Abstract
This paper is devoted to the rigorous study of the low temperature properties of the two-dimensional weakly interacting Hubbard model on the honeycomb lattice in which the renormalized chemical potential \(\mu \) has been fixed such that the Fermi surface consists of a set of exact triangles. Using renormalization group analysis around the Fermi surface, we prove that this model is not a Fermi liquid in the mathematically precise sense of Salmhofer. The main result is proved in two steps. First we prove that the perturbation series for Schwinger functions as well as the self-energy function have non-zero radius of convergence when the temperature T is above an exponentially small value, namely \({T_0\sim \exp {(-C|\lambda |^{-1/2})}}\). Then we prove the necessary lower bound for second derivatives of self-energy w.r.t. the external momentum and achieve the proof.
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Notes
Although summation over all scales of the tadpole terms is not absolutely convergent for \(k_0\rightarrow \infty \), this sum can be controlled by using the explicit expression of the single scale propagator.
References
Abdesselam, A., Rivasseau, V.: Trees, forests and jungles: a botanical garden for cluster expansions. In: Constructive Physics, Lecture Notes in Physics 446. Springer (1995)
Afchain, S., Magnen, J., Rivasseau, V.: Renormalization of the 2-point function of the Hubbard model at half-filling. Ann. Henri Poincaré 6, 399–448 (2005)
Afchain, S., Magnen, J., Rivasseau, V.: The two dimensional Hubbard model at half-filling, part III: the lower bound on the self-energy. Ann. Henri Poincaré 6, 449–483 (2005)
Benfatto, G., Gallavotti, G.: Perturbation theory of the Fermi surface in a quantum liquid. A general quasiparticle formalism and one dimensional systems. J. Stat. Phys. 59, 541–664 (1990)
Benfatto, G., Gallavotti, G.: Renormalization Group, Physics Notes, vol. 1. Princeton University Press, Princeton (1995)
Benfatto, G., Giuliani, A., Mastropietro, V.: Fermi liquid behavior in the 2D Hubbard model at low temperatures. Ann. Henri Poincaré 7, 809–898 (2006)
Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. North-Holland, Amsterdam (1976)
Brydges, D., Kennedy, T.: Mayer expansions and the Hamilton–Jacobi equation. J. Stat. Phys. 48, 19 (1987)
Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics, 2nd edn. Springer, Berlin (2002)
Baeriswyl, D., Campbell, D., Carmelo, J., Guinea, F., Louis, E.: The Hubbard Model, Nato ASI series, vol. 343. Springer Science+Business Media, New York (1995)
Castro Neto, A.H., Guinea, F., Peres, N.M.R., Novoselov, K.S., Geim, A.K.: The electronic properties of graphene. Rev. Mod. Phys. 81, 109–162 (2009)
Das Sarma, S., Adam, S., Hwang, E.H., Rossi, E.: Electronic transport in two-dimensional graphene. Rev. Mod. Phys. 83, 407–470 (2011)
Disertori, M., Rivasseau, V.: Interacting Fermi liquid in two dimensions at finite temperature, Part I—convergent attributions. Commun. Math. Phys. 215, 251–290 (2000)
Disertori, M., Rivasseau, V.: Interacting Fermi liquid in two dimensions at finite temperature, Part II—renormalization. Commun. Math. Phys. 215, 291–341 (2000)
Fefferman, C., Weinstein, M.: Honeycomb lattice potentials and Dirac points. J. Am. Math. Soc. 25, 1169–1220 (2012)
Fefferman, C., Lee-Thorp, J.P., Weinstein, M.: Honeycomb Schrödinger operators in the strong binding regime. Commun. Pure Appl. Math. 71, 1178–1270 (2018)
Feldman, J., Magnen, J., Rivasseau, V., Trubowitz, E.: An infinite volume expansion for many fermions Green functions. Helv. Phys. Acta 65, 679–721 (1992)
Feldman, J., Knörrer, H., Trubowitz, E.: A two dimensional fermi liquid. Commun. Math. Phys 247, 1–319 (2004)
Feldman, J., Salmhofer, M.: Singular Fermi surfaces. I. General power counting and higher dimensional cases. Rev. Math. Phys. 20, 233–274 (2008)
Feldman, J., Salmhofer, M.: Singular Fermi surfaces. II. The two-dimensional case. Rev. Math. Phys. 20, 275–334 (2008)
