Skip to main content
SpringerLink
Log in

Spontaneous Symmetry Breaking of a Hyperbolic Sigma Model in Three Dimensions

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Non-linear sigma models that arise from the supersymmetric approach to disordered electron systems contain a non-compact bosonic sector. We study the model with target space H2, the two-hyperboloid with isometry group SU(1,1), and prove that in three dimensions moments of the fields are finite in the thermodynamic limit. Thus the non-compact symmetry SU(1,1) is spontaneously broken. The bound on moments is compatible with the presence of extended states.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Brascamp, H., Lieb, E.: On extensions of the Brunn-Minkowski and Prekopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Func. Anal. 22, 366–389 (1976)

    Article  MATH  Google Scholar 

  2. Disertori, M., Pinson, H., Spencer, T.: Density of states of random band matrices. Commun. Math. Phys. 232, 83–124 (2002)

    Article  MATH  Google Scholar 

  3. Efetov, K.B.: Supersymmetry and theory of disordered metals. Adv. Phys. 32, 53–127 (1983)

    MathSciNet  Google Scholar 

  4. Efetov, K.B.: Supersymmetry in disorder and chaos. Cambridge: Cambridge University Press, 1987

  5. Fyodorov, Y.V.: Negative moments of characteristic polynomials of random matrices: Ingham-Siegel integral as an alternative to Hubbard-Stratonovich transformation. Nucl. Phys. B 621, 643–674 (2002)

    Article  MATH  Google Scholar 

  6. Fyodorov, Y.V., Mirlin, A.D.: Scaling properties of localization in random band matrices: a σ-model approach. Phys. Rev. Lett. 67, 2405–2409 (1991)

    Article  MATH  Google Scholar 

  7. Helgason, S.: Differential geometry, Lie groups and symmetric spaces. New York: Academic Press, 1978

  8. Niedermaier, M., Seiler, E.: Non-amenability and spontaneous symmetry breaking – the hyperbolic spin-chain. http://arxiv.org/abs/hep-th/0312293, 2003; Duncan, A., Niedermaier, M., Seiler, E.: Vacuum orbit and spontaneous symmetry breaking in hyperbolic sigma models. http://arxiv.org/abs/hep-th/0405163, 2004

  9. Schäfer, L., Wegner, F.: Disordered system with n orbitals per site: Lagrange formulation, hyperbolic symmetry, and Goldstone modes. Z. Phys. B 38, 113–126 (1980)

    Google Scholar 

  10. Wegner, F.: The mobility edge problem: continuous symmetry and a conjecture. Z. Phys. B 35, 207–210 (1979)

    Google Scholar 

  11. Wegner, F.J.: Disordered system with n orbitals per site: n = ∞ limit. Phys. Rev. B 19, 783–792 (1979)

    Article  Google Scholar 

  12. Zirnbauer, M.R.: The supersymmetry method of random-matrix theory. http://arxiv.org/abs/math-ph/ 0404057, 2004

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Institute for Advanced Study, Princeton, NJ, 08540, USA

    T. Spencer

  2. Institut für Theoretische Physik, Universität zu Köln, Germany

    M.R. Zirnbauer

Authors
  1. T. Spencer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M.R. Zirnbauer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. Spencer.

Additional information

Communicated by M. Aizenman

Dedicated to Freeman Dyson on the Occasion of his eightieth birthday

Acknowledgement T. Spencer would like to thank M. Disertori, K. Gawedzki, G. Papanicolau and S.R.S. Varadhan for helpful comments.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Spencer, T., Zirnbauer, M. Spontaneous Symmetry Breaking of a Hyperbolic Sigma Model in Three Dimensions. Commun. Math. Phys. 252, 167–187 (2004). https://doi.org/10.1007/s00220-004-1223-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-004-1223-3

Keywords

Search

Navigation