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The ground state construction of bilayer graphene

Alessandro Giuliani

Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, L.go S.L. Murialdo, 1, 00146 Roma, Italy

E-mail Address: [email protected]

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and

Ian Jauslin

Dipartimento di Fisica, University of Rome “Sapienza”, P.le Aldo Moro, 2, 00185 Rome, Italy

E-mail Address: [email protected]

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https://doi.org/10.1142/S0129055X16500185Cited by:1

Abstract

We consider a model of half-filled bilayer graphene, in which the three dominant Slonczewski–Weiss–McClure hopping parameters are retained, in the presence of short-range interactions. Under a smallness assumption on the interaction strength $U$ as well as on the inter-layer hopping $ϵ$ , we construct the ground state in the thermodynamic limit, and prove that the pressure and two-point Schwinger function, away from its singularities, are analytic in $U$ , uniformly in $ϵ$ . The interacting Fermi surface is degenerate, and consists of eight Fermi points, two of which are protected by symmetries, while the locations of the other six are renormalized by the interaction, and the effective dispersion relation at the Fermi points is conical. The construction reveals the presence of different energy regimes, where the effective behavior of correlation functions changes qualitatively. The analysis of the crossover between regimes plays an important role in the proof of analyticity and in the uniform control of the radius of convergence. The proof is based on a rigorous implementation of fermionic renormalization group methods, including determinant estimates for the renormalized expansion.

Keywords:

