Communications in Mathematical Physics

, Volume 286, Issue 1, pp 163–177 | Cite as

Periodic Minimizers in 1D Local Mean Field Theory

  • Alessandro Giuliani
  • Joel L. Lebowitz
  • Elliott H. Lieb
Article

Abstract

There are not many physical systems where it is possible to demonstate rigorously that energy minimizers are periodic. Using reflection positivity techniques we prove, for a class of mesoscopic free-energies representing 1D systems with competing interactions, that all minimizers are either periodic, with zero average, or of constant sign. Examples of both phenomena are given. This extends our previous work where such results were proved for the ground states of lattice systems with ferromagnetic nearest neighbor interactions and dipolar type antiferromagnetic long range interactions.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arlett J., Whitehead J.P., MacIsaac A.B., De’Bell K.: Phase diagram for the striped phase in the two-dimensional dipolar Ising model. Phys. Rev. B 54, 3394 (1996)CrossRefADSGoogle Scholar
  2. 2.
    Alberti G., Müller S.: A new approach to variational problems with multiple scales. Comm. Pure Appl. Math. 54, 761–825 (2001)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Benois O., Bodineau T., Buttà P., Presutti E.: On the validity of van der Waals theory of surface tension. Markov Process. Related Fields 3, 175–198 (1997)MATHMathSciNetGoogle Scholar
  4. 4.
    Benois O., Bodineau T., Presutti E.: Large deviations in the van der Waals limit. Stochastic Process. Appl. 75, 89–104 (1998)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Borue V.Y., Erukhimovich I.Y.: A Statistical Theory of Weakly Charged Polyelectrolytes: Fluctuations, Equation of State, and Microphase Separation. Macromolecules 21, 3240 (1988)CrossRefGoogle Scholar
  6. 6.
    Buttà P., Lebowitz J.L.: Local Mean Field Models of Uniform to Nonuniform Density (fluid-crystal) Transitions. J. Phys. Chem. B 109, 6849–6854 (2005)CrossRefGoogle Scholar
  7. 7.
    Brazovskii S.A.: Phase transition of an isotropic system to a non uniform state. Zh. Eksp. Teor. Fiz. 68, 175 (1975)Google Scholar
  8. 8.
    Carlen E., Carvalho M.C., Esposito R., Lebowitz J.L., Marra R.: Phase Transitions in Equilibrium Systems: Microscopic Models and Mesoscopic Free Energies. J. Mole. Phys. 103, 3141–3151 (2005)CrossRefADSGoogle Scholar
  9. 9.
    Chen X., Oshita Y.: Periodicity and Uniqueness of Global Minimizers of an Energy Functional Containing a Long-Range Interaction. SIAM J. Math. Anal. 37, 1299–1332 (2006)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    DeSimone, A., Kohn, R.V., Otto, F., Müller, S.: Recent analytical developments in micromagnetics. In: The Science of Hysteresis II: Physical Modeling, Micromagnetics, and Magnetization Dynamics, Bertotti G., Mayergoyz I. (eds) Amsterdam: Elsevier (2006), pp. 269–381Google Scholar
  11. 11.
    Dupuis P., Ellis R.S.: A weak convergence approach to the theory of large deviations. John Wiley & Sons, Inc., Wiley Series in Probability and Statistics, New York (1997)MATHGoogle Scholar
  12. 12.
    Emery V.J., Kivelson S.A.: Frustrated electronic phase separation and high-temperature superconductors. Physica C 209, 597 (1993)CrossRefADSGoogle Scholar
  13. 13.
    Frohlich J., Israel R., Lieb E.H., Simon B.: Phase Transitions and Reflection Positivity. I. General Theory and Long Range Lattice Models. Commun. Math. Phys. 62, 1 (1978)CrossRefADSMathSciNetGoogle Scholar
  14. 14.
    Garel T., Doniach S.: Phase transitions with spontaneous modulation-the dipolar Ising ferromagnet. Phys. Rev. B 26, 325 (1982)CrossRefADSGoogle Scholar
