Differential equation for partition functions and a duality pseudo-forest

TL;DR

This paper introduces a differential equation framework for calculating partition functions of quantum systems with mixed commutation relations, revealing a duality pseudo-forest structure connecting various quantum models.

Contribution

It derives a second-order differential equation for an extended partition function and constructs a duality pseudo-forest illustrating relationships between quantum systems.

Findings

The differential equation enables systematic computation of partition functions.

The pseudo-forest structure maps dualities and self-dualities among quantum systems.

Application to disordered systems demonstrates the method's versatility.

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 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.