Multi-valuedness of the Luttinger-Ward functional in the Fermionic and Bosonic System with Replicas

We study the properties of the Luttinger-Ward functional (LWF) in a simplified Hubbard-type model without time or spatial dimensions, but with $N$ identical replicas located on a single site. The simplicity of this $(0+0)d$ model permits an exact solution for all $N$ and for both bosonic and fermionic statistics. We show that fermionic statistics are directly linked to the fact that multiple values of the noninteracting Green function $G_0$ map to the same value of the interacting Green function $G$, i.e. the mapping $G_0 \mapsto G$ is non-injective. This implies that with fermionic statistics the $(0+0)d$ model has a multiply-valued LWF. The number of LWF values in the fermionic model increases proportionally to the number of replicas $N$, while in the bosonic model the LWF has a single value regardless of $N$. We also discuss the formal connection between the $(0+0)d$ model and the $(0+1)d$ model which was used in previous studies of LWF multivaluedness.