Automatic routing of Goldstone diagrams using genetic algorithms

Abstract

This paper presents an algorithm for an automatic transformation (=routing) of time ordered topologies of Goldstone diagrams (i.e. Wick contractions) into graphical representations of these topologies. Since there is no hard criterion for an optimal routing, the proposed algorithm minimizes an empirically chosen cost function over a set of parameters. Some of the latter are naturally of discrete type (e.g. interchange of particle/hole lines due to antisymmetry) while others (e.g. x,y-position of nodes) are naturally continuous. In order to arrive at a manageable optimization problem the position space is artificially discretized. In terms of the (i) cost function, (ii) the discrete vertex placement, (iii) the interchange of particle/hole lines the routing problem is now well defined and fully discrete. However, it shows an exponential complexity with the number of vertices suggesting to apply a genetic algorithm for its solution. The presented algorithm is capable of routing non trivial (several loops and crossings) Goldstone diagrams. The resulting diagrams are qualitatively fully equivalent to manually routed ones. The proposed algorithm is successfully applied to several Coupled Cluster approaches and a perturbative (fixpoint iterative) CCSD expansion with repeated diagram substitution.