Differential equation for partition functions and a duality pseudo-forest

Abstract

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 that contains necessary and sufficient information for computing the full quantum partition function. We derive a second-order differential equation for an extended partition function z[β, β′, J] that describes a transformation from a “parent” partition function z[0, β′, J] (or anti-commutativity graph) to a “child” partition function z[β, 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.