Journal of Statistical Physics

, Volume 157, Issue 4–5, pp 755–829 | Cite as

Renormalization Theory of a Two Dimensional Bose Gas: Quantum Critical Point and Quasi-Condensed State

Article

Abstract

We present a renormalization group construction of a weakly interacting Bose gas at zero temperature in the two-dimensional continuum, both in the quantum critical regime and in the presence of a condensate fraction. The construction is performed within a rigorous renormalization group scheme, borrowed from the methods of constructive field theory, which allows us to derive explicit bounds on all the orders of renormalized perturbation theory. Our scheme allows us to construct the theory of the quantum critical point completely, both in the ultraviolet and in the infrared regimes, thus extending previous heuristic approaches to this phase. For the condensate phase, we solve completely the ultraviolet problem and we investigate in detail the infrared region, up to length scales of the order \((\lambda ^3\rho _0)^{-1/2}\) (here \(\lambda \) is the interaction strength and \(\rho _0\) the condensate density), which is the largest length scale at which the problem is perturbative in nature. We exhibit violations to the formal Ward Identities, due to the momentum cutoff used to regularize the theory, which suggest that previous proposals about the existence of a non-perturbative non-trivial fixed point for the infrared flow should be reconsidered.

Keywords

Two-dimensional Bose gas Quantum criticality Bose–Einstein condensation Bogoliubov theory Renormalization group Ward identities 

Notes

Acknowledgments

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme ERC Starting Grant CoMBoS (Grant Agreement No. 239694). We thank Giuseppe Benfatto and Vieri Mastropietro for several discussions and illuminating comments. S.C. acknowledges the Hausdorff Center for Mathematics in Bonn for financial support.

