Mapping the landscape of frustration-free models

Frustration-free models are of great interest because they are amenable to specialized techniques and their understanding is more complete among the general quantum spin models. In this talk, I will establish an almost bijective relation between frustration-free families of projections and a subclass of hereditary subalgebras defined by an intrinsic property. This relation sets further synergies between frustration-free models and open projections in double duals, and subsets of pure states spaces. These connections enable a better understanding of the class of frustration-free models. For example, the open projections in the double dual derived from frustration-free models is dense in the norm-topology in the space of generic open projections, thus assuring us that, for many purposes, we can choose to work with frustration-free models without losing generality. Furthermore, the Cuntz semigroup, originally designed to classify the positive elements of C*-algebra, has been proven to also classify the open projections. Given the mentioned connections, we now have a new device to investigate the ground states of quantum spin models.