New Methods and Results for the Topological Susceptibility

There are important models of quantum field theory, where the
 configurations are divided into topological sectors. In these
 models the topological susceptibility is a prominent, fully
 non-perturbative observable.
 We first discuss its definition, and its meaning in QCD and
 in axion physics. Then we address the difficulty in its numerical
 measurement. In this regard, we describe a new method - the
 "slab method" - which is applicable even when the Monte Carlo
 history is confined to a single topological sector. We present
 results for the quantum rotor, the 2d Heisenberg model and
 2-flavor QCD. In the latter case, a modern smoothing procedure
is involved, the Gradient Flow.
 In the second part we focus on the Heisenberg model and the
 millennium question whether or not its topological susceptibility
 scales to a finite continuum limit. According to the paradigm
 of the late 20th century this is not the case, which implies that
 this famous model suffers from a conceptual disease. We have
 revisited this issue by involving for the first time the
 Gradient Flow, along with a powerful cluster algorithm.