• DOI:10.1007/s00220-014-1923-2
• Corpus ID: 73651908

Formation of Stripes and Slabs Near the Ferromagnetic Transition

@article{Giuliani2013FormationOS,
  title={Formation of Stripes and Slabs Near the Ferromagnetic Transition},
  author={Alessandro Giuliani and Elliott H. Lieb and Robert Seiringer},
  journal={Communications in Mathematical Physics},
  year={2013},
  volume={331},
  pages={333-350}
}

• A. Giuliani, E. Lieb, R. Seiringer
• Published 23 April 2013
• Physics
• Communications in Mathematical Physics

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 Jc, then the ground state is homogeneous. It has been conjectured that when J is smaller than but close to Jc, the ground state is periodic and striped, with stripes of constant width h = h(J), and h → ∞ as $${J\to J_c^-}$$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 e0(J)/eS(J) tends to 1 as $${J\to J_c^-}$$J→Jc- , with eS(J) being the energy per site of the optimal periodic striped/slabbed state and e0(J) the actual ground state energy per site of the system. Our proof comes with explicit bounds on the difference e0(J)−eS(J) at small but positive Jc−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.