Micro-to-macro variational analysis of lattice energies for classical spin systems in a non-flat environment: emergence and interaction of defects

• Applicant: Professor Dr. Marco Cicalese
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 529250402

The main goal of this project is the variational analysis of energy-driven physical systems characterised by the emergence and the interaction of defects. We will focus on some classical spin model in dimension two. We will mainly distinguish between two classes of spin systems depending on whether the spin field maps the lattice points to a discrete or a continuous set. Prototypical examples of systems of the first type are Ising or Potts systems, while examples of systems belonging to the second class are XY spins or nematic liquid crystals. The two classes of spin systems are usually characterised by the emergence of defected structures of different codimension. In fact, while Ising and Potts systems form defects of codimension 1 (here named geometric singularities), in XY systems defects of codimension 2 (here named topological singularities) may arise. In more complex systems both kind of singularities coexist. Prototypical examples of such systems are liquid or plastic crystals. In the case of liquid crystals such singularities represent vortices, disclinations and string defects. For plastic crystals they represent dislocations, partial dislocations and stacking faults.In this project we are going to focus on the mathematical analysis of those more complex lattice models introduced above. We refer to such models as to MCS (multiple codimension singularities) models. The guiding example we have in mind is the so-called generalised XY -model. Here the spin field can form both codimension 2 defects (vortices) and codimension 1 defects (strings) whose behaviour and possible interaction characterize the ground states of the system. The analysis of the emergence of defects and of their interactions requires the understanding of the energy landscape at several scales. This leads to a complicated picture that can often be simplified by a coarse-graining procedure at various energy scales. From a rigorous variational point of view the coarse-graining is addressed by computing the Gamma-limits of (possibly rescaled versions of) the energy as the lattice spacing vanishes. For the generalised XY -model such a Gamma-convergence result has been recently obtained in the flat setting and it has raised many questions, some of which will be considered in this proposal. The first question we plan to address is the analysis of MCS models on compact manifolds. We expect that the topology of the manifold affects the emergence and the interaction of the singularities. This analysis requires the extension (or a better understanding) of several mathematical tools exploited in the flat space to the curved setting. The second question concerns the analysis of MCS models in composite materials, where the coarse-graining is expected to be coupled with a homogenisation process.

• DFG Programme: Research Grants