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.