite dynamic mass by vacuum condensations and break the original chiral symmetry contemporarily. Through the theoretical anal-ysis and calculation of the Dyson-Schwinger (DS) equation, we have identified the conditions of the occurrence of DCSB in the HTSC and the graphene, pointed out that the long-range nature of gauge (Coulomb) interaction play a key role in DCSB. The results are also used to discuss the physical effects induced by DCSB. After the specific calculations, we found that temperature, chemical potential, impurity scattering even the mass of gauge field will make the interaction become short-ranged and prevent the generation of dynamic mass, then reduce the fermion critical flavor number Nc and dynamical mass m.
Furthermore, in order to get a better understanding of the physical effects induced by chiral phase transition in graphene, we calculate the specific heat and susceptibil-ity of the system and show that they exhibit distinct behaviors in the semimetal and insulator phases. In the semimetal phase, the Coulomb interaction will lead to non-Fermi liquid behavior of specific heat and susceptibility; while in the insulating phase, the specific heat and susceptibility are strongly suppressed because of finite gap of fermion excitations. Apparently, both specific heat and susceptibility manifest quite different behaviors in these two phases and can distinguish them clearly, which pro-vides important information on experiment research on chiral phase transition. These quantities can be compared with experimental results and hence may help to understand the physical consequence of chiral phase transition.
引文
[1]L. D. Landau, E. M. Lifshits, and L. P. Pitaevskii. Statistical physics. Pergamon Press,1980.
[2]Gordon Baym and Christopher Pethick. Landau Fermi-liquid theory:concepts and applications. Wiley,1991.
[3]C. M. Varma, Z. Nussinov, and W. van Saarloos. Singular or non-fermi liquids. Physics Reports-Review Section of Physics Letters,361(5-6):267-417,2002.
[4]Xiao-Gang Wen. Quantum field theory of many-body systems:from the origin of sound to an origin of light and electrons. Oxford University Press,2004.
[5]H. J. Schulz. Fermi liquids and non-fermi liquids. Mesoscopic Quantum Physics, 61:533-603791,1995.
[6]冯端,金国钧.凝聚态物理学上卷.北京-高等教育出版社,2003.
[7]闫守胜.固体物理基础第二版.北京大学出版社,2003.
[8]H. J. Schulz, G. Cuniberti, and P. Pieri. Fermi liquids and luttinger liquids. Field Theories for Low-Dimensional Condensed Matter Systems,131:9-81275,2000.
[9]H. von Lohneysen, A. Rosch, M. Vojta, and P. Wolfle. Fermi-liquid instabilities at magnetic quantum phase transitions. Reviews of Modern Physics,79(3):1015-1075,2007.
[10]N. F. Mott and R. Peierls. Discussion of the paper by de boer and verwey. Proceedings of the Physical Society,49:72-73,1937.
[11]Wikipedia. High-temperature superconductivity-wikipedia, the free en-cyclopedia.2010. URL http://en.wikipedia.org/w/index. php?title=High-temperature_superconductivity & oldid= 346945237.
[12]J. G. Bednorz and K. A. Mueller. Possible high tc superconductivity in the ba-la-cu-o system. Z. Phys. B,64(189-193),1986.
[13]Wikipedia. Iron-based superconductor - wikipedia, the free encyclope-dia.2010. URL http://en.wikipedia.org/w/index.php?title= Iron-based_superconductor & oldid=342436699.
[14]Y. Kamihara, H. Hiramatsu, M. Hirano, R. Kawamura, H. Yanagi, T. Kamiya, and H. Hosono. Iron-based layered superconductor:Laofep. Journal of the American Chemical Society,128(31):10012-10013,2006.
[15]Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono. Iron-based layered super-conductor la[o1-xfx]feas (x=0.05-0.12) with tc=26 k. Journal of the American Chemical Society,130(11):3296-+,2008.
[16]向涛.d波超导体.科学出版社,2007.
[17]W. N. Hardy, D. A. Bonn, D. C. Morgan, Ruixing Liang, and Kuan Zhang. Precision measurements of the temperature dependence of λ in yba2cu3o6.95: Strong evidence for nodes in the gap function. Physical Review Letters,70: 3999-4002,1993.
[18]Z.-X. Shen, D. S. Dessau, B. O. Wells, D. M. King, W. E. Spicer, A. J. Arko, D. Marshall, L. W. Lombardo, A. Kapitulnik, and P. Dickinson. Anomalously large gap anisotropy in the a-b plane of bi2sr2cacu2o(8+δ). Physical Review Letters, 70:1553-1556,1993.
[19]Z.-X. Shen and D. S. Dessau. Electronic structure and photoemission studies of late transition-metal oxides-mott insulators and high-temperature supercon-ductors. Physics Reports,253:1-162,1995.
[20]H. Ding, M. R. Norman, J. C. Campuzano, M. Randeria, A. F. Bellman, T. Yokoya, T. Takahashi, T. Mochiku, and K. Kadowaki. Angle-resolved photoemission spectroscopy study of the superconducting gap anisotropy in bi2sr2cacu2o8+x. Physical Review B,54:9678,1996.
[21]G. E. Volovik. Superconductivity with lines of gap nodes:density of states in the vortex. Soviet Journal of Experimental and Theoretical Physics Letters,58: 469,1993.
[22]K. A. Moler, D. J. Baar, J. S. Urbach, Ruixing Liang, W. N. Hardy, and A. Kapitulnik. Magnetic field dependence of the density of states of yba2cu3o695 as determined from the specific heat. Physical Review Letters,73:2744-2747, 1994.
[23]Kathryn A. Moler, David L. Sisson, Jeffrey S. Urbach, Malcolm R. Beasley, Aharon Kapitulnik, David J. Baar, Ruixing Liang, and Walter N. Hardy. Specific heat of yba2cu3o7-d. Physical Review B,55:3954-3965,1997.
[24]P. A. Lee. Localized states in a d-wave superconductor. Physical Review Letters, 71(12):1887,1993.
[25]Louis Taillefer, Benoit Lussier, Robert Gagnon, Kamran Behnia, and HerveAubin. Universal heat conduction in yba2cu3o6.9. Physical Review Letters, 79:483-486,1997.
[26]D. A. Wollman, D. J. van Harlingen, W. C. Lee, D. M. Ginsberg, and A. J. Leggett. Experimental determination of the superconducting pairing state in ybco from the phase coherence of ybco-pb dc squids. Physical Review Letters, 71:2134-2137,1993.
[27]D. A. Wollman, D. J. van Harlingen, J. Giapintzakis, and D. M. Ginsberg. Ev-idence for dx2-y2 pairing from the magnetic field modulation of yba2cu3o7-pb josephson junctions. Physical Review Letters,74:797-800,1995.
[28]A. Mathai, Y. Gim, R. C. Black, A. Amar, and F. C. Wellstood. Experimental proof of a time-reversal-invariant order parameter with a p shift in yba2cu3O7-d. Physical Review Letters,74:4523-4526,1995.
[29]D. J. Scalapino. The case for dx2-y2 pairing in the cuprate superconductors. Physics Reports-Review Section of Physics Letters,250(6):330-365,1995.
[30]D. J. Vanharlingen. Phase-sensitive tests of the symmetry of the pairing state in the high-temperature superconductors-evidence for dx2-y2 symmetry. Reviews of Modern Physics,67(2):515-535,1995.
[31]C. C. Tsuei and J. R. Kirtley. Pairing symmetry in cuprate superconductors. Reviews of Modern Physics,72(4):969-1016,2000.
[32]A. C. Durst and P. A. Lee. Impurity-induced quasiparticle transport and universal-limit wiedemann-franz violation in d-wave superconductors. Physi-cal Review B,62(2):1270-1290,2000.
[33]Gabriel Kotliar and Jialin Liu. Superexchange mechanism and d-wave super-conductivity. Physical Review B,38:5142-5145,1988.
[34]L. P. Gor'kov and P. A. Kalugin. Defects and an unusual superconductivity. Soviet Journal of Experimental and Theoretical Physics Letters,41:253,1985.
[35]P. W. Anderson. The resonating valence bond state in la2cuo4 and superconduc-tivity. Science,235:1196-1198,1987.
