Review of electron–electron interaction effects in planar Dirac liquids

S. Teber ; A. V. Kotikov

Teoretičeskaâ i matematičeskaâ fizika, Tome 200 (2019) no. 2, pp. 343-360

Résumé

We review field theory studies devoted to understanding electron–electron interaction effects in condensed matter systems such as planar Dirac liquids, for example, graphene and graphene-like systems, surface states of some topological insulators, and possibly half-filled fractional quantum Hall systems. These liquids are characterized by gapless bands, strong electron–electron interactions, and emergent Lorentz invariance deep in the infrared. We address several important issues raised by experiments on these systems covering subjects of wide current interest in low-energy (condensed matter) and also high-energy (particle) physics. In particular, we consider the subtle influence of interactions on transport properties and their supposedly crucial influence on a potential dynamical mass generation. The resolution of these problems guide us from a thorough examination of the perturbative structure of gauge field theories to the development and application of nonperturbative approaches known from quantum electro/chromo-dynamics to address strong coupling issues.

Détail

Mots-clés : graphene
Keywords: perturbation theory, electron–electron interaction.

Bibliographie

[1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, A. A. Firsov, “Electric field effect in atomically thin carbon films”, Science, 306:5696 (2004), 666–669, arXiv: cond-mat/0410550 | DOI

[2] P. R. Wallace, “The band theory of graphite”, Phys. Rev., 71:9 (1947), 622–634 | DOI

[3] G. W. Semenoff, “Condensed-matter simulation of a three-dimensional anomaly”, Phys. Rev. Lett., 53:26 (1984), 2449–2452 | DOI

[4] J. B. Marston, I. Affleck, “Large-$n$ limit of the Hubbard–Heisenberg model”, Phys. Rev. B, 39:16 (1989), 11538–11558 | DOI | MR

[5] L. B. Ioffe, A. I. Larkin, “Gapless fermions and gauge fields in dielectrics”, Phys. Rev. B, 39:13 (1989), 8988–8999 | DOI

[6] G. E. Volovik, The Universe in a Helium Droplet, Oxford Univ. Press, Oxford, 2009 | MR

[7] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene”, Nature, 438:10 (2005), 197–200, arXiv: cond-mat/0509330 | DOI

[8] A. M. J. Schakel, “Relativistic quantum Hall effect”, Phys. Rev. D, 43:4 (1991), 1428–1431 | DOI

[9] V. P. Gusynin, S. G. Sharapov, “Unconventional integer quantum Hall effect in graphene”, Phys. Rev. Lett., 95:14 (2005), 146801, 4 pp., arXiv: cond-mat/0506575 | DOI

[10] J. C. Meyer, A. K. Geim, M. I. Katsnelson, K. S. Novoselov, T. J. Booth, S. Roth, “The structure of suspended graphene sheets”, Nature, 446 (2007), 60–63, arXiv: cond-mat/0701379 | DOI

[11] D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, M. Z. Hasan, “A topological Dirac insulator in a quantum spin Hall phase”, Nature, 452 (2008), 970–974, arXiv: 0910.2420 | DOI

[12] Y. Xia, D. Qian, D. Hsieh, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, M. Z. Hasan, “Observation of a large-gap topological-insulator class with a single Dirac cone on the surface”, Nature Phys., 5:6 (2009), 398–402, arXiv: 0908.3513 | DOI

[13] M. Z. Hasan, C. L. Kane, “Colloquim: Topological insulators”, Rev. Modern Phys., 82:4 (2010), 3045–3068, arXiv: 1002.3895 | DOI

[14] M. Polini, F. Guinea, M. Lewenstein, H. C. Manoharan, V. Pellegrini, “Artificial honeycomb lattices for electrons, atoms and photons”, Nature Nanotech., 8:9 (2013), 625–633, arXiv: 1304.0750 | DOI

[15] Z. K. Liu, B. Zhou, Y. Zhang, Z. J. Wang, H. M. Weng, D. Prabhakaran, S.-K. Mo, Z. X. Shen, Z. Fang, X. Dai, Z. Hussain, Y. L. Chen, “Discovery of a three-dimensional topological Dirac semimetal, Na${}_3$Bi”, Science, 343:6173 (2014), 864–867, arXiv: 1310.0391 | DOI

