{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/exact-solution-of-the-many-body-problem-with","title":"Exact solution of the many-body problem with a $\\mathcal{O}\\left(n^6\\right)$ complexity","arxiv_id":"2111.15281","date":"2021-11-30","proceeding":null,"authors":["Thierry Deutsch"],"abstract":"In this article, we define a new mathematical object, called a pair $D=\\left(A,C\\right)$ of anti-commutation matrices (ACMP) based on the anti-commutation relation $a^\\dag_{i}a_{j} + a_{j}a^\\dag_{i} = \\delta_{ij}$ applied to the scalar product between the many-body wavefunctions. This ACMP explicitly separates the different levels of correlation. The one-body correlations are defined by a ACMP $D^0=\\left(A^0,C^0\\right)$ and the two-body ones by a set of $n$ ACMPs $D^i=\\left(A^i,C^i\\right)$ where $n$ is the number of states. We show that we can have a compact and exact parametrization with $n^4$ parameters of the two-body reduced density matrix (\\TRDM) of any pure or mixed $N$-body state to determine the ground state energy with a $\\mathcal{O}\\left(n^6\\right)$ complexity.","url_abs":"https://arxiv.org/abs/2111.15281v3","url_pdf":"https://arxiv.org/pdf/2111.15281v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"exact-solution-of-the-many-body-problem-with","repo_url":"https://gitlab.com/thierry-deutsch/many-body-method-acdm","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}