Exact solution of the many-body problem with a O(n⁶) complexity

2021/11/30 by Thierry Deutsch, Deutsch, Thierry
Computer Science · Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Chemical Physics (physics.chem-ph) #FOS: Physical sciences #Mathematical Physics (math-ph) #Noncommutative and Quantum Gravity Theories #Nuclear physics research studies #Quantum Information and Cryptography #Quantum Mechanics and Non-Hermitian Physics #Quantum Physics (quant-ph) #Strongly Correlated Electrons (cond-mat.str-el)

paper · pdf · doi:10.48550/arxiv.2111.15281

openalex publication_date 2021/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article, we define a new mathematical object, called a pair D=(A,C) of anti-commutation matrices (ACMP) based on the anti-commutation relation a^†_ia_j + a_ja^†_i = δ_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 D⁰=(A⁰,C⁰) and the two-body ones by a set of n ACMPs Dⁱ=(Aⁱ,Cⁱ) where n is the number of states. We show that we can have a compact and exact parametrization with n⁴ 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 O(n⁶) complexity.

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