Abstract
We consider Ising models in two and three dimensions, with short range ferromagnetic and long range, power-law decaying, antiferromagnetic interactions. We let J be the ratio between the strength of the ferromagnetic to antiferromagnetic interactions. The competition between these two kinds of interactions induces the system to form domains of minus spins in a background of plus spins, or vice versa. If the decay exponent p of the long range interaction is larger than d + 1, with d the space dimension, this happens for all values of J smaller than a critical value J c (p), beyond which the ground state is homogeneous. In this paper, we give a characterization of the infinite volume ground states of the system, for p > 2d and J in a left neighborhood of J c (p). In particular, we prove that the quasi-one-dimensional states consisting of infinite stripes (d = 2) or slabs (d = 3), all of the same optimal width and orientation, and alternating magnetization, are infinite volume ground states. Our proof is based on localization bounds combined with reflection positivity.
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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)
Ball P.: The Self-Made Tapestry: Pattern Formation in Nature. Oxford University Press, Oxford (1999)
Biskup M., Chayes L., Kivelson S.A.: On the absence of ferromagnetism in typical 2D ferromagnets. Commun. Math. Phys. 274, 217–231 (2007)
Cannas S.A., Michelon M.F., Stariolo D.A., Tamarit F.A.: Ising nematic phase in ultrathin magnetic films: a Monte Carlo study. Phys. Rev. B 73, 184425 (2006)
Chakrabarty S., Dobrosavljevic V., Seidel A., Nussinov Z.: Universality of modulation length and time exponents. Phys. Rev. E 86, 041132 (2012)
Chakrabarty S., Nussinov Z.: Modulation and correlation lengths in systems with competing interactions. Phys. Rev. B 84, 144402 (2011)
Chayes L., Emery V., Kivelson S., Nussinov Z., Tarjus G.: Avoided critical behavior in a uniformly frustrated system. Phys. A 225, 129 (1996)
Cinti F., Portmann O., Pescia D., Vindigni A.: One-dimensional Ising ferromagnet frustrated by long-range interactions at finite temperatures. Phys. Rev. B 79, 214434 (2009)
Czech R., Villain J.: Instability of two-dimensional Ising ferromagnets with dipole interactions. J. Phys. Condens. Matter 1, 619 (1989)
DeBell K., MacIsaac A.B., Whitehead J.P.: Dipolar effects in magnetic thin films and quasi-two-dimensional systems. Rev. Mod. Phys. 72, 225 (2000)
Edlund E., Nilsson Jacobi M.: Universality of striped morphologies. Phys. Rev. Lett. 105, 137203 (2010)
Emery V.J., Kivelson S.A., Tranquada J.M.: Stripe phases in high-temperature superconductors. Proc. Natl. Acad. Sci. USA 96, 8814–8817 (1999)
Flatley L.C., Theil F.: Face-centered cubic crystallization of atomistic configurations. Arch. Ration. Mech. Anal. 218, 363–416 (2015)
Fradkin E., Kivelson S.A.: Liquid-crystal phases of quantum Hall systems. Phys. Rev. B 59, 8065 (1999)
Giuliani A., Lebowitz J., Lieb E.: Ising models with long-range dipolar and short range ferromagnetic interactions. Phys. Rev. B 74, 064420 (2006)
Giuliani A., Lebowitz J., Lieb E.: Striped phases in two-dimensional dipole systems. Phys. Rev. B 76, 184426 (2007)
Giuliani A., Lebowitz J., Lieb E.: Periodic minimizers in 1D local mean field theory. Commun. Math. Phys. 286, 163177 (2009)
Giuliani A., Lebowitz J., Lieb E.: Modulated phases of a 1D sharp interface model in a magnetic field. Phys. Rev. B 80, 134420 (2009)
Giuliani A., Lebowitz J., Lieb E.: Checkerboards, stripes and corner energies in spin models with competing interactions. Phys. Rev. B 84, 064205 (2011)
