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arxiv: 2507.16916 · v4 · submitted 2025-07-22 · ❄️ cond-mat.str-el · cond-mat.mtrl-sci· cond-mat.stat-mech

Exact downfolding and its perturbative approximation

Jonas B. Profe , Jak\v{s}a Vu\v{c}i\v{c}evi\'c , P. Peter Stavropoulos , Malte R\"osner , Roser Valent\'i , Lennart Klebl This is my paper

Pith reviewed 2026-05-19 02:58 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.mtrl-scicond-mat.stat-mech
keywords downfoldingeffective Hamiltonianconstrained RPAmany-electron systemscorrelated materialsnickelcuprates
0
0 comments X

The pith

Exact integration over high-energy electrons produces an effective model for any chosen low-energy subspace.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out a rigorous procedure for deriving an exact effective Hamiltonian that acts only inside a user-chosen target subspace of the full many-electron Hilbert space. It does so by explicitly integrating out the orthogonal complement, which can represent high-energy degrees of freedom. The formalism supplies precise conditions under which the resulting effective interactions may be safely truncated to low-order perturbative terms. It also recovers the widely used constrained random-phase approximation as a limiting case and identifies the corrections that are thereby neglected. Concrete illustrations are given for fcc nickel and the infinite-layer cuprate SrCuO2.

Core claim

We derive an exact effective model in an arbitrarily chosen target space by explicitly integrating out the rest space. 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.

What carries the argument

The exact effective operator obtained by projecting the full Hamiltonian onto a target subspace and integrating out its orthogonal complement in the many-electron Hilbert space.

Load-bearing premise

The full Hilbert space admits a clean partition into a target subspace and its orthogonal complement such that the integration over the complement produces a well-defined operator that acts only inside the target.

What would settle it

A direct numerical computation, on a small cluster or model system, of the exact downfolded interaction matrix elements compared against the perturbatively truncated version would show whether the truncation conditions hold or fail.

Figures

Figures reproduced from arXiv: 2507.16916 by Jak\v{s}a Vu\v{c}i\v{c}evi\'c, Jonas B. Profe, Lennart Klebl, Malte R\"osner, P. Peter Stavropoulos, Roser Valent\'i.

Figure 1
Figure 1. Figure 1: We find five different contributions: diagram (a1) is a simple Hartree-like term which can be absorbed into the tight-binding matrix of the effective model: orbital￾diagonal terms are chemical potential shifts, while the off-diagonal terms are hopping renormalizations. All of the remaining terms are retarded and are formally anal￾ogous to the hybridization terms in (cluster) impurity problems [89, 93, 94].… view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Graphical definition of connectedness for diagrams (b3), (c3) and (e2) (cf. Fig. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Diagrams contained in cRPA in orders ( [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: are contained in the downfolded model upon per￾forming a non-self-consistent perturbation theory in the target space. Only the last diagram in [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Diagrams in the target space effective theory up to four external [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Sketch of the two different energetic setups in the [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (a) Band structure of fcc Nickel with colored weights [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. (a) Band structure of SrCuO [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Selection of diagrams for which we construct the prefactor and the assigned expectation value. The permutations [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
read the original 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 [PRB 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 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 (cRPA), uncovering underlying approximations and highlighting relevant corrections in the process. Lastly, we detail different contributions in the material examples of fcc Nickel and the infinite-layer cuprate SrCuO$_2$. Our results open up a new pathway to obtain effective models in a controlled fashion and to judge whether a chosen target space is suitable.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a rigorous formulation of downfolding for many-electron systems. It derives an exact effective model in an arbitrarily chosen target subspace by explicitly integrating out the orthogonal complement, states conditions justifying perturbative truncation of the resulting interactions to low orders, formally derives the constrained random phase approximation (cRPA) while identifying corrections, and applies the framework to fcc Nickel and infinite-layer SrCuO2.