  15. 15.
    Gates D.J., Penrose O.: The van der Waals limit for classical systems. I. A variational principle. Commun. Math. Phys. 15, 255–276 (1969)MATHCrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Gates D.J., Penrose O.: The van der Waals limit for classical systems. III. Deviation from the van der Waals-Maxwell theory. Commun. Math. Phys. 17, 194–209 (1970)CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Giuliani A., Lebowitz J.L., Lieb E.H.: Ising models with long-range dipolar and short range ferromagnetic interactions. Phys. Rev. B 74, 064420 (2006)CrossRefADSGoogle Scholar
  18. 18.
    Giuliani A., Lebowitz J.L., Lieb E.H.: Striped phases in two dimensional dipole systems. Phys. Rev. B 76, 184426 (2007)CrossRefADSGoogle Scholar
  19. 19.
    Grousson M., Tarjus G., Viot P.: Phase diagram of an Ising model with long-range frustrating interactions: A theoretical analysis. Phys. Rev. E 62, 7781 (2000)CrossRefADSGoogle Scholar
  20. 20.
    Hohenberg P.C., Swift J.B.: Metastability in fluctuation-driven first-order transitions: Nucleation of lamellar phases. Phys. Rev. E 52, 1828 (1995)CrossRefADSGoogle Scholar
  21. 21.
    Lieb E.H., Loss M.: Analysis.Second Edition. Amer. Math. Soc., providence RI (2001)MATHGoogle Scholar
  22. 22.
    Lebowitz J.L., Penrose O.: Rigorous Treatment of the Van Der Waals-Maxwell Theory of the Liquid-Vapor Transition. J. Math. Phys. 7, 98–113 (1966)CrossRefADSMathSciNetGoogle Scholar
  23. 23.
    Leibler L.: Theory of Microphase Separation in Block Copolymers. Macromolecules 13, 1602 (1980)CrossRefGoogle Scholar
  24. 24.
    MacIsaac A.B., Whitehead J.P., Robinson M.C., De’Bell K.: Striped phases in two-dimensional dipolar ferromagnets. Phys. Rev. B 51, 16033 (1995)CrossRefADSGoogle Scholar
  25. 25.
    McMillian W.L.: Landau theory of charge-density waves in transition-metal dichalcogenides. Phys. Rev. B 12, 1187 (1975)CrossRefADSGoogle Scholar
  26. 26.
    Muratov C.B.: Theory of domain patterns in systems with long-range interactions of Coulomb type. Phys. Rev. E 66, 066108 (2002)CrossRefADSMathSciNetGoogle Scholar
  27. 27.
    Müller S.: Singular perturbations as a selection criterion for periodic minimizing sequences. Calc. Var. Partial Differ. Eq. 1, 169–204 (1993)MATHCrossRefGoogle Scholar
  28. 28.
    Nussinov, Z.: Commensurate and Incommensurate O(n) Spin Systems: Novel Even-Odd Effects, A Generalized Mermin-Wagner-Coleman Theorem, and Ground States. http://arxiv.org/list/cond-mat/0105253
  29. 29.
    Ohta T., Kawasaki K.: Equilibrium morphology of block polymer melts. Macromolecules 19, 2621–2632 (1986)CrossRefGoogle Scholar
  30. 30.
    Seul M., Andelman D.: Domain Shapes and Patterns: The Phenomenology of Modulated Phases. Science 267, 476 (1995)CrossRefADSGoogle Scholar
  31. 31.
    Spivak B., Kivelson S.A.: Phases intermediate between a two-dimensional electron liquid and Wigner crystal. Phys. Rev. B 70, 155114 (2004)CrossRefADSGoogle Scholar
  32. 32.
    Spivak B., Kivelson S.A.: Transport in two dimensional electronic micro-emulsions. Ann. Phys. (N.Y.) 321, 2071 (2006)MATHCrossRefADSGoogle Scholar
  33. 33.
    Stoycheva A.D., Singer S.J.: Phys. Rev. Lett. 84, 4657 (2000)CrossRefADSGoogle Scholar

Copyright information

© The Author(s) 2008

Authors and Affiliations

  • Alessandro Giuliani
    • 1
  • Joel L. Lebowitz
    • 2
  • Elliott H. Lieb
    • 3
  1. 1.Department of PhysicsPrinceton UniversityPrincetonUSA
  2. 2.Departments of Mathematics and PhysicsRutgers UniversityPiscatawayUSA
  3. 3.Departments of Mathematics and PhysicsPrinceton UniversityPrincetonUSA

Personalised recommendations