[16] M. Neupane, S.-Y. Xu, R. Sankar, N. Alidoust, G. Bian, C. Liu, I. Belopolski, T.-R. Chang, H.-T. Jeng, H. Lin, A. Bansil, F. Chou, M. Z. Hasan, “Observation of a three-dimensional topological Dirac semimetal phase in high-mobility Cd$_3$As$_2$”, Nature Commun., 5 (2014), 3786, 8 pp., arXiv: 1309.7892 | DOI

[17] S. Borisenko, Q. Gibson, D. Evtushinsky, V. Zabolotnyy, B. Büchner, R. J. Cava, “Experimental realization of a three-dimensional Dirac semimetal”, Phys. Rev. Lett., 113:2 (2014), 027603, 5 pp., arXiv: 1309.7978 | DOI

[18] S.-M. Huang, S.-Y. Xu, I. Belopolski, C.-C. Lee, G. Chang, B. Wang, N. Alidoust, G. Bian, M. Neupane, C. Zhang, S. Jia, A. Bansil, H. Lin, M. Z. Hasan, “A Weyl Fermion semimetal with surface Fermi arcs in the transition metal monopnictide TaAs class”, Nature Commun., 6 (2015), 7373, 6 pp. | DOI

[19] S.-Y. Xu, N. Alidoust, I. Belopolski, Z. Yuan, G. Bian, T.-R. Chang, H. Zheng, V. N. Strocov, D. S. Sanchez, G. Chang, C. Zhang, D. Mou, Y. Wu, L. Huang, C.-C. Lee, S.-M. Huang, B. Wang, A. Bansil, H.-T. Jeng, T. Neupert, A. Kaminski, H. Lin, S. Jia, M. Z. Hasan, “Discovery of a Weyl fermion state with Fermi arcs in niobium arsenide”, Nature Phys., 11 (2015), 748–745 | DOI

[20] A. Bansil, H. Lin, T. Das, “Colloquium: Topological band theory”, Rev. Modern Phys., 88:2 (2016), 021004, 37 pp., arXiv: 1603.03576 | DOI

[21] T. O. Wehling, A. M. Black-Schaffer, A. V. Balatsky, “Dirac materials”, Adv. Phys., 63:1 (2014), 1–76, arXiv: 1405.5774 | DOI

[22] W. Pan, W. Kang, K. W. Baldwin, K. W. West, L. N. Pfeiffer, D. C. Tsui, “Berry phase and anomalous transport of the composite fermions at the half-filled Landau level”, Nature Phys., 13 (2017), 1168–1172, arXiv: 1702.07307 | DOI

[23] J. González, F. Guinea, M. A. H. Vozmediano, “NonFermi liquid behavior of electrons in the half filled honeycomb lattice (a renormalization group approach)”, Nucl. Phys. B, 424:3 (1994), 595–618, arXiv: hep-th/9311105 | DOI

[24] D. C. Elias, R. V. Gorbachev, A. S. Mayorov, S. V. Morozov, A. A. Zhukov, P. Blake, L. A. Ponomarenko, I. V. Grigorieva, K. S. Novoselov, F. Guinea, A. K. Geim, “Dirac cones reshaped by interaction effects in suspended graphene”, Nature Phys., 7 (2011), 701–704, arXiv: 1104.1396 | DOI

[25] D. A. Siegel, C.-H. Parka, C. Hwangb, J. Deslippe, A. V. Fedorov, S. G. Louie, A. Lanzara, “Many-body interactions in quasi-freestanding graphene”, Proc. Natl. Acad. Sci. USA, 108:28 (2011), 11365–11369, arXiv: 1106.5822 | DOI

[26] A. A. Abrikosov, S. D. Beneslavskii, “O vozmozhnosti suschestvovaniya veschestv, promezhutochnykh mezhdu metallami i dielektrikami”, ZhETF, 59:4 (1971), 1280–1298

