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.
Similar content being viewed by others
References
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)
Alberti G., Müller S.: A new approach to variational problems with multiple scales. Comm. Pure Appl. Math. 54, 761–825 (2001)
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)
Benois O., Bodineau T., Presutti E.: Large deviations in the van der Waals limit. Stochastic Process. Appl. 75, 89–104 (1998)
Borue V.Y., Erukhimovich I.Y.: A Statistical Theory of Weakly Charged Polyelectrolytes: Fluctuations, Equation of State, and Microphase Separation. Macromolecules 21, 3240 (1988)
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)
Brazovskii S.A.: Phase transition of an isotropic system to a non uniform state. Zh. Eksp. Teor. Fiz. 68, 175 (1975)
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)
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)
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. 269–381
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)
Emery V.J., Kivelson S.A.: Frustrated electronic phase separation and high-temperature superconductors. Physica C 209, 597 (1993)
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)
Garel T., Doniach S.: Phase transitions with spontaneous modulation-the dipolar Ising ferromagnet. Phys. Rev. B 26, 325 (1982)
Gates D.J., Penrose O.: The van der Waals limit for classical systems. I. A variational principle. Commun. Math. Phys. 15, 255–276 (1969)
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)
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)
Giuliani A., Lebowitz J.L., Lieb E.H.: Striped phases in two dimensional dipole systems. Phys. Rev. B 76, 184426 (2007)
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)
Hohenberg P.C., Swift J.B.: Metastability in fluctuation-driven first-order transitions: Nucleation of lamellar phases. Phys. Rev. E 52, 1828 (1995)
Lieb E.H., Loss M.: Analysis.Second Edition. Amer. Math. Soc., providence RI (2001)
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)
Leibler L.: Theory of Microphase Separation in Block Copolymers. Macromolecules 13, 1602 (1980)
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)
McMillian W.L.: Landau theory of charge-density waves in transition-metal dichalcogenides. Phys. Rev. B 12, 1187 (1975)
Muratov C.B.: Theory of domain patterns in systems with long-range interactions of Coulomb type. Phys. Rev. E 66, 066108 (2002)
Müller S.: Singular perturbations as a selection criterion for periodic minimizing sequences. Calc. Var. Partial Differ. Eq. 1, 169–204 (1993)
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
Ohta T., Kawasaki K.: Equilibrium morphology of block polymer melts. Macromolecules 19, 2621–2632 (1986)
Seul M., Andelman D.: Domain Shapes and Patterns: The Phenomenology of Modulated Phases. Science 267, 476 (1995)
Spivak B., Kivelson S.A.: Phases intermediate between a two-dimensional electron liquid and Wigner crystal. Phys. Rev. B 70, 155114 (2004)
Spivak B., Kivelson S.A.: Transport in two dimensional electronic micro-emulsions. Ann. Phys. (N.Y.) 321, 2071 (2006)
Stoycheva A.D., Singer S.J.: Phys. Rev. Lett. 84, 4657 (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by H. Spohn
© 2008 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.
Rights and permissions
About this article
Cite this article
Giuliani, A., Lebowitz, J.L. & Lieb, E.H. Periodic Minimizers in 1D Local Mean Field Theory. Commun. Math. Phys. 286, 163–177 (2009). https://doi.org/10.1007/s00220-008-0589-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-008-0589-z