Interfacial Roughening in Field Theory

  • Michael H. Köpf
  • Gernot MünsterEmail author


In the rough phase, the width of interfaces separating different phases of statistical systems increases logarithmically with the system size. This phenomenon is commonly described in terms of the capillary wave model, which deals with fluctuating, infinitely thin membranes, requiring ad hoc cut-offs in momentum space. We investigate the interface roughening in a unified approach, which does not rely on joining different models, namely in the framework of the Landau-Ginzburg model, that is renormalized field theory, in the one-loop approximation. The interface profile and width are calculated analytically, resulting in finite expressions with definite coefficients. They are valid in the scaling region and depend on the known renormalized coupling constant.


Interfaces Field theory 


  1. 1.
    Rowlinson, J., Widom, B.: Molecular Theory of Capillarity. Clarendon Press, Oxford (1982) Google Scholar
  2. 2.
    Huang, J.S., Webb, W.W.: Viscous damping of thermal excitations on the interface of critical fluid mixtures. Phys. Rev. Lett. 23, 160–163 (1969) CrossRefADSGoogle Scholar
  3. 3.
    Langevin, D. (ed.): Light Scattering by Liquid Surfaces and Complementary Techniques. Dekker, New York (1992) Google Scholar
  4. 4.
    McClain, B.R., Yoon, M., Litster, J.D., Mochrie, S.G.J.: Interfacial roughness in a near-critical binary fluid mixture: X-ray reflectivity and near-specular diffuse scattering. Eur. Phys. J. B 10, 45–52 (1999) ADSCrossRefGoogle Scholar
  5. 5.
    Thomas, R.K.: Neutron reflection from liquid interfaces. Annu. Rev. Phys. Chem. 55, 391–426 (2004) CrossRefGoogle Scholar
  6. 6.
    Werner, A., Schmid, F., Müller, M., Binder, K.: Anomalous size-dependence of interfacial profiles between coexisting phases of polymer mixtures in thin film geometry: a Monte-Carlo simulation. J. Chem. Phys. 107, 8175–8188 (1997) CrossRefADSGoogle Scholar
  7. 7.
    Werner, A., Schmid, F., Müller, M., Binder, K.: “Intrinsic” profiles and capillary waves at homopolymer interfaces: a Monte Carlo study. Phys. Rev. E 59, 728–738 (1999) CrossRefADSGoogle Scholar
  8. 8.
    Dünweg, B., Landau, D.P., Milchev, A. (eds.): Computer Simulations of Surfaces and Interfaces. Kluwer Acad. Publ., Dordrecht (2003) zbMATHGoogle Scholar
  9. 9.
    Buff, F., Lovett, R., Stillinger, F.: Interfacial density profile for fluids in the critical region. Phys. Rev. Lett. 15, 621–623 (1965) CrossRefADSGoogle Scholar
  10. 10.
    Jasnow, D.: Critical phenomena at interfaces. Rep. Prog. Phys. 47, 1059–1132 (1984) CrossRefADSGoogle Scholar
  11. 11.
    Widom, B.: In: Domb, C., Green, M. (eds.) Phase Transitions and Critical Phenomena, vol. 2. Academic Press, New York (1972) Google Scholar
  12. 12.
    Ohta, T., Kawasaki, K.: Renormalization group approach to the interfacial order parameter profile near the critical point. Prog. Theor. Phys. 58, 467–481 (1977) CrossRefADSGoogle Scholar
  13. 13.
    Rudnick, J., Jasnow, D.: ε expansion for the interfacial profile. Phys. Rev. B 17, 1351–1354 (1978) CrossRefADSGoogle Scholar
  14. 14.
    Sikkenk, J.H., van Leeuwen, J.M.J.: ε-expansion for the interfacial profile in an external field. Physica A 137, 156–177 (1986) CrossRefADSGoogle Scholar
  15. 15.
    Parisi, G.: Field-theoretic approach to second-order phase transition in two-and three-dimensional systems. J. Stat. Phys. 23, 49–82 (1980) CrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Le Guillou, J.C., Zinn-Justin, J.: Critical exponents from field theory. Phys. Rev. B 21, 3976–3998 (1980) CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Münster, G.: Interface tension in three-dimensional systems from field theory. Nucl. Phys. B 340, 559–567 (1990) CrossRefADSGoogle Scholar