[27] A. Bostwick, T. Ohta, T. Seyller, K. Horn, E. Rotenberg, “Quasiparticle dynamics in graphene”, Nature Phys., 3:1 (2007), 36–40 | DOI

[28] G. Li, A. Luican, E. Y. Andrei, “Scanning tunneling spectroscopy of graphene on graphite”, Phys. Rev. Lett., 102:17 (2009), 176804, 4 pp., arXiv: 0803.4016 | DOI

[29] J. González, F. Guinea, M. A. H. Vozmediano, “Unconventional quasiparticle lifetime in graphite”, Phys. Rev. Lett., 77:15 (1996), 3589–3592, arXiv: cond-mat/9603113 | DOI

[30] J. González, F. Guinea, M. A. H. Vozmediano, “Marginal-Fermi-liquid behavior from two-dimensional Coulomb interaction”, Phys. Rev. B, 59:4 (1999), R2474–R2478, arXiv: cond-mat/9807130 | DOI

[31] J. Hofmann, E. Barnes, S. Das Sarma, “Why does graphene behave as a weakly interacting system?”, Phys. Rev. Lett., 113:10 (2014), 105502, 5 pp., arXiv: 1405.7036 | DOI

[32] V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, A. H. Castro Neto, “Electron-electron interactions in graphene: current status and perspectives”, Rev. Modern Phys., 84:3 (2012), 1067–1125, arXiv: 1012.3484 | DOI

[33] W. Kohn, “Cyclotron resonance and de Haas-van Alphen oscillations of an interacting electron gas”, Phys. Rev., 123:4 (1961), 1242–1244 | DOI

[34] R. E. Throckmorton, S. Das Sarma, “Failure of Kohn's theorem and $f$-sum-rule in intrinsic Dirac–Weyl materials in the presence of a filled Fermi sea”, Phys. Rev. B, 98:15 (2018), 155112, 8 pp., arXiv: 1805.03650 | DOI

[35] R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, A. K. Geim, “Fine structure constant defines visual transparency of graphene”, Science, 320:5881 (2008), 1308, 5 pp. | DOI

[36] K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, T. F. Heinz, “Measurement of the optical conductivity of graphene”, Phys. Rev. Lett., 101:19 (2008), 196405, 4 pp., arXiv: 0810.1269 | DOI

[37] N. M. R. Peres, “Colloquim: The transport properties of graphene: an introduction”, Rev. Modern Phys., 82:3 (2010), 2673–2700, arXiv: 1007.2849 | DOI

[38] V. P. Gusynin, S. G. Sharapov, J. P. Carbotte, “Unusual microwave response of Dirac quasiparticles in graphene”, Phys. Rev. Lett., 96:25 (2006), 256802, 4 pp., arXiv: cond-mat/0603267 | DOI

[39] V. P. Gusynin, S. G. Sharapov, J. P. Carbotte, “On the universal ac optical background in graphene”, New J. Phys., 11:9 (2009), 095013, 17 pp., arXiv: 0908.2803 | DOI

[40] S. Teber, A. V. Kotikov, “Interaction corrections to the minimal conductivity of graphene via dimensional regularization”, Europhys. Lett., 107:5 (2014), 57001, arXiv: 1407.7501 | DOI

[41] S. Teber, A. V. Kotikov, “Metod unikalnostei i opticheskaya provodimost grafena: novoe primenenie moschnoi tekhniki mnogopetlevykh vychislenii”, TMF, 190:3 (2017), 519–532 | DOI | DOI | MR

[42] S. Teber, A. V. Kotikov, “Field theoretic renormalization study of interaction corrections to the universal ac conductivity of graphene”, JHEP, 07 (2018), 082, 22 pp., arXiv: 1802.09898 | DOI

[43] S. Teber, Field theoretic study of electron-electron interaction effects in Dirac liquids, arXiv: 1810.08428

[44] A. Giuliani, V. Mastropietro, M. Porta, “Absence of interaction corrections in the optical conductivity of graphene”, Phys. Rev. B, 83:19 (2011), 195401, 6 pp., arXiv: 1010.4461 | DOI

