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

**Title: Striped phases in two dimensional dipole systems
**

**Abstract**:
We prove that a system of discrete 2D in-plane dipoles with four possible orientations, interacting via a 3D dipole-dipole interaction
plus a nearest neighbor ferromagnetic term, has periodic striped ground states. As the strength of the ferromagnetic term is increased,
the size of the stripes in the ground state increases, becoming infinite, i.e., giving a ferromagentic ground state, when the
ferromagentic interaction exceeds a certain critical value. We also give a rigorous proof of the reorientation transition in the ground
state of a 2D system of discrete dipoles with six possible orientations, interacting via a 3D dipole-dipole interaction plus a nearest
neighbor antiferromagnetic term. As the strength of the antiferromagnetic term is increased the ground state flips from being striped and
in-plane to being staggered and out-of-plane. An example of a rotator model with a sinusoidal ground state is also discussed.

**Keywords**:
Discrete dipole models; striped ground states; reflection positivity.

Alessandro Giuliani,

Dipartimento di Matematica,
Universita' di Roma Tre,
L.go S. Leonardo Murialdo 1, 00146 Roma, Italy

giuliani@princeton.edu

Joel L. Lebowitz,

Department of Mathematics and Physics, 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@princeton.edu