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Exact downfolding and its perturbative approximation

Jonas B. Profe1,*, Jakša Vučičević2, P. Peter Stavropoulos1, Malte Rösner3,4, Roser Valentí1, and Lennart Klebl5

  • 1Institute for Theoretical Physics, Goethe University Frankfurt, Max-von-Laue-Straße 1, 60438 Frankfurt am Main, Germany
  • 2Scientific Computing Laboratory, Center for the Study of Complex Systems, Institute of Physics Belgrade, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia
  • 3Institute for Molecules and Materials, Radboud University, Nijmegen, The Netherlands
  • 4Faculty of Physics, Bielefeld University, 33501 Bielefeld, Germany
  • 5Institut für Theoretische Physik und Astrophysik and Würzburg-Dresden Cluster of Excellence ct.qmat, Universität Würzburg, 97074 Würzburg, Germany

  • *Contact author: Profe@itp.uni-frankfurt.de
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Phys. Rev. Research 8, 023091 – Published 29 April, 2026

DOI: https://doi.org/10.1103/npv1-w4gt

Abstract

Solving the many-electron problem, even approximately, is one of the most challenging and simultaneously most important problems in contemporary condensed matter physics with various connections to other fields. The standard approach is to follow a divide and conquer strategy that combines various numerical and analytical techniques. A crucial step in this strategy is the derivation of an effective model for a subset of degrees of freedom by a procedure called downfolding, which often corresponds to integrating out energy scales far away from the Fermi level. In this work, we present a rigorous formulation of this downfolding procedure, which complements the renormalization group picture put forward by Honerkamp [Phys. Rev. B 85, 195129 (2012)]. We derive an exact effective model in an arbitrarily chosen target space (e.g., low-energy degrees of freedom) by explicitly integrating out the rest space (e.g., high-energy degrees of freedom). Within this formalism, we state conditions that justify a perturbative truncation of the downfolded effective interactions to just a few low-order terms. Furthermore, we utilize the exact formalism to formally derive the widely used constrained random phase approximation, uncovering underlying approximations and highlighting relevant corrections in the process. Finally, we detail different contributions in the material examples of fcc nickel and the infinite-layer cuprate SrCuO2. Our results open up a pathway to obtain effective models in a controlled fashion and to judge whether a chosen target space is suitable.

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Phys. Rev. Research 8, 023091– Published 29 April, 2026
  • Received 12 August 2025
  • Accepted 30 March 2026
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