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

**Title:
Modulated phases of a 1D sharp interface model in a magnetic field
**

**Abstract**:
We investigate the ground states of 1D continuum models
having short-range ferromagnetic type interactions and a wide class of
competing longer-range antiferromagnetic type interactions.
The model is defined in terms of an energy functional, which can be thought of
as the Hamiltonian of a coarse-grained microscopic system or as a mesoscopic
free energy functional describing various materials.
We prove that the ground state is simple periodic whatever the prescribed total
magnetization might be. Previous studies of this model of frustrated
systems
assumed this simple periodicity but, as in many examples in condensed matter
physics, it is neither obvious nor always true that ground states do not
have a more complicated, or even chaotic structure.

**Keywords**:
Striped order, periodic ground state, Ising model, reflection positivity,
magnetic field, 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