[45] E. Fradkin, “Critical behavior of disordered degenerate semiconductors. II. Spectrum and transport properties in mean-field theory”, Phys. Rev. B, 33:5 (1986), 3263–3268 | DOI

[46] P. A. Lee, “Localized states in a $d$-wave superconductor”, Phys. Rev. Lett., 71:12 (1993), 1887–1890 | DOI

[47] A. W. W. Ludwig, M. P. A. Fisher, R. Shankar, G. Grinstein, “Integer quantum Hall transition: an alternative approach and exact results”, Phys. Rev. B, 50:11 (1994), 7526–7552 | DOI

[48] E. G. Mishchenko, “Minimal conductivity in graphene: Interaction corrections and ultraviolet anomaly”, Europhys. Lett., 83:1 (2008), 17005, 6 pp., arXiv: 0709.4245 | DOI

[49] A. V. Kotikov, S. Teber, “Critical behaviour of reduced $\mathrm{QED}_{4,3}$ and dynamical fermion gap generation in graphene”, Phys. Rev. D, 94:11 (2016), 114010, 8 pp., arXiv: 1610.00934 | DOI

[50] M. S. Nevius, M. Conrad, F. Wang, A. Celis, M. N. Nair, A. Taleb-Ibrahimi, A. Tejeda, E. H. Conrad, “Semiconducting graphene from highly ordered substrate interactions”, Phys. Rev. Lett., 115:13 (2015), 136802, 5 pp., arXiv: 1505.00435 | DOI

[51] S. Teber, “Electromagnetic current correlations in reduced quantum electrodynamics”, Phys. Rev. D, 86:2 (2012), 025005, 9 pp., arXiv: 1204.5664 | DOI

[52] A. V. Kotikov, S. Teber, “Note on an application of the method of uniqueness to reduced quantum electrodynamics”, Phys. Rev. D, 87:8 (2013), 087701, 5 pp., arXiv: 1302.3939 | DOI

[53] A. V. Kotikov, S. Teber, “Two-loop fermion self-energy in reduced quantum electrodynamics and application to the ultrarelativistic limit of graphene”, Phys. Rev. D, 89:6 (2014), 065038, 24 pp., arXiv: 1312.2430 | DOI

[54] S. Teber, A. V. Kotikov, “Field theoretic renormalization study of reduced quantum electrodynamics and applications to the ultrarelativistic limit of Dirac liquids”, Phys. Rev. D, 97:7 (2018), 074004, 16 pp., arXiv: 1801.10385 | DOI

[55] R. D. Pisarski, “Chiral-symmetry breaking in three-dimensional electrodynamics”, Phys. Rev. D, 29:10 (198), 2423–2426 | DOI

[56] T. Appelquist, D. Nash, L. C. R. Wijewardhana, “Critical behavior in $(2+1)$-dimensional QED”, Phys. Rev. Lett., 60:25 (1988), 2575–2578 | DOI | MR

[57] D. Nash, “Higher-order corrections in $(2+1)$-dimensional QED”, Phys. Rev. Lett., 62:26 (1989), 3024–3026 | DOI

[58] A. V. Kotikov, “O kriticheskom povedenii trekhmernoi elektrodinamiki”, Pisma v ZhETF, 58:10 (1993), 785–789

[59] A. V. Kotikov, “On the sritical behavior of $(2+1)$-dimensional QED”, YaF, 75:7 (2012), 945–947, arXiv: 1104.3888 | DOI

[60] A. V. Kotikov, V. I. Shilin, S. Teber, “Critical behaviour of $(2+1)$-dimensional QED: $1/N_f$-corrections in the Landau gauge”, Phys. Rev. D, 94:5 (2016), 056009, 6 pp., arXiv: 1605.01911 | DOI | MR

[61] A. V. Kotikov, S. Teber, “Critical behavior of $(2+1)$-dimensional QED: $1/N_f$ corrections in an arbitrary nonlocal gauge”, Phys. Rev. D, 94:11 (2016), 114011, 9 pp., arXiv: 1609.06912 | DOI

