Differential equation for partition functions and a duality pseudo-forest

We consider finite quantum systems defined by a mixed set of commutation and anti-commutation relations between components of the Hamiltonian operator. These relations are represented by an anti-commutativity graph which contains a necessary and sufficient information for computing the full quantum partition function. We derive a second-order differential equation for an extended partition function $z[\beta, \beta', J]$ which describes a transformation from a ``parent'' partition function $z[0, \beta', J]$ (or anti-commutativity graph) to a ``child'' partition function $z[\beta, 0, J]$ (or anti-commutativity graph). The procedure can be iterated and then one forms a pseudo-forest of duality transformations between quantum systems, i.e. a directed graph in which every vertex (or quantum system) has at most one incoming edge (from its parent system). The pseudo-forest has a single tree connected to a constant partition function, many pseudo-trees connected to self-dual systems and all other pseudo-trees connected to closed cycles of transformations between mutually dual systems. We also show how the differential equation for the extended partition function can be used to study disordered systems.
Comment: 18 pages, 7 figures, accepted for publication in the Journal of Mathematical Physics