# Formation of stripes and slabs near the ferromagnetic transition

Giuliani A, Lieb É, Seiringer R. 2014. Formation of stripes and slabs near the ferromagnetic transition. Communications in Mathematical Physics. 331, 333–350.

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Author

Giuliani, Alessandro;
Lieb, Élliott;
Seiringer, Robert

^{ISTA}^{}Department

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 → Jc -. (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 → Jc -, 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.

Publishing Year

Date Published

2014-10-01

Journal Title

Communications in Mathematical Physics

Acknowledgement

2014 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.
The research leading to these results has received funding from the European Research
Council under the European Union’s Seventh Framework Programme ERC Starting Grant CoMBoS (Grant Agreement No. 239694; A.G. and R.S.), the U.S. National Science Foundation (Grant PHY 0965859; E.H.L.), the Simons Foundation (Grant # 230207; E.H.L) and the NSERC (R.S.). The work is part of a project started in collaboration with Joel Lebowitz, whom we thank for many useful discussions and for his constant encouragement.

Volume

331

Page

333 - 350

ISSN

eISSN

IST-REx-ID

### Cite this

Giuliani A, Lieb É, Seiringer R. Formation of stripes and slabs near the ferromagnetic transition.

*Communications in Mathematical Physics*. 2014;331:333-350. doi:10.1007/s00220-014-1923-2Giuliani, A., Lieb, É., & Seiringer, R. (2014). Formation of stripes and slabs near the ferromagnetic transition.

*Communications in Mathematical Physics*. Springer. https://doi.org/10.1007/s00220-014-1923-2Giuliani, Alessandro, Élliott Lieb, and Robert Seiringer. “Formation of Stripes and Slabs near the Ferromagnetic Transition.”

*Communications in Mathematical Physics*. Springer, 2014. https://doi.org/10.1007/s00220-014-1923-2.A. Giuliani, É. Lieb, and R. Seiringer, “Formation of stripes and slabs near the ferromagnetic transition,”

*Communications in Mathematical Physics*, vol. 331. Springer, pp. 333–350, 2014.Giuliani, Alessandro, et al. “Formation of Stripes and Slabs near the Ferromagnetic Transition.”

*Communications in Mathematical Physics*, vol. 331, Springer, 2014, pp. 333–50, doi:10.1007/s00220-014-1923-2.**All files available under the following license(s):**

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arXiv 1304.6344