Transitions in the Unitary Matrix Models

  • C. B. Wang

Abstract

The first-, second- and third-order phase transitions, or discontinuities, in the unitary matrix models will be discussed in this chapter. The Gross-Witten third-order phase transition is described in association with the string equation in the unitary matrix model, and it will be generalized by considering the higher degree potentials. The critical phenomena (second-order divergences) and third-order divergences are discussed similarly to the critical phenomenon in the planar diagram model, but a different Toda lattice and string equation will be applied here by using the double scaling method. The discontinuous property in the first-order transition model of the Hermitian matrix model discussed before will recur in the first-order transition model of the unitary matrix model, indicating a common mathematical background behind the first-order discontinuities. The purpose of this chapter is to further confirm that the string equation method can be widely applied to study phase transition problems in matrix models, and that the expansion method based on the string equations can work efficiently to find the power-law divergences considered in the transition problems.

Keywords

Divergence Gross-Witten transition Multi-cut Painleve equation Unitary matrix model 

References

  1. 1.
    Cresswell, C., Joshi, N.: The discrete first, second and thirty-fourth Painlevé hierarchies. J. Phys. A 32, 655–669 (1999) MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Foerster, D.: On condensation of extended structures. Phys. Lett. B 77, 211–213 (1978) CrossRefGoogle Scholar
  3. 3.
    Gross, D.J., Witten, E.: Possible third-order phase transition in the large-N lattice gauge theory. Phys. Rev. D 21, 446–453 (1980) CrossRefGoogle Scholar
  4. 4.
    Hisakado, M.: Unitary matrix model and the Painlevé III. Mod. Phys. Lett. A 11, 3001–3010 (1996) MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Hisakado, M., Wadati, M.: Matrix models of two-dimensional gravity and discrete Toda theory. Mod. Phys. Lett. A 11, 1797–1806 (1996) MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Its, A.R., Novokshenov, Yu.: The Isomonodromy Deformation Method in the Theory of Painlevé Equations. Lecture Notes in Mathematics, vol. 1191. Springer, Berlin (1986) Google Scholar
  7. 7.
    Janke, W., Kleinert, H.: How good is the Villain approximation? Nucl. Phys. B 270, 135–153 (1986) CrossRefGoogle Scholar
  8. 8.
    Klebanov, I.R., Maldacena, J., Seiberg, N.: Unitary and complex matrix models as 1D type 0 strings. Commun. Math. Phys. 252, 275–323 (2004) MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Korepin, V.E., Bogoliubov, N.M., Izergin, A.G.: Quantum Inverse Scattering Method and Correlation Functions. Cambridge University Press, Cambridge (1993) CrossRefMATHGoogle Scholar
  10. 10.
    McLeod, J.B., Wang, C.B.: Discrete integrable systems associated with the unitary matrix model. Anal. Appl. 2, 101–127 (2004) MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    McLeod, J.B., Wang, C.B.: Eigenvalue density in Hermitian matrix models by the Lax pair method. J. Phys. A, Math. Theor. 42, 205205 (2009) MathSciNetCrossRefGoogle Scholar
  12. 12.
    Morozov, A.Yu.: Unitary integrals and related matrix models. Theor. Math. Phys. 162, 1–33 (2010) CrossRefMATHGoogle Scholar
  13. 13.
    Periwal, V., Shevitz, D.: Unitary-matrix models as exactly solvable string theories. Phys. Rev. Lett. 64, 1326–1329 (1990) CrossRefGoogle Scholar
  14. 14.
    Periwal, V., Shevitz, D.: Exactly solvable unitary matrix models: multicritical potentials and correlations. Nucl. Phys. B 344, 731–746 (1990) MathSciNetCrossRefGoogle Scholar
  15. 15.
    Polyakov, A.M.: String representations and hidden symmetries for gauge fields. Phys. Lett. B 82, 247–250 (1979) CrossRefGoogle Scholar
  16. 16.
    Szabo, R.J., Tierz, M.: Chern-Simons matrix models, two-dimensional Yang-Mills theory and the Sutherland model. J. Phys. A 43, 265401 (2010) MathSciNetGoogle Scholar
  17. 17.
    Wang, C.B.: Orthonormal polynomials on the unit circle and spatially discrete Painlevé II equation. J. Phys. A 32, 7207–7217 (1999) MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Yeomans, J.M.: Statistical Mechanics of Phase Transitions. Oxford University Press, London (1994) Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • C. B. Wang
    • 1
  1. 1.Institute of AnalysisTroyUSA

Personalised recommendations