[36]F. C. Zhang and T. M. Rice. Effective hamiltonian for the superconducting cu oxides. Physical Review B,37:3759-3761,1988.
[37]N. Read and D. M. Newns. A new functional integral formalism for the de-generate anderson model. Journal of Physics C Solid State Physics,16:1055, 1983.
[38]G. Baskaran and P. W. Anderson. Gauge theory of high-temperature supercon-ductors and strongly correlated fermi systems. Physical Review B,37(1):580, 1988.
[39]M. U. Ubbens and P. A. Lee. Superconductivity phase-diagram in the gauge-field description of the t-j model. Physical Review B,49(10):6853-6863,1994.
[40]B. L. Altshuler, L. B. Ioffe, and A. J. Millis. Theory of the spin gap in high-temperature superconductors. Physical Review B,53(1):415-424,1996.
[41]P. A. Lee and N. Nagaosa. Gauge-theory of the normal state of high-tc super-conductors. Physical Review B,46(9):5621-5639,1992.
[42]X. G. Wen and P. A. Lee. Theory of underdoped cuprates. Physical Review Letters,76(3):503-506,1996.
[43]P. A. Lee, N. Nagaosa, T. K. Ng, and X. G. Wen. Su(2) formulation of the t-j model:Application to underdoped cuprates. Physical Review B,57(10):6003-6021,1998.
[44]S. Reich, J. Maultzsch, C. Thomsen, and P. Ordejon. Tight-binding description of graphene. Physical Review B,66(3),2002.
[45]P. R. Wallace. The band theory of graphite. Physical Review,71:622-634,1947.
[46]Neil W. Ashcroft and N. David Mermin. Solid state physics. Holt,1976.
[47]K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov. Two-dimensional gas of massless dirac fermions in graphene. Nature,438(7065):197-200,2005.
[48]Gordon W. Semenoff. Condensed-matter simulation of a three-dimensional anomaly. Physical Review Letters,53:2449-2452,1984.
[49]Y. Nambu and G. Jona-Lasinio. Dynamical model of elementary particles based on an analogy with superconductivity. i. Physical Review,122:345-358,1961.
[50]Y. Nambu and G. Jona-Lasinio. Dynamical model of elementary particles based on an analogy with superconductivity. ii. Physical Review,124:246-254,1961.
[51]R. D. Pisarski. Chiral-symmetry breaking in 3-dimensional electrodynamics. Physical Review D,29(10):2423-2426,1984.
[52]R. D. Pisarski. Finite-temperature qcd at large n. Physical Review D,29(6): 1222-1227,1984.
[53]Thomas W. Appelquist, Mark Bowick, Dimitra Karabali, and L. C. R. Wije-wardhana. Spontaneous chiral-symmetry breaking in three-dimensional qed. Physical Review D,33(12):3704,1986.
[54]T. Appelquist, D. Carrier, L. C. R. Wijewardhana, and W. Zheng. Numerical studies of enhanced chiral condensates. Physical Review Letters,60(12):1114, 1988.
[55]Daniel Nash. Higher-order corrections in (2+1)-dimensional qed. Physical Re-view Letters,62:3024-3026,1989.
[56]Elbio Dagotto, John B. Kogut, and Aleksandar Kocic. Computer simulation of chiral-symmetry breaking in (2+1)-dimensional qed with n flavors. Physical Review Letters,62:1083-1086,1989.
[57]Elbio Dagotto, Aleksandar Kocic, and J. B. Kogut. Chiral symmetry breaking in three dimensional qed with nflavors. Nuclear Physics B,334:279-301,1990.
[58]D. Atkinson, P. W. Johnson, and P. Maris. Dynamic mass generation in 3-dimensional qed-improved vertex function. Physical Review D,42(2):602-609, 1990.
[59]I. J. R. Aitchison, N. Dorey, M. Kleinkreisler, and N. E. Mavromatos. Phase-structure of qed3 at finite temperature. Physics Letters B,294(1):91-99,1992.
[60]D. C. Curtis, M. R. Pennington, and D. Walsh. Dynamical mass generation in qed3 and the 1/n expansion. Physics Letters B,295:313-319,1992.
[61]M. R. Pennington and D. Walsh. Masses from nothing. a non-perturbative study of qed3. Physics Letters B,253:246-251,1991.
[62]P. Maris. Confinement and complex singularities in 3-dimensional qed. Physical Review D,52(10):6087-6097,1995.
[63]P. Maris. Influence of the full vertex and vacuum polarization on the fermion propagator in (2+1)-dimensional qed. Physical Review D,54(6):4049-4058, 1996.
[64]V. P. Gusynin, A. H. Hams, and M. Reenders. (2+1)-dimensional qed with dy-namically massive fermions in vacuum polarization. Physical Review D,53(4): 2227,1996.
[65]C. S. Fischer, R. Alkofer, T. Dahm, and P. Maris. Dynamical chiral symmetry breaking in unquenched qed3. Physical Review D,70(7),2004.
[66]S. J. Hands, J. B. Kogut, and C. G. Strouthos. Non-compact qed3 with n-f(?)= 2. Nuclear Physics B,645(1-2):321-336,2002.
[67]S. J. Hands, J. B. Kogut, L. Scorzato, and C. G. Strouthos. Noncompact three-dimensional quantum electrodynamics with nf=1 and n-f=4. Physical Review B,70(10),2004.
[68]V. P. Gusynin and A. Reenders. Infrared cutoff dependence of the critical flavor number in three-dimensional qed. Physical Review D,68(2),2003.
[69]Craig D. Roberts and Anthony G. Williams. Dyson-schwinger equations and their application to hadronic physics. Progress in Particle and Nuclear Physics, 33:477-575,1994.
[70]T. Appelquist, J. Terning, and L. C. R. Wijewardhana. (2+1)-dimensional qed and a novel phase-transition. Physical Review Letters,75(11):2081-2084,1995.
[71]Richard E. Prange and Steven M. Girvin. The Quantum Hall effect. Springer-Verlag,2nd edition,1990.
[72]D. V. Khveshchenko. Ghost excitonic insulator transition in layered graphite. Physical Review Letters,87(24),2001.
[73]E. V. Gorbar, V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy. Magnetic field driven metal-insulator phase transition in planar systems. Physical Review B,66 (4):22,2002.
[74]D. V. Khveshchenko and W. F. Shively. Excitonic pairing between nodal fermions. Physical Review B,73(11),2006.
[75]D. T. Son. Quantum critical point in graphene approached in the limit of in-finitely strong coulomb interaction. Physical Review B,75(23),2007.
[76]S. Hands and C. Strouthos. Quantum critical behavior in a graphenelike model. Physical Review B,78(16),2008.
[77]Michael Edward Peskin and Daniel V. Schroeder. An introduction to quantum field theory. Addison-Wesley Pub. Co.,1995.
[78]S. P. Klevansky. The nambu-jona-lasinio model of quantum chromodynamics. Reviews of Modern Physics,64:649-708,1992.
[79]David J. Gross and AndreNeveu. Dynamical symmetry breaking in asymptoti-cally free field theories. Physical Review D,10:3235-3253,1974.
[80]Thomas Appelquist, Mark J. Bowick, Dimitra Karabali, and L. C. R. Wijew-ardhana. Spontaneous breaking of parity in (2+1)-dimensional qed. Physical Review D,33:3774,1986.
[81]B. Lake, G. Aeppli, K. N. Clausen, D. F. McMorrow, K. Lefmann, N. E. Hussey, N. Mangkorntong, M. Nohara, H. Takagi, T. E. Mason, and A. Schr?der. Spins in the vortices of a high-temperature superconductor. Science,291:1759-1762, 2001.
[82]B. Lake, H. M. R?nnow, N. B. Christensen, G. Aeppli, K. Lefmann, D. F. Mc-Morrow, P. Vorderwisch, P. Smeibidl, N. Mangkorntong, T. Sasagawa, M. No-hara, H. Takagi, and T. E. Mason. Antiferromagnetic order induced by an ap-plied magnetic field in a high-temperature superconductor. Nature,415:299-302,2002.
