Abstract
We study the interfaces of ground states of ferromagnetic Ising models with external fields. We show that, if the coefficients of the interaction and the magnetic field are periodic, the magnetic field has zero flux over a period and is small enough, then for every plane, we can find a ground state whose interface lies at a bounded distance of the plane. This bound on the width of the interface can be chosen independent of the plane. We also study the average energy of the plane-like interfaces as a function of the direction. We show that there is a well defined thermodynamic limit for the energy of the interface and that it enjoys several convexity properties.
Similar content being viewed by others
References
S. Aubry P. Y. Le Daeron (1983) ArticleTitleThe discrete Frenkel-Kontorova model and its extension. I. exact results for the ground-states Phys. D 8 IssueID3 381–422
T. Bodineau D. Ioffe Y. Velenik (2000) ArticleTitleRigorous probabilistic analysis of equilibrium crystal shapes J. Math. Phys. 41 IssueID3 1033–1098
A. Candel R. de la Llave (1998) ArticleTitleOn the Aubry-Mather theory in statistical mechanics Comm. Math. Phys. 192 IssueID3 649–669
L. A. Caffarelli Rafael de la Llave (2001) ArticleTitlePlanelike minimizers in periodic media Comm. Pure Appl. Math. 54 IssueID14 1403–1441
L. A. Caffarelli and Rafael de la Llave. Planelike minimizers in periodic media ii: consequences of the maximum principle. 2003. Manuscript.
L. A. Caffarelli and Rafael de la Llave. Quasiperiodic minimizers for periodic variational problems. 2003. Manuscript.
Marzio Cassandro Enza Orlandi Pierre Picco (2002) ArticleTitleThe optimal interface profile for a non-local model of phase separation Nonlinearity 15 IssueID15 1621–1651
Enrico Giusti. Minimal Surfaces and Functions of Bounded Variation (Birkhäuser Verlag, Basel, 1984).
G. A. Hedlund (1932) ArticleTitleGeodesics on a two-dimensional Riemannian manifold with periodic coefficients Ann. Math. 33 719–739
R. B. Israel. Convexity in the Theory of Lattice Gases, Princeton Series in Physics, With an introduction by Arthur S. Wightman (Princeton University Press, Princeton, N.J., 1979)
A. Katok (1982) ArticleTitleSome remarks of Birkhoff and Mather twist map theorems Ergodic Theory Dyn. Sys. 2 IssueID2 185–194
A. Katok. Periodic and quasiperiodic orbits for twist maps. in Dynamical systems and chaos (sitges/ Barcelona, 1982), volume 179 of Lecture Notes in Phys., (Springer, Berlin, 1983), pp. 47–65.
J. N. Mather (1982) ArticleTitleExistence of quasiperiodic orbits for twist homeomorphisms of the annulus Topology 21 IssueID4 457–467
M. Morse (1924) ArticleTitleA fundamental class of geodesics on any closed surface of genus greater than one Trans. AMS. 26 25–60
Jürgen Moser (1986) ArticleTitleMinimal solutions of variational problems on a torus Ann. Inst. H. Poincaré Anal. Non Linéaire 3 IssueID3 229–272
David Ruelle. Statistical Mechanics. (World Scientific Publishing Co. Inc., River Edge, NJ, 1999) Rigorous results, Reprint of the 1989 edition.
Barry Simon. The Statistical Mechanics of Lattice Gases. Vol. I Princeton Series in Physics (Princeton University Press, Princeton, NJ, 1993).
Ya. G. Sinai (Eds) (1991) Mathematical Problems of Statistical Mechanics, volume 2 of Advanced Series in Nonlinear Dynamics World Scientific Publishing Co. Inc. Teaneck, NJ
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Caffarelli, L.A., Llave, R. Interfaces of Ground States in Ising Models with Periodic Coefficients. J Stat Phys 118, 687–719 (2005). https://doi.org/10.1007/s10955-004-8825-1
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10955-004-8825-1