Feldman, J., Salmhofer, M., Trubowitz, E.: Perturbation theory around nonnested Fermi surfaces. I. Kee** the Fermi surface fixed. J. Stat. Phys. 84, 1209–1336 (1996)
Feldman, J., Salmhofer, M., Trubowitz, E.: An inversion theorem in Fermi surface theory. Commun. Pure Appl. Math. 53, 1350–1384 (2000)
Feldman, J., Trubowitz, E.: Perturbation theory for many fermion systems. Helv. Phys. Acta 63, 156–260 (1990)
Gallavotti, G., Nicolò, F.: Renormalization theory for four dimensional scalar fields. Part I. Commun. Math. Phys. 100, 545–590 (1985)
Gallavotti, G., Nicolò, F.: Renormalization theory for four dimensional scalar fields. Part II. Commun. Math. Phys. 101, 471–562 (1985)
Gawedzki, K., Kupiainen, A.: Gross–Neveu model through convergent perturbation expansions. Commun. Math. Phys. 102, 1–30 (1985)
Giuliani, A., Mastropietro, V.: The two-dimensional Hubbard model on the honeycomb lattice. Commun. Math. Phys. 293, 301–346 (2010)
Hubbard, J.: Electron correlations in narrow energy bands. Proc. R. Soc. (Lond.) A276, 238–257 (1963)
Itzykson, C., Zuber, J.-B.: Quantum Field Theory. Dover Publications, New York (2005)
Kotov, V.N., Uchoa, B., Pereira, V.M., Guinea, F., Castro Neto, A.H.: Electron–electron interactions in graphene: current status and perspectives. Rev. Mod. Phys. 84, 1067–1125 (2012)
Lifshitz, I.M.: Anomalies of electron characteristics of a metal in the high pressure. Sov. Phys. JETP 11, 1130 (1960)
Lint, S., et al.: Introducing strong correlation effects into graphene by gadolinium interaction. Phys. Rev. B 100, 121407(R) (2019)
Krajewski, T., Rivasseau, V., Tanasa, A., Wang, Z.: Topological graph polynomials and quantum field theory, Part I: Heat kernel theories. J. Noncommut. Geom. 4, 29 (2010)
Lesniewski, A.: Effective action for the Yukawa\(_2\) quantum field theory. Commun. Math. Phys. 108, 437–467 (1987)
Mastropietro, V.: Non-Perturbative Renormalization. World Scientific, Singapore (2008)
Novoselov, K.S., Geim, A.K., Morozov, S.V., Jiang, D., Zhang, Y., Dubonos, S.V., Grigorieva, I.V., Firsov, A.A.: Electric field effect in atomically thin carbon films. Science 306, 666 (2004)
Rivasseau, V.: The two dimensional Hubbard model at half-filling. I. Convergent contributions. J. Stat. Phys. 106, 693–722 (2002)
Rivasseau, V.: From Perturbative Renormalization to Constructive Renormalization. Princeton University Press, Princeton
McChesney, J.L., et al.: Extended van Hove singularity and superconducting instability in doped graphene. Phys. Rev. Lett. 104, 136803 (2010)
Rosenberg, P., et al.: Tuning the do** level of graphene in the vicinity of the Van Hove singularity via ytterbium intercalation. Phys. Rev. B 100, 035445 (2019)
Rosenberg, P., et al.: Overdo** graphene beyond the van Hove singularity. Phys. Rev. Lett. 125, 176403 (2020)
Rivasseau, V., Wang, Z.: How to Resum Feynman graphs. Ann. Henri Poincare 15(11), 2069 (2014)
Salmhofer, M.: Continuous renormalization for Fermions and Fermi liquid theory. Commun. Math. Phys. 194, 249–295 (1998)
Wallace, P.R.: The band theory of graphite. Phys. Rev. 71, 622–634 (1947)
Wang, Z.: On the sector counting lemma. Lett. Math. Phys. 111, 128 (2021)
Acknowledgements
Zhituo Wang is very grateful to Horst Knörrer for useful discussions and encouragements, and to Alessandro Giuliani and Vieri Mastropietro for useful discussions. Part of this work has been finished during Zhituo Wang’s visit to the Institute of Mathematics, University of Zurich. He is also very grateful to Benjamin Schlein for invitation and hospitality. We are grateful to the anonymous referee for his comments and suggestions, which lead to significant improvements of the first manuscript. Zhituo Wang is supported by NSFC Nos. 12071099 and 11701121. Vincent Rivasseau is supported by Paris-Saclay University and the IJCLab of the CNRS.
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Rivasseau, V., Wang, Z. Honeycomb Hubbard Model at van Hove Filling. Commun. Math. Phys. 401, 2569–2642 (2023). https://doi.org/10.1007/s00220-023-04696-8
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DOI: https://doi.org/10.1007/s00220-023-04696-8