[62] V. P. Gusynin, P. K. Pyatkovskiy, “Critical number of fermions in three-dimensional QED”, Phys. Rev. D, 94:12 (2016), 125009, 14 pp., arXiv: 1607.08582 | DOI | MR

[63] H. Isobe, N. Nagaosa, “Coulomb interaction effect in Weyl Fermions with tilted energy dispersion in two dimensions”, Phys. Rev. Lett., 116:11 (2016), 116803, 5 pp., arXiv: 1512.08704 | DOI

[64] E. Barnes, E. H. Hwang, R. E. Throckmorton, S. Das Sarma, “Effective field theory, three-loop perturbative expansion, and their experimental implications in graphene many-body effects”, Phys. Rev. B, 89:23 (2014), 235431, 57 pp., arXiv: 1401.7011 | DOI

[65] E. V. Gorbar, V. P. Gusynin, V. A. Miransky, “Dynamical chiral symmetry breaking on a brane in reduced QED”, Phys. Rev. D, 64:10 (2001), 105028, 16 pp., arXiv: hep-ph/0105059 | DOI

[66] A. V. Kotikov, S. Teber, “Multi-loop techniques for massless Feynman diagram calculations”, Phys. Part. Nucl., 50:1 (2019), 1–41, arXiv: 1805.05109 | DOI

[67] S. Laporta, “High-precision calculation of multiloop Feynman integrals by difference equations”, Internat. J. Modern Phys. A, 15:32 (2000), 5087–5159, arXiv: hep-ph/0102033 | DOI | MR

[68] R. N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv: ;; “LiteRed 1.4: a powerful tool for reduction of multiloop integrals”, J. Phys. Conf.: Ser., 523:1 (2014), 012059, 8 pp., arXiv: 1212.26851310.1145 | DOI

[69] B. Ruijl, T. Ueda, J. A. M. Vermaseren, Forcer, a FORM program for the parametric reduction of four-loop massless propagator diagrams, arXiv: 1704.06650

[70] K. G. Chetyrkin, F. V. Tkachov, “Integration by parts: the algorithm to calculate $\beta$-functions in 4 loops”, Nucl. Phys. B, 192:1 (1981), 159–204 | DOI

[71] K. G. Chetyrkin, G. Falcioni, F. Herzog, J. A. M. Vermaseren, “The method of global $R*$ and its applications”, PoS (RADCOR 2017), 290 (2018), 004, 10 pp., arXiv: 1801.03024 | DOI

[72] S. Hu, O. Schnetz, J. Shaw, K. Yeats, Further investigations into the graph theory of $\phi^4$-periods and the $c_2$ invariant, arXiv: 1812.08751

[73] J. A. Gracey, T. Luthe, Y. Schröder, “Four loop renormalization of the Gross–Neveu model”, Phys. Rev. D, 94:12 (2016), 125028, 18 pp., arXiv: 1609.05071 | DOI | MR

[74] P. A. Baikov, K. G. Chetyrkin, J. H. Kühn, “Five-loop running of the QCD coupling constant”, Phys. Rev. Lett., 118:8 (2017), 082002, 4 pp., arXiv: 1606.08659 | DOI

[75] T. Luthe, A. Maier, P. Marquard, Y. Schröder, “Five-loop quark mass and field anomalous dimensions for a general gauge group”, JHEP, 01 (2017), 081, 14 pp., arXiv: 1612.05512 | DOI

[76] T. Luthe, A. Maier, P. Marquard, Y. Schröder, “Complete renormalization of QCD at five loops”, JHEP, 03 (2017), 020, 15 pp., arXiv: 1701.07068 | DOI

[77] F. Herzog, B. Ruijl, T. Ueda, J. A. M. Vermaseren, A. Vogt, “The five-loop beta function of Yang–Mills theory with fermions”, JHEP, 02 (2017), 090, 18 pp., arXiv: 1701.01404 | DOI | MR

[78] K. G. Chetyrkin, G. Falcioni, F. Herzog, J. A. M. Vermaseren, “Five-loop renormalisation of QCD in covariant gauges”, JHEP, 10 (2017), 179, 18 pp. | DOI | MR