[83]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 sur-rounding vortex cores in bi2sr2cacu2o8+δ. Science,295:466-469,2002.
[84]G. Z. Liu and G. Cheng. Extension of the anderson-higgs mechanism. Physical Review B,65(13),2002.
[85]G. Z. Liu and G. Cheng. Effect of gauge boson mass on chiral symmetry break-ing in three-dimensional qed. Physical Review D,67(6),2003.
[86]Elbio Dagotto, Eduardo Fradkin, and Adriana Moreo. Su(2) gauge invariance and order parameters in strongly coupled electronic systems. Physical Review B,38:2926-2929,1988.
[87]J. B. Marston. Instantons and massless fermions in (2+1)-dimensional lattice qed and antiferromagnets. Physical Review Letters,64:1166-1169,1990.
[88]D. H. Kim and P. A. Lee. Theory of spin excitations in undoped and underdoped cuprates. Annals of Physics,272(1):130-164,1999.
[89]I. F. Herbut. Antiferromagnetism from phase disordering of a d-wave supercon-ductor. Physical Review Letters,88(4),2002.
[90]Z. Tesanovic, O. Vafek, and M. Franz. Chiral symmetry breaking and phase fluctuations:A qed3 theory of the pseudogap state in cuprate superconductors. Physical Review B,65(18),2002.
[91]G. Z. Liu, H. Jiang, W. Li, and G. Cheng. Quantum electrodynamics in three dimensions:Dynamical chiral symmetry breaking, confinement, and disorder effects. Physical Review B,79(1),2009.
[92]O. Vafek. Anomalous thermodynamics of coulomb-interacting massless dirac fermions in two spatial dimensions. Physical Review Letters,98(21),2007.
[93]A. K. Geim and K. S. Novoselov. The rise of graphene. Nature Materials,6(3): 183-191,2007.
[94]A. K. Geim. Graphene:Status and prospects. Science,324(5934):1530-1534, 2009.
[95]A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim. The electronic properties of graphene. Reviews of Modern Physics,81(1):109-162,2009.
[96]J. Gonzalez, F. Guinea, and M. A. H. Vozmediano. Non-fermi liquid behav-ior of electrons in the half-filled honeycomb lattice-(a renormalization-group approach). Nuclear Physics B,424(3):595-618,1994.
[97]I. F. Herbut. Interactions and phase transitions on graphene's honeycomb lattice. Physical Review Letters,97(14),2006.
[98]G. Z. Liu, W. Li, and G. Cheng. Interaction and excitonic insulating transition in graphene. Physical Review B,79(20),2009.
[99]Thomas Appelquist, Kenneth Lane, and Uma Mahanta. Ladder approximation for spontaneous chiral-symmetry breaking. Physical Review Letters,61(14): 1553,1988.
[100]Thomas Appelquist, Daniel Nash, and L. C. R. Wijewardhana. Critical behavior in (2+1)-dimensional qed. Physical Review Letters,60(25):2575,1988.
[101]Thomas Appelquist and L. C. R. Wijewardhana. Phase structure of non-compact qed3 and the abelian higgs model. ArXiv:hep-ph\0403250,2004.
[102]G. Z. Liu. Confinement of matter fields in compact (2+1)-dimensional qed the-ory of high-t-c superconductors. Physical Review B,71(17),2005.
[103]G. Z. Liu, W. Li, and G. Cheng. Matter fields near quantum critical point in (2+1)-dimensional u(1) gauge theory. Nuclear Physics B,825(3):303-319, 2010.
[104]W. Li and G. Z. Liu. Dynamical chiral symmetry breaking in qed3 at finite density and impurity potential. Physical Review D,81(4),2010.
[105]H. T. Feng, F. Y. Hou, X. He, W. M. Sun, and H. S. Zong. Influence of finite chemical potential on the critical number of fermion flavors in qed3. Physical Review D,73(1),2006.
[106]H. T. Feng, W. M. Sun, D. K. He, and H. S. Zong. Influence of finite chemical potential on the fermion chiral condensate in qed3. Physics Letters B,661(1): 57-65,2008.
[107]P. M. Ostrovsky, I. V. Gornyi, and A. D. Mirlin. Electron transport in disordered graphene. Physical Review B,74(23),2006.
[108]A. A. Nersesyan, A. M. Tsvelik, and F. Wenger. Disorder effects in 2-dimensional fermi systems with conical spectrum-exact results for the density-of-states. Nuclear Physics B,438(3):561-588,1995.
[109]V. P. Gusynin and S. G. Sharapov. Transport of dirac quasiparticles in graphene: Hall and optical conductivities. Physical Review B,73(24),2006.
[110]V. P. Gusynin and V. A. Miransky. Thermal conductivity and competing orders in d-wave superconductors. European Physical Journal B,37(3):363-368,2004.
[111]N. Dorey and N. E. Mavromatos. Qed3 and 2-dimensional superconductivity without parity violation. Nuclear Physics B,386(3):614-680,1992.
[112]D. Atkinson, V. P. Gusynin, and P. Maris. Chiral symmetry-breaking with the curtis-pennington vertex. Physics Letters B,303(1-2):157-162,1993.
[113]G. Cheng and T. K. Kuo. Bifurcation solutions of the schwinger-dyson equation. Journal of Mathematical Physics,35(12):6270-6290,1994.
[114]G. Cheng and T. K. Kuo. A generalized parameter imbedding method. Journal of Mathematical Physics,35(12):6693-6702,1994.
[115]P. A. Lee, N. Nagaosa, and X. G. Wen. Doping a mott insulator:Physics of high-temperature superconductivity. Reviews of Modern Physics,78(1):17-85, 2006.
[116]Ian Affleck and J. Brad Marston. Large-n limit of the heisenberg-hubbard model: Implications for high-tc superconductors. Physical Review B,37(7):3774-3777, 1988.
[117]L. B. Ioffe and A. I. Larkin. Gapless fermions and gauge fields in dielectrics. Physical Review B,39(13):8988,1989.
[118]D. H. Kim, P. A. Lee, and X. G. Wen. Massless dirac fermions, gauge fields, and underdoped cuprates. Physical Review Letters,79(11):2109-2112,1997.
[119]W. Rantner and X. G. Wen. Electron spectral function and algebraic spin liquid for the normal state of underdoped high t-c superconductors. Physical Review Letters,86(17):3871-3874,2001.
[120]T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and M. P. A. Fisher. Quantum criticality beyond the landau-ginzburg-wilson paradigm. Physical Review B,70 (14):33-2004.
[121]T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher. Decon-fined quantum critical points. Science,303(5663):1490-1494,2004.
[122]R. K. Kaul, A. Kolezhuk, M. Levin, S. Sachdev, and T. Senthil. Hole dynamics in an antiferromagnet across a deconfined quantum critical point. Physical Review B,75(23),2007.
[123]R. K. Kaul, Y. B. Kim, S. Sachdev, and T. Senthil. Algebraic charge liquids. Nature Physics,4(1):28-31,2008.
[124]R. K. Kaul and R. G. Melko. Large-n estimates of universal amplitudes of the cpn-1 theory and comparison with a s=1/2 square-lattice model with competing four-spin interactions. Physical Review B,78(1),2008.
[125]C. J. Burden, J. Praschifka, and C. D. Roberts. Photon polarization tensor and gauge dependence in 3-dimensional quantum electrodynamics. Physical Review D,46(6):2695-2702,1992.
[126]M. Hermele, T. Senthil, M. P. A. Fisher, P. A. Lee, N. Nagaosa, and X. G. Wen. Stability of u(1) spin liquids in two dimensions. Physical Review B,70(21):9, 2004.
[127]F. S. Nogueira and H. Kleinert. Compact quantum electrodynamics in 2+1 di-mensions and spinon deconfinement:A renormalization group analysis. Physi-cal Review B,77(4),2008.
[128]S. Takashima, I. Ichinose, and T. Matsui. Deconfinement of spinons on critical points:Multiflavor cp1+u(1) lattice gauge theory in three dimensions. Physical Review B,73(7),2006.
[129]A. M. Polyakov. Gauge fields and strings. Harwood Academic Publishers,1987.
