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Frustrated variational lattice problems

Course Type: 
PhD Course
Academic Year: 
January - September
30 h

We consider Ising systems on a lattice involving antiferromagnetic interactions; that is, systems defined on spin functions (taking the values -1 or 1) with an energy favouring spins with alternating signs. Depending on the lattice geometry, ground states with alternating sign may (e.g., on the square lattice) or may not (e.g., on the triangular lattice) exist. In the latter case we say that the system is frustrated. We face the problem of the overall behaviour of a system involving antiferromagnetic and possibly ferromagnetic interactions (that is, favouring uniform states) through a discrete-to-continuum approach, defining (if possible) an approximating continuum energy with interfaces. We show that in general such a description may not exist, give conditions under which it does, and study a number of model cases leading to open problems. We mainly study periodic systems (that is, with a periodic arrangement of interactions), hinting at some random cases.

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