[79] T. Luthe, A. Maier, P. Marquard, Y. Schröder, “The five-loop beta function for a general gauge group and anomalous dimensions beyond Feynman gauge”, JHEP, 10 (2017), 166, 19 pp., arXiv: 1709.0771 | DOI | MR

[80] D. V. Batkovich, K. G. Chetyrkin, M. V. Kompaniets, “Six loop analytical calculation of the field anomalous dimension and the critical exponent $\eta$ in $O(n)$-symmetric $\varphi^4$ model”, Nucl. Phys. B, 906 (2016), 147–167, arXiv: 1601.01960 | DOI | MR

[81] M. Kompaniets, E. Panzer, “Renormalization group functions of $\phi^4$ theory in the MS-scheme to six loops”, PoS (LL2016), 260 (2016), 038, 16 pp., arXiv: 1606.09210 | DOI

[82] M. V. Kompaniets, E. Panzer, “Minimally subtracted six loop renormalization of $O(n)$-symmetric $\phi^4$ theory and critical exponents”, Phys. Rev. D, 96:3 (2017), 036016, 26 pp., arXiv: 1705.06483 | DOI | MR

[83] O. Schnetz, “Numbers and functions in quantum field theory”, Phys. Rev. D, 97:8 (2018), 085018, 20 pp., arXiv: 1606.08598 | DOI

[84] C. Marboe, V. Velizhanin, “Twist-2 at seven loops in planar $\mathcal{N} $=4 SYM theory: full result and analytic properties”, JHEP, 11 (2016), 013, 18 pp., arXiv: 1607.06047 | DOI | MR

[85] D. I. Kazakov, “Vychislenie feinmanovskikh integralov metodom ‘unikalnostei’ ”, TMF, 58:3 (1984), 343–353 | DOI | MR

[86] R. Jackiw, S. Templeton, “How super-renormalizable interactions cure their infrared divergences”, Phys. Rev. D, 23:10 (1981), 2291–2304 | DOI

[87] E. I. Guendelman, Z. M. Radulovic, “Loop expansion in massless three-dimensional QED”, Phys. Rev. D, 27:2 (1983), 357–365 | DOI

[88] E. I. Guendelman, Z. M. Radulovic, “Infrared divergences in three-dimensional gauge theories”, Phys. Rev. D, 30:6 (1984), 1338–1348 | DOI

[89] T. Appelquist, R. D. Pisarski, “High-temperature Yang–Mills theories and three-dimensional quantum chromodynamics”, Phys. Rev. D, 23:10 (1981), 2305–2317 | DOI

[90] T. Appelquist, U. W. Heinz, “Three-dimensional $O(n)$ theories at large distances”, Phys. Rev. D, 24:8 (1981), 2169–2181 | DOI

[91] A. V. Kotikov, S. Teber, “Addendum to `Critical behaviour of $(2+1)$-dimensional QED: $1/N_f$-corrections in an arbitrary non-local gauge' ”, Phys. Rev. D, 99:5 (2019), 059902, 5 pp., arXiv: 1902.03790 | DOI

[92] E. E. Boos, A. I. Davydychev, “Metod vychisleniya massivnykh feinmanovskikh integralov”, TMF, 89:1 (1991), 56–72 | DOI | MR

[93] A. V. Kotikov, “Differential equations method: the calculation of vertex type Feynman diagrams”, Phys. Lett. B, 259:3 (1991), 314–322 | DOI | MR

[94] N. N. Bogoliubov, O. S. Parasiuk, “Über die Multiplikation der Kauzalfunktionen der Quantentheorie der Felder”, Acta Math., 97 (1957), 227–266 | DOI | MR

[95] K. Hepp, “Proof of the Bogolyubov–Parasiuk theorem on renormalization”, Commun. Math. Phys., 2:1 (1966), 301–326 | DOI

[96] W. Zimmermann, “Convergence of Bogoliubov's method of renormalization in momentum space”, Commun. Math. Phys., 15:3 (1969), 208–234 | DOI | MR