[130]LD Landau and IM Khalatnikov. The gauge transformation of the green's func-tion for charged particles. JETP,1956.
[131]Bruno Zumino. Gauge properties of propagators in quantum electrodynamics. Journal of Mathematical Physics,1:1-7,1960.
[132]H. Georgi, E. H. Simmons, and A. G. Cohen. Finding gauges where z(p) equals one. Physics Letters B,236(2):183-186,1990.
[133]T. Kugo and M. G. Mitchard. The chiral ward-takahashi identity in the ladder approximation. Physics Letters B,282(1-2):162-170,1992.
[134]K. I. Kondo, T. Ebihara, T. Lizuka, and E. Tanaka. Dynamical breakdown of chirality and parity in (2+1)-dimensional qed. Nuclear Physics B,434(1-2): 85-108,1995.
[135]K. I. Kondo. Running coupling constant of a gauge theory in the framework of the schwinger-dyson equation:Infrared behavior of three-dimensional quantum electrodynamics. Physical Review D,55(12):7826-7839,1997.
[136]D. J. Lee and I. F. Herbut. Velocity anisotropy and the antiferromagnetic in-stability in the qed3 theory of underdoped cuprates. Physical Review B,66(9), 2002.
[137]O. Vafek, Z. Tesanovic, and M. Franz. Relativity restored:Dirac anisotropy in qed3. Physical Review Letters,89(15),2002.
[138]Y. Ran, M. Hermele, P. A. Lee, and X. G. Wen. Projected-wave-function study of the spin-1/2 heisenberg model on the kagome lattice. Physical Review Letters, 98(11),2007.
[139]C. Mudry and E. Fradkin. Separation of spin and charge quantum numbers in strongly correlated systems. Physical Review B,49(8):5200-5219,1994.
[140]S. Yamashita, Y. Nakazawa, M. Oguni, Y. Oshima, H. Nojiri, Y. Shimizu, K. Miyagawa, and K. Kanoda. Thermodynamic properties of a spin-1/2 spin-liquid state in a kappa-type organic salt. Nature Physics,4(6):459-462,2008.
[141]M. Yamashita, N. Nakata, Y. Kasahara, T. Sasaki, N. Yoneyama, N. Kobayashi, S. Fujimoto, T. Shibauchi, and Y. Matsuda. Thermal-transport measurements in a quantum spin-liquid state of the frustrated triangular magnet κ-(bedt-ttf)2cu2(cn)3. Nature Physics,5(1):44-47,2009.
[142]K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsev. Electric field effect in atomically thin carbon films. Science,306(5296):666-669,2004.
[143]Linus Pauling. The chemical bond; a brief introduction to modern structural chemistry. Cornell University Press,1967.
[144]D. V. Khveshchenko and H. Leal. Excitonic instability in layered degenerate semimetals. Nuclear Physics B,687(3):323-331,2004.
[145]O. Vafek and M. J. Case. Renormalization group approach to two-dimensional coulomb interacting dirac fermions with random gauge potential. Physical Re-view B,77(3),2008.
[146]I. L. Aleiner, D. E. Kharzeev, and A. M. Tsvelik. Spontaneous symmetry break-ing in graphene subjected to an in-plane magnetic field. Physical Review B,76 (19):195415,2007.
[147]D. E. Sheehy and J. Schmalian. Quantum critical scaling in graphene. Physical Review Letters,99(22),2007.
[148]A. H. C. Neto. Pauling's dreams for graphene. Physics,2:30,2009.
[149]Linus Pauling. The nature of the chemical bond and the structure of molecules and crystals; an introduction to modern structural chemistry. Cornell University Press,3d edition,1960.
[150]J. E. Drut and T. A. Lahde. Is graphene in vacuum an insulator? Physical Review Letters,102(2):4,2009.
[151]J. E. Drut and T. A. Lahde. Lattice field theory simulations of graphene. Physical Review B,79(16):165425,2009.
[152]J. G. Checkelsky, L. Li, and N. P. Ong. Zero-energy state in graphene in a high magnetic field. Physical Review Letters,100(20):206801,2008.
[153]D. V. Khveshchenko. Massive dirac fermions in single-layer graphene. Journal of Physics-Condensed Matter,21(7):7,2009.
[154]O. V. Gamayun, E. V. Gorbar, and V. P. Gusynin. Supercritical coulomb center and excitonic instability in graphcnc. Physical Review B,80(16),2009.
[155]Wes Armour, Simon Hands, and Costas Strouthos. Monte carlo simulation of the semimetal-insulator phase transition in monolayer graphene. Phys. Rev. B, 81(12):125105,2010.
[156]J. E. Drut and D. T. Son. Renormalization group flow of quartic perturbations in graphene:Strong coupling and large-n limits. Physical Review B,77(7):075115, 2008.
[157]M. Y. Kharitonov and K. B. Efetov. Electron screening and excitonic condensa-tion in double-layer graphene systems. Physical Review B,78(24),2008.
[158]R. Bistritzer, H. Min, J. J. Su, and A. H. MacDonald. Comment on "elec-tron screening and excitonic condensation in double-layer graphene systems". Arxiv:con-mat\0810.0331,2008.
[159]A. Altland, B. D. Simons, and M. R. Zirnbauer. Theories of low-energy quasi-particle states in disordered d-wave superconductors. Physics Reports-Review Section of Physics Letters,359(4):283-354,2002.
[160]Eduardo Fradkin. Critical behavior of disordered degenerate semiconductors, ii. spectrum and transport properties in mean-field theory. Physical Review B,33: 3263-3268,1986.
[161]Matthew P. A. Fisher and Eduardo Fradkin. Localization in a magnetic field: Tight binding model with one-half of a flux quantum per plaquette. Nuclear Physics B,251:457-471,1985.
[162]A. W. W. Ludwig, M. P. A. Fisher, R. Shankar, and G. Grinstein. Integer quantum hall transition-an alternative approach and exact results. Physical Review B,50 (11):7526-7552,1994.
[163]F. Evers and A. D. Mirlin. Anderson transitions. Reviews of Modern Physics, 80(4):1355-1417,2008.
[164]Baruch Rosenstein, Brian J. Warr, and Seon H. Park. Dynamical symmetry breaking in four-fermion interaction models. Physics Reports,205(2):59-108, 1991.
[165]J. Gonzalez, F. Guinea, and M. A. H. Vozmediano. Unconventional quasiparticle lifetime in graphite. Physical Review Letters,77(17):3589-3592,1996.
[166]S. Das Sarma, E. H. Hwang, and W. K. Tse. Many-body interaction effects in doped and undoped graphene:Fermi liquid versus non-fermi liquid. Physical Review B,75(12):121406,2007.
[167]C. K. Xu, Y. Qi, and S. Sachdev. Experimental observables near a nematic quantum critical point in the pnictide and cuprate superconductors. Physical Review B,78(13):8,2008.
[168]V. P. Gusynin, V. A. Miransky, S. G. Sharapov, and I. A. Shovkovy. Excitonic gap, phase transition, and quantum hall effect in graphene. Physical Review B, 74(19),2006.
[169]V. N. Kotov, B. Uchoa, and A. H. C. Neto.1/n expansion in correlated graphene. Physical Review B,80(16),2009.
[170]A. Qaiumzadeh and R. Asgari. The effect of sublattice symmetry breaking on the electronic properties of doped graphene. New Journal of Physics,11:-,2009.
[171]H. Jiang, G. Z. Liu, and G. Cheng. Pairing instability in the mixed state of a d-wave superconductor. Physical Review B,79(17),2009.
[172]R. K. Kaul and S. Sachdev. Quantum criticality of u(1) gauge theories with fermionic and bosonic matter in two spatial dimensions. Physical Review B,77 (15),2008.
[173]Joseph I. Kapusta and Charles Gale. Finite-temperature field theory:principles and applications. Cambridge University Press,2nd edition,2006.
[174]M. R. Ramezanali, M. M. Vazifeh, R. Asgari, M. Polini, and A. H. MacDonald. Finite-temperature screening and the specific heat of doped graphene sheets. Journal of Physics a-Mathematical and Theoretical,42(21),2009.