Giuliani A., Lieb E.H., Seiringer R.: Formation of stripes and slabs near the ferromagnetic transition. Comm. Math. Phys. 331, 333–350 (2014)
Giuliani A., Lieb E.H., Seiringer R.: Realization of stripes and slabs in two and three dimensions. Phys. Rev. B 88, 064401 (2013)
Giuliani A., Müller S.: Striped periodic minimizers of a two-dimensional model for martensitic phase transitions. Commun. Math. Phys. 309, 313–339 (2012)
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)
Harrison C. et al.: Mechanisms of ordering in striped patterns. Science 24, 1558 (2000)
Heitmann R.C., Radin C.: The ground state for sticky disks. J. Stat. Phys. 22, 281 (1980)
Kennedy T., Lieb E.H.: An itinerant electron model with crystalline or magnetic long range order. Phys. A 138, 320 (1986)
Kohn R.V., Müller S.: Branching of twins near an austenite–twinned-martensite interface. Philos. Mag. A 66, 697 (1992)
Low U., Emery V.J., Fabricius K., Kivelson S.A.: Study of an Ising model with competing long- and short-range interactions. Phys. Rev. Lett. 72, 1918 (1994)
MacIsaac A.B., Whitehead J.P., Robinson M.C., DeBell K.: Striped phases in two-dimensional dipolar ferromagnets. Phys. Rev. B 51, 16033 (1995)
Mendoza-Coto A., Stariolo D.A., Nicolao L.: Nature of long range order in stripe forming systems with long range repulsive interactions. Phys. Rev. Lett. 114, 116101 (2015)
Nielsen E., Bhatt R.N., Huse D.A.: Modulated phases in magnetic models frustrated by long-range interactions. Phys. Rev. B 77, 054432 (2008)
Okamoto M., Maruyama T., Yabana K., Tatsumi T.: Nuclear “pasta” structures in low-density nuclear matter and properties of the neutron-star crust. Phys. Rev. C 88, 025801 (2013)
Osenda O., Tamarit F.A., Cannas S.A.: Nonequilibrium structures and slow dynamics in a two-dimensional spin system with competing long-range and short-range interactions. Phys. Rev. E 80, 021114 (2009)
Pighin S.A., Cannas S.A.: Phase diagram of an Ising model for ultrathin magnetic films: comparing mean field and Monte Carlo predictions. Phys. Rev. B 75, 224433 (2007)
Portmann O., Golzer A., Saratz N., Billoni O.V., Pescia D., Vindigni A.: Scaling hypothesis for modulated systems. Phys. Rev. B 82, 184409 (2010)
Rastelli E., Regina S., Tassi A.: Phase transitions in a square Ising model with exchange and dipole interactions. Phys. Rev. B 73, 144418 (2006)
Seul M., Wolf R.: Evolution of disorder in two-dimensional stripe patterns: “Smectic” instabilities and disclination unbinding. Phys. Rev. Lett. 68, 2460 (1992)
Spivak B., Kivelson S.A.: Phases intermediate between a two-dimensional electron liquid and Wigner crystal. Phys. Rev. B 70, 155114 (2004)
Stoycheva A.D., Singer S.J.: Stripe melting in a two-dimensional system with competing interactions. Phys. Rev. Lett. 84, 4657 (2000)
Süto A.: Crystalline ground states for classical particles. Phys. Rev. Lett. 95, 265501 (2005)
Theil F.: A Proof of crystallization in two dimensions. Commun. Math. Phys. 262, 209 (2006)
Vindigni A., Saratz N., Portmann O., Pescia D., Politi P.: Stripe width and nonlocal domain walls in the two-dimensional dipolar frustrated Ising ferromagnet. Phys. Rev. B 77, 092414 (2008)
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Communicated by L. Erdös
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Giuliani, A., Seiringer, R. Periodic Striped Ground States in Ising Models with Competing Interactions. Commun. Math. Phys. 347, 983–1007 (2016). https://doi.org/10.1007/s00220-016-2665-0
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DOI: https://doi.org/10.1007/s00220-016-2665-0