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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Cresswell, C., Joshi, N.: The discrete first, second and thirty-fourth Painlevé hierarchies. J. Phys. A 32, 655–669 (1999)
Foerster, D.: On condensation of extended structures. Phys. Lett. B 77, 211–213 (1978)
Gross, D.J., Witten, E.: Possible third-order phase transition in the large-N lattice gauge theory. Phys. Rev. D 21, 446–453 (1980)
Hisakado, M.: Unitary matrix model and the Painlevé III. Mod. Phys. Lett. A 11, 3001–3010 (1996)
Hisakado, M., Wadati, M.: Matrix models of two-dimensional gravity and discrete Toda theory. Mod. Phys. Lett. A 11, 1797–1806 (1996)
Its, A.R., Novokshenov, Yu.: The Isomonodromy Deformation Method in the Theory of Painlevé Equations. Lecture Notes in Mathematics, vol. 1191. Springer, Berlin (1986)
Janke, W., Kleinert, H.: How good is the Villain approximation? Nucl. Phys. B 270, 135–153 (1986)
Klebanov, I.R., Maldacena, J., Seiberg, N.: Unitary and complex matrix models as 1D type 0 strings. Commun. Math. Phys. 252, 275–323 (2004)
Korepin, V.E., Bogoliubov, N.M., Izergin, A.G.: Quantum Inverse Scattering Method and Correlation Functions. Cambridge University Press, Cambridge (1993)
McLeod, J.B., Wang, C.B.: Discrete integrable systems associated with the unitary matrix model. Anal. Appl. 2, 101–127 (2004)
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)
Morozov, A.Yu.: Unitary integrals and related matrix models. Theor. Math. Phys. 162, 1–33 (2010)
Periwal, V., Shevitz, D.: Unitary-matrix models as exactly solvable string theories. Phys. Rev. Lett. 64, 1326–1329 (1990)
Periwal, V., Shevitz, D.: Exactly solvable unitary matrix models: multicritical potentials and correlations. Nucl. Phys. B 344, 731–746 (1990)
Polyakov, A.M.: String representations and hidden symmetries for gauge fields. Phys. Lett. B 82, 247–250 (1979)
Szabo, R.J., Tierz, M.: Chern-Simons matrix models, two-dimensional Yang-Mills theory and the Sutherland model. J. Phys. A 43, 265401 (2010)
Wang, C.B.: Orthonormal polynomials on the unit circle and spatially discrete Painlevé II equation. J. Phys. A 32, 7207–7217 (1999)
Yeomans, J.M.: Statistical Mechanics of Phase Transitions. Oxford University Press, London (1994)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Wang, C.B. (2013). Transitions in the Unitary Matrix Models. In: Application of Integrable Systems to Phase Transitions. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38565-0_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-38565-0_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-38564-3
Online ISBN: 978-3-642-38565-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)