References

  1. 1.
    Balaban, T.: Ultraviolet stability of three-dimensional lattice pure gauge field theories. Commun. Math. Phys. 102(2), 175–347 (1985)CrossRefGoogle Scholar
  2. 2.
    Balaban, T.: Renormalization group approach to lattice gauge field theories. I. Generation of effective actions in a small field approximation and a coupling constant renormalization in four dimensions. Commun. Math. Phys. 109(2), 177–352 (1987)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Balaban, T.: A low temperature expansion for classical N-vector models. I. A renormalization group flow. Commun. Math. Phys. 167(1), 103–154 (1995)MathSciNetCrossRefMATHADSGoogle Scholar
  4. 4.
    Balaban, T., Feldman, J., Knörrer, H., and Trubowitz E.: The temporal ultraviolet limit. In: Proceedings of the Les Houches Summer School “Quantum theory from small to large scales”, pp. 99–170 (2010).Google Scholar
  5. 5.
    Beliaev, S.T.: Application of the methods of quantum field theory to a system of bosons. Sov. Phys. JETP 7(2), 289–299 (1958)MathSciNetGoogle Scholar
  6. 6.
    Benfatto, G.: Renormalization group approach to zero temperature Bose condensation. In: Proceedings of the workshop “Constructive results in Field Theory, Statistical Mechanics and Condensed Matter Physics”, Palaiseau, July 25–27, pp. 219–247 (1994)Google Scholar
  7. 7.
    Benfatto, G.: On the ultraviolet problem for the 2D weakly interacting Fermi gas. Ann. Henri Poincaré 10(1), 1–17 (2009)MathSciNetCrossRefMATHADSGoogle Scholar
  8. 8.
    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(3–4), 541–664 (1990)MathSciNetCrossRefMATHADSGoogle Scholar
  9. 9.
    Benfatto, G., Gallavotti, G.: Renormalization Group. Princeton University Press, Princeton, NJ (1995)MATHGoogle Scholar
  10. 10.
    Benfatto, G., Mastropietro, V.: Renormalization group, hidden symmetries and approximate Ward identities in the XYZ model. Rev. Math. Phys. 13, 1323–1435 (2001)Google Scholar
  11. 11.
    Benfatto, G., Mastropietro, V.: Ward identities and chiral anomaly in the Luttinger liquid. Commun. Math. Phys. 258(3), 609–655 (2005)MathSciNetCrossRefMATHADSGoogle Scholar
  12. 12.
    Benfatto, G., Giuliani, A., Mastropietro, V.: Fermi liquid behavior in the 2d Hubbard model at low temperatures. Ann. Henri Poincaré 7(5), 809–898 (2006)MathSciNetCrossRefMATHADSGoogle Scholar
  13. 13.
    Benfatto, G., Gallavotti, G., Procacci, A., Scoppola, B.: Beta function and Schwinger functions for a many fermions system in one dimension. Anomaly of the fermi surface. Commun. Math. Phys. 160(1), 93–171 (1994)MathSciNetCrossRefMATHADSGoogle Scholar
  14. 14.
    Bogoliubov, NN: On the theory of superfluidity. Eng. Trans. J. Phis. (USSR) 11, 23 (1947)Google Scholar
  15. 15.
    Castellani, C., Di Castro, C., Pistolesi, F., Strinati, G.C.: Infrared behavior of interacting bosons at zero temperature. Phys. Rev. Lett. 78(9), 1612–1615 (1997)CrossRefADSGoogle Scholar
  16. 16.
    Cenatiempo, S.: Low dimensional interacting bosons. PhD thesis, Scuola di dottorato in Scienze MM.FF.NN., Sapienza Università di Roma (2013)Google Scholar
  17. 17.
    Dupuis, N.: Infrared behavior and spectral function of a Bose superfluid at zero temperature. Phys. Rev. A 80, 043627 (2009)MathSciNetCrossRefADSGoogle Scholar
  18. 18.
    Dupuis, N.: Unified picture of superfluidity: from Bogoliubov’s approximation to Popov’s hydrodynamic theory. Phys. Rev. Lett. 102, 190401 (2009)CrossRefADSGoogle Scholar
  19. 19.
    Dupuis, N.: Infrared behavior in systems with a broken continuous symmetry: classical O(N) model versus interacting bosons. Phys. Rev. E 83, 031120 (2011)MathSciNetCrossRefADSGoogle Scholar
  20. 20.
    Dupuis, N., Sengupta, K.: Non-perturbative renormalization group approach to zero-temperature bose systems. EPL (Europhysics Letters) 80(5), 50007 (2007)MathSciNetCrossRefADSGoogle Scholar
  21. 21.
    Fisher, D.S., Hohenberg, P.C.: Dilute Bose gas in two dimensions. Phys. Rev. B 37, 4936–4943 (1988)CrossRefADSGoogle Scholar
  22. 22.
    Fisher, M.P.A., Weichman, P.B., Grinstein, G., Fisher, D.S.: Boson localization and the superfluid-insulator transition. Phys. Rev. B 40, 546–570 (1989)CrossRefADSGoogle Scholar
  23. 23.
    Gallavotti, G.: Renormalization theory and ultraviolet stability for scalar fields via renormalization group methods. Rev. Mod. Phys. 57(2), 471–562 (1985)MathSciNetCrossRefADSGoogle Scholar
