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 $G_0\mapsto G$ is noninjective. This implies that with fermionic statistics the $(0+0)dN$-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.

中文翻译：

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具有复制品的费米和玻色子系统中Luttinger-Ward泛函的多值性

我们在没有时间或空间维度，但具有时间或空间维度的简化Hubbard型模型中研究Luttinger-Ward函数（LWF）的属性 $ñ$位于单个站点上的相同副本。这个简单$（0+0）d$ 模型允许为所有对象提供精确的解决方案 $ñ$同时适用于玻色子和费米子统计。我们表明，铁离子统计量与非相互作用格林函数的多个值直接相关$G_0$ 映射到相互作用的格林函数的相同值 $G$; 也就是说，映射$G_0\mapsto G$是非内射的。这意味着在费米离子统计中，$（0+0）dñ$-replica模型的乘积值为LWF。费米子模型中LWF值的数量与复制品的数量成比例地增加$ñ$，而在bosonic模型中，LWF具有单个值，无论 $ñ$。我们还讨论了$ñ$复制模型和 $（0+\mathrm{1个}）d$ 以前在LWF多值性研究中使用的Hubbard原子。