[175]A. V. Chubukov, D. L. Maslov, S. Gangadharaiah, and L. I. Glazman. Singular perturbation theory for interacting fermions in two dimensions. Physical Review B,71(20):205112,2005.
[176]Steven Weinberg. The quantum theory of fields. Cambridge University Press, 1995.
[177]J. Alicea and M. P. A. Fisher. Graphene integer quantum hall effect in the ferro-magnetic and paramagnetic regimes. Physical Review B,74(7):075422,2006.
[178]O. V. Gamayun, E. V. Gorbar, and V. P. Gusynin. Gap generation and semimetal-insulator phase transition in graphene. Physical Review B,81(7),2010.
[2]Gordon Baym and Christopher Pethick. Landau Fermi-liquid theory:concepts and applications. Wiley,1991.
[3]C. M. Varma, Z. Nussinov, and W. van Saarloos. Singular or non-fermi liquids. Physics Reports-Review Section of Physics Letters,361(5-6):267-417,2002.
[4]Xiao-Gang Wen. Quantum field theory of many-body systems:from the origin of sound to an origin of light and electrons. Oxford University Press,2004.
[5]H. J. Schulz. Fermi liquids and non-fermi liquids. Mesoscopic Quantum Physics, 61:533-603791,1995.
[6]冯端,金国钧.凝聚态物理学上卷.北京-高等教育出版社,2003.
[7]闫守胜.固体物理基础第二版.北京大学出版社,2003.
[8]H. J. Schulz, G. Cuniberti, and P. Pieri. Fermi liquids and luttinger liquids. Field Theories for Low-Dimensional Condensed Matter Systems,131:9-81275,2000.
[9]H. von Lohneysen, A. Rosch, M. Vojta, and P. Wolfle. Fermi-liquid instabilities at magnetic quantum phase transitions. Reviews of Modern Physics,79(3):1015-1075,2007.
[10]N. F. Mott and R. Peierls. Discussion of the paper by de boer and verwey. Proceedings of the Physical Society,49:72-73,1937.
[11]Wikipedia. High-temperature superconductivity-wikipedia, the free en-cyclopedia.2010. URL http://en.wikipedia.org/w/index. php?title=High-temperature_superconductivity & oldid= 346945237.
[12]J. G. Bednorz and K. A. Mueller. Possible high tc superconductivity in the ba-la-cu-o system. Z. Phys. B,64(189-193),1986.
[13]Wikipedia. Iron-based superconductor - wikipedia, the free encyclope-dia.2010. URL http://en.wikipedia.org/w/index.php?title= Iron-based_superconductor & oldid=342436699.
[14]Y. Kamihara, H. Hiramatsu, M. Hirano, R. Kawamura, H. Yanagi, T. Kamiya, and H. Hosono. Iron-based layered superconductor:Laofep. Journal of the American Chemical Society,128(31):10012-10013,2006.
[15]Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono. Iron-based layered super-conductor la[o1-xfx]feas (x=0.05-0.12) with tc=26 k. Journal of the American Chemical Society,130(11):3296-+,2008.
[16]向涛.d波超导体.科学出版社,2007.
[17]W. N. Hardy, D. A. Bonn, D. C. Morgan, Ruixing Liang, and Kuan Zhang. Precision measurements of the temperature dependence of λ in yba2cu3o6.95: Strong evidence for nodes in the gap function. Physical Review Letters,70: 3999-4002,1993.
[18]Z.-X. Shen, D. S. Dessau, B. O. Wells, D. M. King, W. E. Spicer, A. J. Arko, D. Marshall, L. W. Lombardo, A. Kapitulnik, and P. Dickinson. Anomalously large gap anisotropy in the a-b plane of bi2sr2cacu2o(8+δ). Physical Review Letters, 70:1553-1556,1993.
[19]Z.-X. Shen and D. S. Dessau. Electronic structure and photoemission studies of late transition-metal oxides-mott insulators and high-temperature supercon-ductors. Physics Reports,253:1-162,1995.
[20]H. Ding, M. R. Norman, J. C. Campuzano, M. Randeria, A. F. Bellman, T. Yokoya, T. Takahashi, T. Mochiku, and K. Kadowaki. Angle-resolved photoemission spectroscopy study of the superconducting gap anisotropy in bi2sr2cacu2o8+x. Physical Review B,54:9678,1996.
[21]G. E. Volovik. Superconductivity with lines of gap nodes:density of states in the vortex. Soviet Journal of Experimental and Theoretical Physics Letters,58: 469,1993.
[22]K. A. Moler, D. J. Baar, J. S. Urbach, Ruixing Liang, W. N. Hardy, and A. Kapitulnik. Magnetic field dependence of the density of states of yba2cu3o695 as determined from the specific heat. Physical Review Letters,73:2744-2747, 1994.
[23]Kathryn A. Moler, David L. Sisson, Jeffrey S. Urbach, Malcolm R. Beasley, Aharon Kapitulnik, David J. Baar, Ruixing Liang, and Walter N. Hardy. Specific heat of yba2cu3o7-d. Physical Review B,55:3954-3965,1997.
[24]P. A. Lee. Localized states in a d-wave superconductor. Physical Review Letters, 71(12):1887,1993.
[25]Louis Taillefer, Benoit Lussier, Robert Gagnon, Kamran Behnia, and HerveAubin. Universal heat conduction in yba2cu3o6.9. Physical Review Letters, 79:483-486,1997.
[26]D. A. Wollman, D. J. van Harlingen, W. C. Lee, D. M. Ginsberg, and A. J. Leggett. Experimental determination of the superconducting pairing state in ybco from the phase coherence of ybco-pb dc squids. Physical Review Letters, 71:2134-2137,1993.
[27]D. A. Wollman, D. J. van Harlingen, J. Giapintzakis, and D. M. Ginsberg. Ev-idence for dx2-y2 pairing from the magnetic field modulation of yba2cu3o7-pb josephson junctions. Physical Review Letters,74:797-800,1995.
[28]A. Mathai, Y. Gim, R. C. Black, A. Amar, and F. C. Wellstood. Experimental proof of a time-reversal-invariant order parameter with a p shift in yba2cu3O7-d. Physical Review Letters,74:4523-4526,1995.
[29]D. J. Scalapino. The case for dx2-y2 pairing in the cuprate superconductors. Physics Reports-Review Section of Physics Letters,250(6):330-365,1995.
[30]D. J. Vanharlingen. Phase-sensitive tests of the symmetry of the pairing state in the high-temperature superconductors-evidence for dx2-y2 symmetry. Reviews of Modern Physics,67(2):515-535,1995.
[31]C. C. Tsuei and J. R. Kirtley. Pairing symmetry in cuprate superconductors. Reviews of Modern Physics,72(4):969-1016,2000.
[32]A. C. Durst and P. A. Lee. Impurity-induced quasiparticle transport and universal-limit wiedemann-franz violation in d-wave superconductors. Physi-cal Review B,62(2):1270-1290,2000.
[33]Gabriel Kotliar and Jialin Liu. Superexchange mechanism and d-wave super-conductivity. Physical Review B,38:5142-5145,1988.
[34]L. P. Gor'kov and P. A. Kalugin. Defects and an unusual superconductivity. Soviet Journal of Experimental and Theoretical Physics Letters,41:253,1985.
[35]P. W. Anderson. The resonating valence bond state in la2cuo4 and superconduc-tivity. Science,235:1196-1198,1987.
[36]F. C. Zhang and T. M. Rice. Effective hamiltonian for the superconducting cu oxides. Physical Review B,37:3759-3761,1988.
[37]N. Read and D. M. Newns. A new functional integral formalism for the de-generate anderson model. Journal of Physics C Solid State Physics,16:1055, 1983.
[38]G. Baskaran and P. W. Anderson. Gauge theory of high-temperature supercon-ductors and strongly correlated fermi systems. Physical Review B,37(1):580, 1988.
[39]M. U. Ubbens and P. A. Lee. Superconductivity phase-diagram in the gauge-field description of the t-j model. Physical Review B,49(10):6853-6863,1994.
