Realization of stripes and slabs in two and three dimensions

Abstract

We consider Ising models in two and three dimensions with nearest neighbor ferromagnetic interactions and long range, power law decaying, antiferromagnetic interactions. If the strength of the ferromagnetic coupling J is larger than a critical value J_c, then the ground state is homogeneous and ferromagnetic. As the critical value is approached from smaller values of J, it is believed that the ground state consists of a periodic array of stripes (d=2) or slabs (d=3), all of the same size and alternating magnetization. Here we prove rigorously that the ground state energy per site converges to that of the optimal periodic striped/slabbed state, in the limit that J tends to the ferromagnetic transition point. While this theorem does not prove rigorously that the ground state is precisely striped/slabbed, it does prove that in any suitably large box the ground state is striped/slabbed with high probability.Comment: 4 pages, 1 figur

Periodic Striped Ground States in Ising Models with Competing Interactions

“…In this paper, we choose the exponent p to satisfy the constraint p > 2d. As discussed in a previous work [20,21], if J > J c , with…”

“…Our estimates allowed us to prove emergence of periodic stripe order in a suitable asymptotic sense, but they were not strong enough to fully control the ground state structure, or to prove breaking of rotational symmetry in the ground state. In this paper we extended the ideas of [19][20][21] and prove that periodic striped states of optimal width are exact infinite volume ground states. Moreover, we give a characterization of infinite volume ground states that are invariant under translations by one (for d = 2) or two (for d = 3) independent fixed lattice vectors.…”

“…In a recent work [20,21], we succeeded in computing the specific ground state energy of such a system, with power-law decaying repulsive interactions and decay exponent p > 2d in d = 2, 3 space dimensions, asymptotically as the ferromagnetic transition line is approached. Our estimates allowed us to prove emergence of periodic stripe order in a suitable asymptotic sense, but they were not strong enough to fully control the ground state structure, or to prove breaking of rotational symmetry in the ground state.…”

“…The proof of Theorem 3 is divided into several steps, and uses many notations and ideas introduced in [20,21], which will be recalled here. In reading the proof, it may be useful to keep in mind that the goal is to derive lower bounds for the energy of the "Peierls contours", i.e., of the connected components of the union of the bad tiles, and to show that we gain in energy by erasing the contours, as in the standard Peierls argument.…”

The crystallization conjecture: a review

“…However, it does not always allow to conclude that the system is periodic. For recent examples of application of this theory to crystallization problems, see for instance [110,111,113,114]. …”