Seminarium KMMF "Teoria Dwoistości"

Lattice spin systems model particles whose positions are confined to the vertices of a graph, representing either crystalline or disordered structures. In such systems, only the internal degrees of freedom are relevant for describing the dynamics. These models can be formulated in the framework of quasi-local C*-algebras, obtained as the inductive limit oflocal algebras associated with finite subsets of vertices. Thermodynamic equilibrium states are then described by algebraic states—positive, normalized linear functionals—satisfying the KMS condition at a given temperature. After introducing this setting, I will present two new results establishing the uniqueness of KMS states at sufficiently high temperatures. Both results apply to classical and quantum spin systems and hold independently of the number of local degrees of freedom. The first isbased on an expansion of the spin space in terms of spherical harmonics and their quantized counterparts, while the second relies on a decomposition of observables into components with vanishing partial trace over finite regions of the lattice. If time allows it, Iwill discuss possible extensions to models with infinite number of single site degrees of freedom and problematic aspects of our approach.