[40]B. L. Altshuler, L. B. Ioffe, and A. J. Millis. Theory of the spin gap in high-temperature superconductors. Physical Review B,53(1):415-424,1996.
[41]P. A. Lee and N. Nagaosa. Gauge-theory of the normal state of high-tc super-conductors. Physical Review B,46(9):5621-5639,1992.
[42]X. G. Wen and P. A. Lee. Theory of underdoped cuprates. Physical Review Letters,76(3):503-506,1996.
[43]P. A. Lee, N. Nagaosa, T. K. Ng, and X. G. Wen. Su(2) formulation of the t-j model:Application to underdoped cuprates. Physical Review B,57(10):6003-6021,1998.
[44]S. Reich, J. Maultzsch, C. Thomsen, and P. Ordejon. Tight-binding description of graphene. Physical Review B,66(3),2002.
[45]P. R. Wallace. The band theory of graphite. Physical Review,71:622-634,1947.
[46]Neil W. Ashcroft and N. David Mermin. Solid state physics. Holt,1976.
[47]K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov. Two-dimensional gas of massless dirac fermions in graphene. Nature,438(7065):197-200,2005.
[48]Gordon W. Semenoff. Condensed-matter simulation of a three-dimensional anomaly. Physical Review Letters,53:2449-2452,1984.
[49]Y. Nambu and G. Jona-Lasinio. Dynamical model of elementary particles based on an analogy with superconductivity. i. Physical Review,122:345-358,1961.
[50]Y. Nambu and G. Jona-Lasinio. Dynamical model of elementary particles based on an analogy with superconductivity. ii. Physical Review,124:246-254,1961.
[51]R. D. Pisarski. Chiral-symmetry breaking in 3-dimensional electrodynamics. Physical Review D,29(10):2423-2426,1984.
[52]R. D. Pisarski. Finite-temperature qcd at large n. Physical Review D,29(6): 1222-1227,1984.
[53]Thomas W. Appelquist, Mark Bowick, Dimitra Karabali, and L. C. R. Wije-wardhana. Spontaneous chiral-symmetry breaking in three-dimensional qed. Physical Review D,33(12):3704,1986.
[54]T. Appelquist, D. Carrier, L. C. R. Wijewardhana, and W. Zheng. Numerical studies of enhanced chiral condensates. Physical Review Letters,60(12):1114, 1988.
[55]Daniel Nash. Higher-order corrections in (2+1)-dimensional qed. Physical Re-view Letters,62:3024-3026,1989.
[56]Elbio Dagotto, John B. Kogut, and Aleksandar Kocic. Computer simulation of chiral-symmetry breaking in (2+1)-dimensional qed with n flavors. Physical Review Letters,62:1083-1086,1989.
[57]Elbio Dagotto, Aleksandar Kocic, and J. B. Kogut. Chiral symmetry breaking in three dimensional qed with nflavors. Nuclear Physics B,334:279-301,1990.
[58]D. Atkinson, P. W. Johnson, and P. Maris. Dynamic mass generation in 3-dimensional qed-improved vertex function. Physical Review D,42(2):602-609, 1990.
[59]I. J. R. Aitchison, N. Dorey, M. Kleinkreisler, and N. E. Mavromatos. Phase-structure of qed3 at finite temperature. Physics Letters B,294(1):91-99,1992.
[60]D. C. Curtis, M. R. Pennington, and D. Walsh. Dynamical mass generation in qed3 and the 1/n expansion. Physics Letters B,295:313-319,1992.
[61]M. R. Pennington and D. Walsh. Masses from nothing. a non-perturbative study of qed3. Physics Letters B,253:246-251,1991.
[62]P. Maris. Confinement and complex singularities in 3-dimensional qed. Physical Review D,52(10):6087-6097,1995.
[63]P. Maris. Influence of the full vertex and vacuum polarization on the fermion propagator in (2+1)-dimensional qed. Physical Review D,54(6):4049-4058, 1996.
[64]V. P. Gusynin, A. H. Hams, and M. Reenders. (2+1)-dimensional qed with dy-namically massive fermions in vacuum polarization. Physical Review D,53(4): 2227,1996.
[65]C. S. Fischer, R. Alkofer, T. Dahm, and P. Maris. Dynamical chiral symmetry breaking in unquenched qed3. Physical Review D,70(7),2004.
[66]S. J. Hands, J. B. Kogut, and C. G. Strouthos. Non-compact qed3 with n-f(?)= 2. Nuclear Physics B,645(1-2):321-336,2002.
[67]S. J. Hands, J. B. Kogut, L. Scorzato, and C. G. Strouthos. Noncompact three-dimensional quantum electrodynamics with nf=1 and n-f=4. Physical Review B,70(10),2004.
[68]V. P. Gusynin and A. Reenders. Infrared cutoff dependence of the critical flavor number in three-dimensional qed. Physical Review D,68(2),2003.
[69]Craig D. Roberts and Anthony G. Williams. Dyson-schwinger equations and their application to hadronic physics. Progress in Particle and Nuclear Physics, 33:477-575,1994.
[70]T. Appelquist, J. Terning, and L. C. R. Wijewardhana. (2+1)-dimensional qed and a novel phase-transition. Physical Review Letters,75(11):2081-2084,1995.
[71]Richard E. Prange and Steven M. Girvin. The Quantum Hall effect. Springer-Verlag,2nd edition,1990.
[72]D. V. Khveshchenko. Ghost excitonic insulator transition in layered graphite. Physical Review Letters,87(24),2001.
[73]E. V. Gorbar, V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy. Magnetic field driven metal-insulator phase transition in planar systems. Physical Review B,66 (4):22,2002.
[74]D. V. Khveshchenko and W. F. Shively. Excitonic pairing between nodal fermions. Physical Review B,73(11),2006.
[75]D. T. Son. Quantum critical point in graphene approached in the limit of in-finitely strong coulomb interaction. Physical Review B,75(23),2007.
[76]S. Hands and C. Strouthos. Quantum critical behavior in a graphenelike model. Physical Review B,78(16),2008.
[77]Michael Edward Peskin and Daniel V. Schroeder. An introduction to quantum field theory. Addison-Wesley Pub. Co.,1995.
[78]S. P. Klevansky. The nambu-jona-lasinio model of quantum chromodynamics. Reviews of Modern Physics,64:649-708,1992.
[79]David J. Gross and AndreNeveu. Dynamical symmetry breaking in asymptoti-cally free field theories. Physical Review D,10:3235-3253,1974.
[80]Thomas Appelquist, Mark J. Bowick, Dimitra Karabali, and L. C. R. Wijew-ardhana. Spontaneous breaking of parity in (2+1)-dimensional qed. Physical Review D,33:3774,1986.
[81]B. Lake, G. Aeppli, K. N. Clausen, D. F. McMorrow, K. Lefmann, N. E. Hussey, N. Mangkorntong, M. Nohara, H. Takagi, T. E. Mason, and A. Schr?der. Spins in the vortices of a high-temperature superconductor. Science,291:1759-1762, 2001.
[82]B. Lake, H. M. R?nnow, N. B. Christensen, G. Aeppli, K. Lefmann, D. F. Mc-Morrow, P. Vorderwisch, P. Smeibidl, N. Mangkorntong, T. Sasagawa, M. No-hara, H. Takagi, and T. E. Mason. Antiferromagnetic order induced by an ap-plied magnetic field in a high-temperature superconductor. Nature,415:299-302,2002.
[83]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 sur-rounding vortex cores in bi2sr2cacu2o8+δ. Science,295:466-469,2002.
[84]G. Z. Liu and G. Cheng. Extension of the anderson-higgs mechanism. Physical Review B,65(13),2002.
[85]G. Z. Liu and G. Cheng. Effect of gauge boson mass on chiral symmetry break-ing in three-dimensional qed. Physical Review D,67(6),2003.
[86]Elbio Dagotto, Eduardo Fradkin, and Adriana Moreo. Su(2) gauge invariance and order parameters in strongly coupled electronic systems. Physical Review B,38:2926-2929,1988.
