Abstract
We consider Ising models in d = 2 and d = 3 dimensions with nearest neighbor ferromagnetic and long-range antiferromagnetic interactions, the latter decaying as (distance)−p, p > 2d, at large distances. If the strength J of the ferromagnetic interaction is larger than a critical value J c , then the ground state is homogeneous. It has been conjectured that when J is smaller than but close to J c , the ground state is periodic and striped, with stripes of constant width h = h(J), and h → ∞ as \({J\to J_c^-}\) . (In d = 3 stripes mean slabs, not columns.) Here we rigorously prove that, if we normalize the energy in such a way that the energy of the homogeneous state is zero, then the ratio e 0(J)/e S(J) tends to 1 as \({J\to J_c^-}\) , with e S(J) being the energy per site of the optimal periodic striped/slabbed state and e 0(J) the actual ground state energy per site of the system. Our proof comes with explicit bounds on the difference e 0(J)−e S(J) at small but positive J c −J, and also shows that in this parameter range the ground state is striped/slabbed in a certain sense: namely, if one looks at a randomly chosen window, of suitable size ℓ (very large compared to the optimal stripe size h(J)), one finds a striped/slabbed state with high probability.
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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)
Biskup M., Chayes L., Kivelson S.A.: On the absence of ferromagnetism in typical 2D ferromagnets. Commun. Math. Phys. 274, 217–231 (2007)
Buttà P., Esposito R., Giuliani A., Marra R.: Froth-like minimizers of a non local free energy functional with competing interactions. Commun. Math. Phys. 322, 593–632 (2013)
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. Physica 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)
Edlund E., Nilsson Jacobi M.: Universality of striped morphologies. Phys. Rev. Lett. 105, 137203 (2010)
Frank R.L., Lieb E.H.: Inversion positivity and the sharp Hardy–Littlewood–Sobolev inequality. Calc. Var. Partial Differ. Equ. 39, 85–99 (2010)
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)
Giuliani, A., Lebowitz, J.L., Lieb, E.H.: Pattern formation in systems with competing interactions. In: AIP Conference Proceedings of the 10th Granada Seminar on Computational Physics, 15–19 Sept. 2008 (2008)
Giuliani A., Lebowitz J.L., Lieb E.H.: Periodic minimizers in 1D local mean field theory. Commun. Math. Phys. 286, 163–177 (2009)
Giuliani A., Lebowitz J.L., Lieb E.H.: Modulated phases of a 1D sharp interface model in a magnetic field. Phys. Rev. B 80, 134420 (2009)
Giuliani A., Lebowitz J.L., Lieb E.H.: Checkerboards, stripes and corner energies in spin models with competing interactions. Phys. Rev. B 84, 064205 (2011)
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)
Jamei R., Kivelson S., Spivak B.: Universal aspects of Coulomb-frustrated phase separation. Phys. Rev. Lett. 94, 056805 (2005)
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., De’Bell K.: Striped phases in two-dimensional dipolar ferromagnets. Phys. Rev. B 51, 16033 (1995)
Nielsen E., Bhatt R.N., Huse D.A.: Modulated phases in magnetic models frustrated by long-range interactions. Phys. Rev. B 77, 054432 (2008)
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)
Spivak B.: Phase separation in the two-dimensional electron liquid in MOSFET’s. Phys. Rev. B 67, 125205 (2003)
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. 321, 2071–2115 (2006)
Stoycheva A.D., Singer S.J.: Stripe melting in a two-dimensional system with competing interactions. Phys. Rev. Lett. 84, 4657 (1999)
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., Lieb, E.H. & Seiringer, R. Formation of Stripes and Slabs Near the Ferromagnetic Transition. Commun. Math. Phys. 331, 333–350 (2014). https://doi.org/10.1007/s00220-014-1923-2
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DOI: https://doi.org/10.1007/s00220-014-1923-2