**Author**
: A. Giuliani, E.H. Lieb, J.L. Lebowitz

**Title: Ising models with long--range dipolar
and short range
ferromagnetic interactions
**

**Abstract**: We study the ground state of
a d--dimensional Ising model with both long range (dipole--like) and
nearest neighbor ferromagnetic (FM) interactions. The long range
interaction is equal to $r^{-p}$, $p>d$, while the FM interaction has
strength $J$. If $p>d+1$ and $J$ is large enough the ground state is
FM, while if $d< p\le d+1$ the FM state is not the ground state for any
choice of $J$. In $d=1$ we show that for any $p>1$ the ground state
has a series of transitions from an antiferromagnetic state of period
2 to $2h$--periodic states of blocks of sizes $h$ with alternating
sign, the size $h$ growing when the FM interaction strength $J$ is
increased (a generalization of this result to the case $0< p\le 1$ is
also discussed). In $d\ge 2$ we prove, for $d< p \le d+1$, that the
dominant asymptotic behavior of the ground state energy agrees for
large $J$ with that obtained from a periodic striped state conjectured
to be the true ground state. The geometry of contours in the ground
state is discussed.

**Keywords**:
ising model; long range interactions; periodic ground states.

Alessandro Giuliani,

Department of Physics,
Princeton University, Princeton 08544 NJ, USA

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