This project aims at contributing towards a better mathematical understanding of ordering phenomena in magnetic materials.

In this field, a particular interest is in the so-called frustrated magnets. Those are magnets in which not all interaction energies between neighboring atoms can be minimized simultaneously, or in other words, not all atoms can be made "happy" at the same time. If this happens due to the geometry of the underlying crystalline lattice, then we talk about geometric frustration. To have a picture in mind, we can think about atoms sitting on a crystalline lattice and attached to every atom we can imagine a unit vector (the spin) representing the local magnetization. If the interactions between nearest neighbors are antiferromagnetic, then neighboring spins want to be anti-parallel to each other. In many crystalline structures this cannot be achieved for all nearest neighbors at the same time, as for example on the 2D triangular lattice.

In the case of frustration, the total energy of the system is minimized by finding suitable "compromises", for example by rotating neighboring spins by a fixed angle. As a consequence, the corresponding ground states show unconventional magnetic ordering. Moreover, there are typically several compromises available at the same time, which leads to a variety of corresponding ground states and thus to a particularly rich energy landscape. This rather complicated picture can often be simplified by passing in a suitable way from a discrete to a continuum model, which will be the main focus of this project.

To achieve the passage from discrete to continuum models, we will work within the framework of the calculus of variations and in particular within the framework of what is known as discrete-to­ continuum variational analysis. The latter is a procedure that allows to characterize continuum limits of discrete energies (defined for example on a crystalline lattice) by using the notion of Gamma­ convergence. More in detail, by taking the Gamma-limit of suitable renormalizations of discrete energies when the lattice spacing vanishes, one can characterize the macroscopic behavior of the system at several energy scales.