AMSC: 81T16, 82B10, 82B28, 82D25

References

1. K. Novoselov, A. Geim, S. Morozov, D. Jiang, Y. Zhang, S. Dubonos, I. Grigorieva and A. Firsov, Electric field effect in atomically thin carbon films, Science 306 (2004) 666–669. Crossref, ISI, ADS, Google Scholar
2. A. Geim, Random walk to graphene, Nobel lecture (2010). Google Scholar
3. K. Novoselov, A. Geim, S. Morozov, D. Jiang, M. Katsnelson, I. Grigorieva, S. Dubonos and A. Firsov, Two-dimensional gas of massless Dirac fermions in graphene, Nature 438(10) (2005) 197–200. Crossref, ADS, Google Scholar
4. Y. Zhang, Y. W. Tan, H. Stormer and P. Kim, Experimental observation of the quantum Hall effect and Berry’s phase in graphene, Nature 438(10) (2005) 201–204. Crossref, ADS, Google Scholar
5. A. Geim and K. Novoselov, The rise of graphene, Nat. Mat. 6 (2007) 183–191. Crossref, ISI, ADS, Google Scholar
6. P. Wallace, The band theory of graphite, Phys. Rev. 71(9) (1947) 622–634. Crossref, ISI, ADS, Google Scholar
7. A. Giuliani and V. Mastropietro, The two-dimensional Hubbard model on the honey-comb lattice, Comm. Math. Phys. 293 (2010) 301–364. Crossref, ISI, ADS, Google Scholar
8. A. Giuliani, V. Mastropietro and M. Porta, Absence of interaction corrections in the optical conductivity of graphene, Phys. Rev. B 83 (2011) 195401. Crossref, ADS, Google Scholar
9. A. Giuliani, V. Mastropietro and M. Porta, Universality of conductivity in interacting graphene, Comm. Math. Phys. 311(2) (2012) 317–355. Crossref, ISI, ADS, Google Scholar
10. G. Benfatto and G. Gallavotti, Perturbation theory of the Fermi surface in a quantum liquid — A general quasiparticle formalism and one-dimensional systems, J. Statist. Phys. 59(3–4) (1990) 541–664. Crossref, ISI, ADS, Google Scholar
11. A. Giuliani, V. Mastropietro and M. Porta, Lattice gauge theory model for graphene, Phys. Rev. B 82 (2010) 121418(R). Crossref, ADS, Google Scholar
12. A. Giuliani, V. Mastropietro and M. Porta, Lattice quantum electrodynamics for graphene, Ann. Phys. 327(2) (2011) 461–511. Crossref, ISI, ADS, Google Scholar
13. J. Slonczewski and P. Weiss, Band structure of graphite, Phys. Rev. 109 (1958) 272–279. Crossref, ISI, ADS, Google Scholar
14. J. McClure, Band structure of graphite and de Haas–van Alphen effect, Phys. Rev. 108 (1957) 612–618. Crossref, ISI, ADS, Google Scholar
15. K. Novoselov, E. McCann, S. Morozov, V. Fal’ko, M. Katsnelson, U. Zeitler, D. Jiang, F. Schedin and A. Geim, Unconventional quantum Hall effect and Berry’s phase of $π$ in bilayer graphene, Nat. Phys. 2 (2006) 177–180. Crossref, ISI, ADS, Google Scholar
16. O. Vafek, Interacting Fermions on the honeycomb bilayer: From weak to strong coupling, Phys. Rev. B 82 (2010) 205106. Crossref, ADS, Google Scholar
17. O. Vafek and K. Yang, Many-body instability of Coulomb interacting bilayer graphene: Renormalization group approach, Phys. Rev. B 81 (2010) 041401. Crossref, ISI, ADS, Google Scholar
18. R. Throckmorton and O. Vafek, Fermions on bilayer graphene: Symmetry breaking for $B = 0$ and $ν = 0$ , Phys. Rev. B 86 (2012) 115447. Crossref, ADS, Google Scholar
19. E. McCann and V. Fal’ko, Landau-level degeneracy and Quantum Hall Effect in a graphite bilayer, Phys. Rev. Lett. 86 (2006) 086805. Crossref, Google Scholar
20. V. Cvetkovic, R. Throckmorton and O. Vafek, Electronic multicriticality in bilayer graphene, Phys. Rev. B 86 (2012) 075467. Crossref, ADS, Google Scholar
21. B. Partoens and F. Peeters, From graphene to graphite: Electronic structure around the $K$ point, Phys. Rev. B 74 (2006) 075404. Crossref, ISI, ADS, Google Scholar
22. G. Benfatto, A. Giuliani and V. Mastropietro, Fermi liquid behavior in the 2D Hubbard model, Ann. Henri Poincaré 7 (2006) 809–898. Crossref, ISI, ADS, Google Scholar
23. M. Dresselhaus and G. Dresselhaus, Intercalation compounds of graphite, Adv. Phys. 51(1) (2002) 1–186. Crossref, ISI, ADS, Google Scholar
24. W. Toy, M. Dresselhaus and G. Dresselhaus, Minority carriers in graphite and the $H$ -point magnetoreflection spectra, Phys. Rev. B 15 (1997) 4077–4090. Crossref, ADS, Google Scholar
25. A. Misu, E. Mendez and M. S. Dresselhaus, Near infrared reflectivity of graphite under hydrostatic pressure, J. Phys. Soc. Japan 47(1) (1979) 199–207. Crossref, ADS, Google Scholar