  24. 24.
    Gallavotti, G., Nicolò, F.: Renormalization theory in four-dimensional scalar fields, (I) and (II). Commun. Math. Phys. 100 and 101:545–590 and 247–282 (1985)Google Scholar
  25. 25.
    Gavoret, J., Noziéres, P.: Structure of the perturbation expansion for the Bose liquid at zero temperature. Ann. Phys. 28(3), 349–399 (1964)Google Scholar
  26. 26.
    Gentile, G., Mastropietro, V.: Renormalization group for one-dimensional fermions: a review on mathematical results. Phys. Rep. 352, 273–437 (2001)Google Scholar
  27. 27.
    Ginibre, J.: On the asymptotic exactness of the Bogoliubov approximation for many boson systems. Commun. Math. Phys. 8(1), 26–51 (1968)MathSciNetCrossRefMATHADSGoogle Scholar
  28. 28.
    Giuliani, A., Mastropietro, V.: Rigorous construction of ground state correlations in graphene: renormalization of the velocities and ward identities. Phys. Rev. B 79(20), 201403 (2009)CrossRefADSGoogle Scholar
  29. 29.
    Giuliani, A., Mastropietro, V.: The two-dimensional Hubbard model on the honeycomb lattice. Commun. Math. Phys. 293(2), 301–346 (2010)MathSciNetCrossRefMATHADSGoogle Scholar
  30. 30.
    Giuliani, A., Mastropietro, V., Porta, M.: Lattice gauge theory model for graphene. Phys. Rev. B 82(12), 121418 (2010)CrossRefADSGoogle Scholar
  31. 31.
    Giuliani, A., Mastropietro, V., Porta, M.: Lattice quantum electrodynamics for graphene. Ann. Phys. 327(2), 461–511 (2012)MathSciNetCrossRefMATHADSGoogle Scholar
  32. 32.
    Hohenberg, P.C.: Existence of long-range order in one and two dimensions. Phys. Rev. 158, 383–386 (1967)CrossRefADSGoogle Scholar
  33. 33.
    Hugenholtz, N.M., Pines, D.: Ground-state energy and excitation spectrum of a system of interacting bosons. Phys. Rev. 116(3), 489–506 (1959)MathSciNetCrossRefMATHADSGoogle Scholar
  34. 34.
    Lee, T.D., Yang, C.N.: Many-body problem in quantum statistical mechanics. V. Degenerate phase in Bose-Einstein condensation. Phys. Rev. 117(4), 897–920 (1960)MathSciNetCrossRefADSGoogle Scholar
  35. 35.
    Lieb, E.H., Seiringer, R., Solovej, J.P., Yngvason, J.: The Mathematics of the Bose Gas and its Condensation. Birkhäuser, Basel (2005)MATHGoogle Scholar
  36. 36.
    Mastropietro, V.: Non-perturbative Renormalization. World Scientific, Singapore (2009)Google Scholar
  37. 37.
    Negele, J.W., Orland, H.: Quantum Many-Particle Systems. Addison-Wesley, Reading, MA (1987)Google Scholar
  38. 38.
    Nepomnyashchii, Y.A., Nepomnyashchii, A.A.: Infrared divergence in field theory of a Bose system with a condensate. Sov. Phys. JETP 48(3), 493–501 (1978)Google Scholar
  39. 39.
    Pistolesi, F., Castellani, C., Di Castro, C., Strinati, G.C.: Renormalization-group approach to the infrared behavior of a zero-temperature Bose system. Phys. Rev. B 69(2), 024513 (2004)CrossRefADSGoogle Scholar
  40. 40.
    Popov, V.N., Seredniakov, A.V.: Low-frequency asymptotic form of the self-energy parts of a superfluid Bose system at \(t=0\). Sov. Phys. JETP 50, 193 (1979)ADSGoogle Scholar
  41. 41.
    Rançon, A., Dupuis, N.: Non-perturbative renormalization group approach to strongly correlated lattice bosons. Phys. Rev. B 84, 174513 (2011)CrossRefADSGoogle Scholar
  42. 42.
    Rançon, A., Dupuis, N.: Non-perturbative renormalization group approach to the Bose-Hubbard model. Phys. Rev. B 83, 172501 (2011)CrossRefADSGoogle Scholar
  43. 43.
    Rançon, A., Dupuis, N.: Universal thermodynamics of a two-dimensional Bose gas. Phys. Rev. A 85, 063607 (2012)CrossRefADSGoogle Scholar
  44. 44.
    Sachdev, S.: Quantum Phase Transitions. Cambridge University Press, Cambridge, MA (1999)Google Scholar
  45. 45.
    Sachdev, S., Senthil, T., Shankar, R.: Finite temperature properties of quantum antiferromagnets in a uniform magnetic field in one and two dimensions. Phys. Rev. B 50, 258–272 (1994)CrossRefADSGoogle Scholar
  46. 46.
    Sinner, A., Hasselmann, N., Kopietz, P.: Functional renormalization group approach to interacting bosons at zero temperature. Phys. Rev. A 82, 063632 (2010)CrossRefADSGoogle Scholar
  47. 47.
    Wetterich, C.: Functional renormalization for quantum phase transitions with nonrelativistic bosons. Phys. Rev. B 77, 064504 (2008)MathSciNetCrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Universität ZürichZürichSwitzerland
  2. 2.Università degli Studi Roma TreRomaItaly

Personalised recommendations