[87]J. B. Marston. Instantons and massless fermions in (2+1)-dimensional lattice qed and antiferromagnets. Physical Review Letters,64:1166-1169,1990.
[88]D. H. Kim and P. A. Lee. Theory of spin excitations in undoped and underdoped cuprates. Annals of Physics,272(1):130-164,1999.
[89]I. F. Herbut. Antiferromagnetism from phase disordering of a d-wave supercon-ductor. Physical Review Letters,88(4),2002.
[90]Z. Tesanovic, O. Vafek, and M. Franz. Chiral symmetry breaking and phase fluctuations:A qed3 theory of the pseudogap state in cuprate superconductors. Physical Review B,65(18),2002.
[91]G. Z. Liu, H. Jiang, W. Li, and G. Cheng. Quantum electrodynamics in three dimensions:Dynamical chiral symmetry breaking, confinement, and disorder effects. Physical Review B,79(1),2009.
[92]O. Vafek. Anomalous thermodynamics of coulomb-interacting massless dirac fermions in two spatial dimensions. Physical Review Letters,98(21),2007.
[93]A. K. Geim and K. S. Novoselov. The rise of graphene. Nature Materials,6(3): 183-191,2007.
[94]A. K. Geim. Graphene:Status and prospects. Science,324(5934):1530-1534, 2009.
[95]A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim. The electronic properties of graphene. Reviews of Modern Physics,81(1):109-162,2009.
[96]J. Gonzalez, F. Guinea, and M. A. H. Vozmediano. Non-fermi liquid behav-ior of electrons in the half-filled honeycomb lattice-(a renormalization-group approach). Nuclear Physics B,424(3):595-618,1994.
[97]I. F. Herbut. Interactions and phase transitions on graphene's honeycomb lattice. Physical Review Letters,97(14),2006.
[98]G. Z. Liu, W. Li, and G. Cheng. Interaction and excitonic insulating transition in graphene. Physical Review B,79(20),2009.
[99]Thomas Appelquist, Kenneth Lane, and Uma Mahanta. Ladder approximation for spontaneous chiral-symmetry breaking. Physical Review Letters,61(14): 1553,1988.
[100]Thomas Appelquist, Daniel Nash, and L. C. R. Wijewardhana. Critical behavior in (2+1)-dimensional qed. Physical Review Letters,60(25):2575,1988.
[101]Thomas Appelquist and L. C. R. Wijewardhana. Phase structure of non-compact qed3 and the abelian higgs model. ArXiv:hep-ph\0403250,2004.
[102]G. Z. Liu. Confinement of matter fields in compact (2+1)-dimensional qed the-ory of high-t-c superconductors. Physical Review B,71(17),2005.
[103]G. Z. Liu, W. Li, and G. Cheng. Matter fields near quantum critical point in (2+1)-dimensional u(1) gauge theory. Nuclear Physics B,825(3):303-319, 2010.
[104]W. Li and G. Z. Liu. Dynamical chiral symmetry breaking in qed3 at finite density and impurity potential. Physical Review D,81(4),2010.
[105]H. T. Feng, F. Y. Hou, X. He, W. M. Sun, and H. S. Zong. Influence of finite chemical potential on the critical number of fermion flavors in qed3. Physical Review D,73(1),2006.
[106]H. T. Feng, W. M. Sun, D. K. He, and H. S. Zong. Influence of finite chemical potential on the fermion chiral condensate in qed3. Physics Letters B,661(1): 57-65,2008.
[107]P. M. Ostrovsky, I. V. Gornyi, and A. D. Mirlin. Electron transport in disordered graphene. Physical Review B,74(23),2006.
[108]A. A. Nersesyan, A. M. Tsvelik, and F. Wenger. Disorder effects in 2-dimensional fermi systems with conical spectrum-exact results for the density-of-states. Nuclear Physics B,438(3):561-588,1995.
[109]V. P. Gusynin and S. G. Sharapov. Transport of dirac quasiparticles in graphene: Hall and optical conductivities. Physical Review B,73(24),2006.
[110]V. P. Gusynin and V. A. Miransky. Thermal conductivity and competing orders in d-wave superconductors. European Physical Journal B,37(3):363-368,2004.
[111]N. Dorey and N. E. Mavromatos. Qed3 and 2-dimensional superconductivity without parity violation. Nuclear Physics B,386(3):614-680,1992.
[112]D. Atkinson, V. P. Gusynin, and P. Maris. Chiral symmetry-breaking with the curtis-pennington vertex. Physics Letters B,303(1-2):157-162,1993.
[113]G. Cheng and T. K. Kuo. Bifurcation solutions of the schwinger-dyson equation. Journal of Mathematical Physics,35(12):6270-6290,1994.
[114]G. Cheng and T. K. Kuo. A generalized parameter imbedding method. Journal of Mathematical Physics,35(12):6693-6702,1994.
[115]P. A. Lee, N. Nagaosa, and X. G. Wen. Doping a mott insulator:Physics of high-temperature superconductivity. Reviews of Modern Physics,78(1):17-85, 2006.
[116]Ian Affleck and J. Brad Marston. Large-n limit of the heisenberg-hubbard model: Implications for high-tc superconductors. Physical Review B,37(7):3774-3777, 1988.
[117]L. B. Ioffe and A. I. Larkin. Gapless fermions and gauge fields in dielectrics. Physical Review B,39(13):8988,1989.
[118]D. H. Kim, P. A. Lee, and X. G. Wen. Massless dirac fermions, gauge fields, and underdoped cuprates. Physical Review Letters,79(11):2109-2112,1997.
[119]W. Rantner and X. G. Wen. Electron spectral function and algebraic spin liquid for the normal state of underdoped high t-c superconductors. Physical Review Letters,86(17):3871-3874,2001.
[120]T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and M. P. A. Fisher. Quantum criticality beyond the landau-ginzburg-wilson paradigm. Physical Review B,70 (14):33-2004.
[121]T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher. Decon-fined quantum critical points. Science,303(5663):1490-1494,2004.
[122]R. K. Kaul, A. Kolezhuk, M. Levin, S. Sachdev, and T. Senthil. Hole dynamics in an antiferromagnet across a deconfined quantum critical point. Physical Review B,75(23),2007.
[123]R. K. Kaul, Y. B. Kim, S. Sachdev, and T. Senthil. Algebraic charge liquids. Nature Physics,4(1):28-31,2008.
[124]R. K. Kaul and R. G. Melko. Large-n estimates of universal amplitudes of the cpn-1 theory and comparison with a s=1/2 square-lattice model with competing four-spin interactions. Physical Review B,78(1),2008.
[125]C. J. Burden, J. Praschifka, and C. D. Roberts. Photon polarization tensor and gauge dependence in 3-dimensional quantum electrodynamics. Physical Review D,46(6):2695-2702,1992.
[126]M. Hermele, T. Senthil, M. P. A. Fisher, P. A. Lee, N. Nagaosa, and X. G. Wen. Stability of u(1) spin liquids in two dimensions. Physical Review B,70(21):9, 2004.
[127]F. S. Nogueira and H. Kleinert. Compact quantum electrodynamics in 2+1 di-mensions and spinon deconfinement:A renormalization group analysis. Physi-cal Review B,77(4),2008.
[128]S. Takashima, I. Ichinose, and T. Matsui. Deconfinement of spinons on critical points:Multiflavor cp1+u(1) lattice gauge theory in three dimensions. Physical Review B,73(7),2006.
[129]A. M. Polyakov. Gauge fields and strings. Harwood Academic Publishers,1987.
[130]LD Landau and IM Khalatnikov. The gauge transformation of the green's func-tion for charged particles. JETP,1956.
[131]Bruno Zumino. Gauge properties of propagators in quantum electrodynamics. Journal of Mathematical Physics,1:1-7,1960.
[132]H. Georgi, E. H. Simmons, and A. G. Cohen. Finding gauges where z(p) equals one. Physics Letters B,236(2):183-186,1990.
[133]T. Kugo and M. G. Mitchard. The chiral ward-takahashi identity in the ladder approximation. Physics Letters B,282(1-2):162-170,1992.