26. R. Doezema, W. Datars, H. Schaber and A. Van Schyndel, Far-infrared magnetospectroscopy of the Landau-level structure in graphite, Phys. Rev. B 19(8) (1979) 4224–4230. Crossref, ISI, ADS, Google Scholar
27. L. Zhang, Z. Li, D. Basov, M. Fogler, Z. Hao and M. Martin, Determination of the electronic structure of bilayer graphene from infrared spectroscopy, Phys. Rev. B 78 (2008) 235408. Crossref, ISI, ADS, Google Scholar
28. L. Malard, J. Nilsson, D. Elias, J. Brant, F. Plentz, E. Alves, A. Castro Neto and M. Pimenta, Probing the electronic structure of bilayer graphene by Raman scattering, Phys. Rev. B 76 (2007) 201401. Crossref, ISI, ADS, Google Scholar
29. L. Lu, Constructive analysis of two dimensional Fermi systems at finite temperature, PhD thesis, Institute for Theoretical Physics, Heidelberg (2013); http://www.ub.uni-heidelberg.de/archiv/14947. Google Scholar
30. W. Pedra and M. Salmhofer, Determinant bounds and the Matsubara UV problem of many-fermion systems, Comm. Math. Phys. 282(3) (2008) 797–818. Crossref, ISI, ADS, Google Scholar
31. D. Brydges and P. Federbush, A new form of the Mayer expansion in classical statistical mechanics, J. Math. Phys. 19 (1978) 2064–2067. Crossref, ISI, ADS, Google Scholar
32. G. Battle and P. Federbush, A note on cluster expansions, tree graph identities, extra $1 / N$ factors!!!, Lett. Math. Phys. 8 (1984) 55–57. Crossref, ISI, ADS, Google Scholar
33. D. Brydges and T. Kennedy, Mayer expansions and the Hamilton–Jacobi equation, J. Statist. Phys. 48(1–2) (1987) 19–49. Crossref, ISI, ADS, Google Scholar
34. A. Abdesselam and V. Rivasseau, Explicit Fermionic tree expansions, Lett. Math. Phys. 44(1) (1998) 77–88. Crossref, ISI, Google Scholar
35. G. Gallavotti and F. Nicolò, Renormalization theory for four dimensional scalar fields, I, Comm. Math. Phys. 100 (1985) 545–590. Crossref, ISI, ADS, Google Scholar
36. G. Gallavotti and F. Nicolò, Renormalization theory for four dimensional scalar fields, II, Comm. Math. Phys. 101 (1985) 247–282. Crossref, ISI, ADS, Google Scholar
37. G. Gentile and V. Mastropietro, Renormalization group for one-dimensional fermions — A review on mathematical results, Phys. Rep. 352 (2001) 273–437. Crossref, ISI, ADS, Google Scholar
38. A. Giuliani, The ground state construction of the two-dimensional Hubbard model on the honeycomb lattice, in Quantum Theory from Small to Large Scales, Lecture Notes of the Les Houches Summer School, Vol. 95 (Oxford University Press, 2010), pp. 553–615. Google Scholar
39. S. Trickey, F. Müller-Plathe, G. Diercksen and J. Boettger, Interplanar binding and lattice relaxation in a graphite dilayer, Phys. Rev. B 45 (1992) 4460–4468. Crossref, ISI, ADS, Google Scholar
40. G. Benfatto and G. Gallavotti, Renormalization Group (Princeton University Press, 1995). Crossref, Google Scholar
41. M. Salmhofer, Renormalization: An Introduction (Springer Science & Business Media, 2013). Google Scholar
42. G. Benfatto and V. Mastropietro, On the density–density critical indices in interacting Fermi systems, Comm. Math. Phys. 231(1) (2002) 97–134. Crossref, ISI, ADS, Google Scholar
43. J. Feldman, H. Knörrer and E. Trubowitz, A two dimensional Fermi liquid. Part 1: Overview, Comm. Math. Phys. 247(1) (2004) 1–47. Crossref, ISI, ADS, Google Scholar
44. J. Feldman, H. Knörrer and E. Trubowitz, A two dimensional Fermi liquid. Part 2: Convergence, Comm. Math. Phys. 247(1) (2004) 49–111. Crossref, ISI, ADS, Google Scholar
45. J. Feldman, H. Knörrer and E. Trubowitz, A two dimensional Fermi liquid. Part 3: The Fermi surface, Comm. Math. Phys. 247(1) (2004) 113–177. Crossref, ISI, ADS, Google Scholar
46. V. Mastropietro, Conductivity between Luttinger liquids: Coupled chains and bilayer graphene, Phys. Rev. B 84 (2011) 035109. Crossref, ISI, ADS, Google Scholar

Vol. 28, No. 08

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History

Received 20 August 2015

Accepted 12 August 2016

Published: 13 September 2016

Keywords

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The ground state construction of bilayer graphene

Abstract

References

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