  18. 18.
    Münster, G., Heitger, J.: Field-theoretic calculation of the universal amplitude ratio of correlation lengths in 3D Ising systems. Nucl. Phys. B 424, 582–594 (1994) CrossRefADSGoogle Scholar
  19. 19.
    Gutsfeld, C., Küster, J., Münster, G.: Calculation of universal amplitude ratios in three-loop order. Nucl. Phys. B 479, 654–662 (1996) CrossRefADSGoogle Scholar
  20. 20.
    Brézin, E., Zinn-Justin, J.: Finite size effects in phase transitions. Nucl. Phys. B 257, 867–893 (1985) CrossRefADSGoogle Scholar
  21. 21.
    Zinn-Justin, J.: Quantum Field Theory and Critical Phenomena. Cambridge University Press, Cambridge (2002) Google Scholar
  22. 22.
    Jasnow, D., Rudnick, J.: Interfacial profile in three dimensions. Phys. Rev. Lett. 41, 698–701 (1978) CrossRefADSGoogle Scholar
  23. 23.
    Le Bellac, M.: Quantum and Statistical Field Theory. Clarendon Press, Oxford (1991) Google Scholar
  24. 24.
    van der Waals, J.: Thermodynamische theorie der capillariteit in de onderstelling van continue dichtheidsverandering. Verhandel. Konink. Akad. Weten. Amsterdam 1 (1893) (in Dutch). English translation: Rowlinson, J.: The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density. J. Stat. Phys. 20, 197–244 (1979) Google Scholar
  25. 25.
    Cahn, J., Hilliard, J.: Free energy of a nonuniform system. J. Chem. Phys. 28, 258–267 (1958) CrossRefADSGoogle Scholar
  26. 26.
    Rajamaran, R.: Non-perturbative semi-classical methods in quantum field theory (a pedagogical review). Phys. Rep. 21C, 227–313 (1975) CrossRefADSGoogle Scholar
  27. 27.
    Gervais, J.L., Sakita, B.: Extended particles in quantum field theories. Phys. Rev. D 11, 2943–2945 (1975) ADSGoogle Scholar
  28. 28.
    Küster, J., Münster, G.: The interfacial profile in two-loop order. J. Stat. Phys. 129, 441–451 (2007) CrossRefADSMathSciNetzbMATHGoogle Scholar
  29. 29.
    Lüscher, M., Weisz, P.: Scaling laws and triviality bounds in the lattice φ 4-theory; one-component model in the phase with spontaneous broken symmetry. Nucl. Phys. B 295, 65–92 (1987) CrossRefGoogle Scholar
  30. 30.
    Köpf, M.H.: Rauhigkeit von Grenzflächen in der Ising-Universalitätsklasse. Diploma thesis, Univ. of Münster, 2007 Google Scholar
  31. 31.
    Fisk, S., Widom, B.: Structure and free energy of the interface between fluid phases in equilibrium near the critical point. J. Chem. Phys. 50, 3219–3227 (1960) CrossRefADSGoogle Scholar
  32. 32.
    Hoppe, P., Münster, G.: The interface tension of the three-dimensional Ising model in two-loop order. Phys. Lett. A 238, 265–269 (1998) CrossRefADSMathSciNetzbMATHGoogle Scholar
  33. 33.
    Caselle, M., Hasenbusch, M.: Universal amplitude ratios in the 3D Ising model. J. Phys. A 30, 4963–4982 (1997) ADSzbMATHGoogle Scholar
  34. 34.
    Mon, K., Landau, D., Stauffer, D.: Interface roughening in the three-dimensional Ising model. Phys. Rev. B 42, 545–547 (1990) CrossRefADSGoogle Scholar
  35. 35.
    Bürkner, E., Stauffer, D.: Monte Carlo study of surface roughening in the three-dimensional Ising model. Z. Phys. B 53, 241–243 (1983) ADSGoogle Scholar
  36. 36.
    Hasenbusch, M., Pinn, K.: Surface tension, surface stiffness, and surface width of the 3-dimensional Ising model on a cubic lattice. Physica A 192, 342–374 (1992) CrossRefADSGoogle Scholar
  37. 37.
    Müller, M., Münster, G.: Profile and width of rough interfaces. J. Stat. Phys. 118, 669–686 (2005) CrossRefADSzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikUniversität MünsterMünsterGermany

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