**Authors**
: Alessandro Giuliani, Joel L. Lebowitz, Elliott H. Lieb

**Title:
Checkerboards, stripes and corner energies in spin models with competing interactions
**

**Abstract**:
We study the zero temperature phase diagram of Ising spin systems in two
dimensions in the presence of competing interactions, long range
antiferromagnetic and nearest neighbor ferromagnetic of
strength J. We first introduce the notion of a "corner energy" which
shows, when the antiferromagnetic interaction decays faster than the
fourth power of the distance, that a striped state is
favored with respect to a checkerboard state when J is close to J_c, the
transition to the ferromagnetic state, i.e., when the length scales of
the uniformly magnetized domains become large. Next,
we perform detailed analytic computations on the energies of the striped
and checkerboard states in the cases of antiferromagnetic interactions
with exponential decay and with power law decay
r^{-p}, p>2, that depend on the Manhattan distance instead of the
Euclidean distance. We prove that the striped phase is always favored
compared to the checkerboard phase when the scale of the
ground state structure is very large. This happens for J\lesssim J_c if
p>3, and for J sufficiently large if 2<p≤ 3. Many of our
considerations
involving rigorous bounds carry over to dimensions
greater than two and to more general short-range ferromagnetic interactions.

**Keywords**:
Striped order, checkerboard states, Ising model, long range competing interactions.

Alessandro Giuliani,

Dipartimento di Matematica,

Universita' di Roma Tre,

L.go S. Leonardo Murialdo 1, 00146 Roma, Italy

giuliani@mat.uniroma3.it

Joel L. Lebowitz,

Department of Mathematics and Physics,

Rutgers University, Rutgers University, Piscataway, NJ 08854 USA

lebowitz@math.rutgers.edu

Elliott H. Lieb,

Department of Mathematics and Physics,

Princeton University, Princeton 08544 NJ, USA

lieb@math.princeton.edu