[134]K. I. Kondo, T. Ebihara, T. Lizuka, and E. Tanaka. Dynamical breakdown of chirality and parity in (2+1)-dimensional qed. Nuclear Physics B,434(1-2): 85-108,1995.
[135]K. I. Kondo. Running coupling constant of a gauge theory in the framework of the schwinger-dyson equation:Infrared behavior of three-dimensional quantum electrodynamics. Physical Review D,55(12):7826-7839,1997.
[136]D. J. Lee and I. F. Herbut. Velocity anisotropy and the antiferromagnetic in-stability in the qed3 theory of underdoped cuprates. Physical Review B,66(9), 2002.
[137]O. Vafek, Z. Tesanovic, and M. Franz. Relativity restored:Dirac anisotropy in qed3. Physical Review Letters,89(15),2002.
[138]Y. Ran, M. Hermele, P. A. Lee, and X. G. Wen. Projected-wave-function study of the spin-1/2 heisenberg model on the kagome lattice. Physical Review Letters, 98(11),2007.
[139]C. Mudry and E. Fradkin. Separation of spin and charge quantum numbers in strongly correlated systems. Physical Review B,49(8):5200-5219,1994.
[140]S. Yamashita, Y. Nakazawa, M. Oguni, Y. Oshima, H. Nojiri, Y. Shimizu, K. Miyagawa, and K. Kanoda. Thermodynamic properties of a spin-1/2 spin-liquid state in a kappa-type organic salt. Nature Physics,4(6):459-462,2008.
[141]M. Yamashita, N. Nakata, Y. Kasahara, T. Sasaki, N. Yoneyama, N. Kobayashi, S. Fujimoto, T. Shibauchi, and Y. Matsuda. Thermal-transport measurements in a quantum spin-liquid state of the frustrated triangular magnet κ-(bedt-ttf)2cu2(cn)3. Nature Physics,5(1):44-47,2009.
[142]K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsev. Electric field effect in atomically thin carbon films. Science,306(5296):666-669,2004.
[143]Linus Pauling. The chemical bond; a brief introduction to modern structural chemistry. Cornell University Press,1967.
[144]D. V. Khveshchenko and H. Leal. Excitonic instability in layered degenerate semimetals. Nuclear Physics B,687(3):323-331,2004.
[145]O. Vafek and M. J. Case. Renormalization group approach to two-dimensional coulomb interacting dirac fermions with random gauge potential. Physical Re-view B,77(3),2008.
[146]I. L. Aleiner, D. E. Kharzeev, and A. M. Tsvelik. Spontaneous symmetry break-ing in graphene subjected to an in-plane magnetic field. Physical Review B,76 (19):195415,2007.
[147]D. E. Sheehy and J. Schmalian. Quantum critical scaling in graphene. Physical Review Letters,99(22),2007.
[148]A. H. C. Neto. Pauling's dreams for graphene. Physics,2:30,2009.
[149]Linus Pauling. The nature of the chemical bond and the structure of molecules and crystals; an introduction to modern structural chemistry. Cornell University Press,3d edition,1960.
[150]J. E. Drut and T. A. Lahde. Is graphene in vacuum an insulator? Physical Review Letters,102(2):4,2009.
[151]J. E. Drut and T. A. Lahde. Lattice field theory simulations of graphene. Physical Review B,79(16):165425,2009.
[152]J. G. Checkelsky, L. Li, and N. P. Ong. Zero-energy state in graphene in a high magnetic field. Physical Review Letters,100(20):206801,2008.
[153]D. V. Khveshchenko. Massive dirac fermions in single-layer graphene. Journal of Physics-Condensed Matter,21(7):7,2009.
[154]O. V. Gamayun, E. V. Gorbar, and V. P. Gusynin. Supercritical coulomb center and excitonic instability in graphcnc. Physical Review B,80(16),2009.
[155]Wes Armour, Simon Hands, and Costas Strouthos. Monte carlo simulation of the semimetal-insulator phase transition in monolayer graphene. Phys. Rev. B, 81(12):125105,2010.
[156]J. E. Drut and D. T. Son. Renormalization group flow of quartic perturbations in graphene:Strong coupling and large-n limits. Physical Review B,77(7):075115, 2008.
[157]M. Y. Kharitonov and K. B. Efetov. Electron screening and excitonic condensa-tion in double-layer graphene systems. Physical Review B,78(24),2008.
[158]R. Bistritzer, H. Min, J. J. Su, and A. H. MacDonald. Comment on "elec-tron screening and excitonic condensation in double-layer graphene systems". Arxiv:con-mat\0810.0331,2008.
[159]A. Altland, B. D. Simons, and M. R. Zirnbauer. Theories of low-energy quasi-particle states in disordered d-wave superconductors. Physics Reports-Review Section of Physics Letters,359(4):283-354,2002.
[160]Eduardo Fradkin. Critical behavior of disordered degenerate semiconductors, ii. spectrum and transport properties in mean-field theory. Physical Review B,33: 3263-3268,1986.
[161]Matthew P. A. Fisher and Eduardo Fradkin. Localization in a magnetic field: Tight binding model with one-half of a flux quantum per plaquette. Nuclear Physics B,251:457-471,1985.
[162]A. W. W. Ludwig, M. P. A. Fisher, R. Shankar, and G. Grinstein. Integer quantum hall transition-an alternative approach and exact results. Physical Review B,50 (11):7526-7552,1994.
[163]F. Evers and A. D. Mirlin. Anderson transitions. Reviews of Modern Physics, 80(4):1355-1417,2008.
[164]Baruch Rosenstein, Brian J. Warr, and Seon H. Park. Dynamical symmetry breaking in four-fermion interaction models. Physics Reports,205(2):59-108, 1991.
[165]J. Gonzalez, F. Guinea, and M. A. H. Vozmediano. Unconventional quasiparticle lifetime in graphite. Physical Review Letters,77(17):3589-3592,1996.
[166]S. Das Sarma, E. H. Hwang, and W. K. Tse. Many-body interaction effects in doped and undoped graphene:Fermi liquid versus non-fermi liquid. Physical Review B,75(12):121406,2007.
[167]C. K. Xu, Y. Qi, and S. Sachdev. Experimental observables near a nematic quantum critical point in the pnictide and cuprate superconductors. Physical Review B,78(13):8,2008.
[168]V. P. Gusynin, V. A. Miransky, S. G. Sharapov, and I. A. Shovkovy. Excitonic gap, phase transition, and quantum hall effect in graphene. Physical Review B, 74(19),2006.
[169]V. N. Kotov, B. Uchoa, and A. H. C. Neto.1/n expansion in correlated graphene. Physical Review B,80(16),2009.
[170]A. Qaiumzadeh and R. Asgari. The effect of sublattice symmetry breaking on the electronic properties of doped graphene. New Journal of Physics,11:-,2009.
[171]H. Jiang, G. Z. Liu, and G. Cheng. Pairing instability in the mixed state of a d-wave superconductor. Physical Review B,79(17),2009.
[172]R. K. Kaul and S. Sachdev. Quantum criticality of u(1) gauge theories with fermionic and bosonic matter in two spatial dimensions. Physical Review B,77 (15),2008.
[173]Joseph I. Kapusta and Charles Gale. Finite-temperature field theory:principles and applications. Cambridge University Press,2nd edition,2006.
[174]M. R. Ramezanali, M. M. Vazifeh, R. Asgari, M. Polini, and A. H. MacDonald. Finite-temperature screening and the specific heat of doped graphene sheets. Journal of Physics a-Mathematical and Theoretical,42(21),2009.
[175]A. V. Chubukov, D. L. Maslov, S. Gangadharaiah, and L. I. Glazman. Singular perturbation theory for interacting fermions in two dimensions. Physical Review B,71(20):205112,2005.
[176]Steven Weinberg. The quantum theory of fields. Cambridge University Press, 1995.
[177]J. Alicea and M. P. A. Fisher. Graphene integer quantum hall effect in the ferro-magnetic and paramagnetic regimes. Physical Review B,74(7):075422,2006.
[178]O. V. Gamayun, E. V. Gorbar, and V. P. Gusynin. Gap generation and semimetal-insulator phase transition in graphene. Physical Review B,81(7),2010.