Anyons are by definition particles with quantum statistics different from those of bosons and fermions. They can occur only in low dimensions, 2D being the most relevant case for this talk. They have hitherto remained hypothetical, but there is good theoretical evidence that certain quasi-particles occuring in quantum Hall physics should behave as anyons.

I shall consider the case of tracer particles immersed in a so-called Laughlin liquid. I will argue that, under certain circumstances, these become anyons. This is made manifest by the emergence of a particular effective Hamiltonian for their motion. The latter is notoriously hard to solve even in simple cases, and well-controled simplifications are highly desirable. I will discuss a possible mean-field approximation, leading to a one-particle energy functional with self-consistent magnetic field.[-]

We consider the quantum ferromagnetic Heisenberg model in three dimensions, for all spins $S= 1/2$. We rigorously prove the validity of the spin-wave approximation for the excitation spectrum, at the level of the first non-trivial contribution to the free energy at low temperatures. The proof combines a bosonic representation of the model introduced by Holstein and Primakoff with probabilistic estimates, localization bounds and functional inequalities.
Joint work with Michele Correggi and Alessandro Giuliani[-]