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

Research output: Contribution to journalArticle

Aaram J. Kim, Vincent Sacksteder

Original languageEnglish
Article number115146
JournalPhysical Review B
Issue number11
Publication statusPublished - 15 Mar 2020

King's Authors


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 G0 map to the same value of the interacting Green's function G; that is, the mapping G0â†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.

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