Multivaluedness of the Luttinger-Ward functional in the fermionic and bosonic system with replicas

Abstract

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's function ${G}_{0}$ map to the same value of the interacting Green's function $G$; that is, the mapping ${G}_{0}\ensuremath{\mapsto}G$ is noninjective. This implies that with fermionic statistics the $(0+0)d\phantom{\rule{4pt}{0ex}}N$-replica 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 $N$-replica model and the $(0+1)d$ Hubbard atom which was used in previous studies of LWF's multivaluedness.