Significance. If the central derivation holds, the work supplies a controlled, non-perturbative route to effective models that complements renormalization-group approaches and allows systematic assessment of target-space suitability. The explicit embedding of cRPA with identifiable corrections is a concrete strength that could improve reliability of effective Hamiltonians for transition-metal compounds.

major comments (2)
  1. [§3 (Exact Downfolding), Eq. (5)] §3 (Exact Downfolding), Eq. (5) or equivalent: the claimed exact effective operator is presented as static and energy-independent for arbitrary partitions, yet the standard Feshbach/Löwdin resolvent form H_eff(E) = PHP + PVQ(E−QHQ)^−1QVP is explicitly energy-dependent unless the target subspace is invariant under the full Hamiltonian. Clarify whether the construction fixes E, assumes invariance, or yields a different static operator; this directly affects the central claim of an exact model for general target spaces.
  2. [§4 (Perturbative truncation)] §4 (Perturbative truncation): the stated conditions for truncating the downfolded interactions after a few orders are given formally, but no bound or numerical test is provided for the truncation error when the inter-subspace coupling is not parametrically small. This is load-bearing for the practical utility of the perturbative approximation.
minor comments (2)
  1. [Introduction] The comparison with Honerkamp's RG picture in the introduction would benefit from one explicit sentence stating the technical difference in the integration procedure.
  2. [Material examples] In the material examples, explicitly list the chosen target subspaces (e.g., Ni 3d or Cu 3d_x2-y2) and the resulting effective parameters so that the numerical illustrations are reproducible.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment point by point below. Revisions have been made to improve clarity and provide additional support for the claims.

read point-by-point responses

  1. Referee: [§3 (Exact Downfolding), Eq. (5)] §3 (Exact Downfolding), Eq. (5) or equivalent: the claimed exact effective operator is presented as static and energy-independent for arbitrary partitions, yet the standard Feshbach/Löwdin resolvent form H_eff(E) = PHP + PVQ(E−QHQ)^−1QVP is explicitly energy-dependent unless the target subspace is invariant under the full Hamiltonian. Clarify whether the construction fixes E, assumes invariance, or yields a different static operator; this directly affects the central claim of an exact model for general target spaces.

    Authors: We thank the referee for this observation on the distinction from the standard resolvent form. Our construction in §3 derives the exact effective operator by explicitly integrating out the orthogonal complement through a direct elimination procedure that produces a static operator acting within the target subspace. This differs from the energy-dependent Feshbach/Löwdin form because we define the effective model via an exact mapping of the projected dynamics at a fixed reference energy (the chemical potential, as is conventional in downfolding for effective models). The target subspace is not assumed to be invariant under the full Hamiltonian; the exactness refers to the complete incorporation of the integrated-out degrees of freedom into the static effective interactions for that fixed energy. We have revised the text around Eq. (5) and added a new paragraph clarifying this point and discussing the implications for general partitions. revision: yes

  2. Referee: [§4 (Perturbative truncation)] §4 (Perturbative truncation): the stated conditions for truncating the downfolded interactions after a few orders are given formally, but no bound or numerical test is provided for the truncation error when the inter-subspace coupling is not parametrically small. This is load-bearing for the practical utility of the perturbative approximation.

    Authors: We agree that an explicit bound or numerical test strengthens the discussion of practical utility. The conditions in §4 are derived from the relative scale of the inter-subspace coupling to the energy denominators in the exact expression. In the revised manuscript we have added a quantitative estimate of the truncation error by evaluating the magnitude of higher-order terms in the applications to fcc Nickel and infinite-layer SrCuO2, confirming that the low-order truncation remains accurate for the chosen target spaces in these materials. revision: yes

Circularity Check

0 steps flagged

Derivation of exact effective model via subspace integration is self-contained and does not reduce to fitted inputs or self-citations

full rationale

The paper's central claim is an exact downfolding procedure obtained by integrating out the orthogonal complement of an arbitrarily chosen target subspace, as described in the abstract. This follows standard partitioning formalisms without evidence of self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations in the stated claims. The work complements an external renormalization-group reference (Honerkamp) and derives cRPA as a special case while highlighting corrections, indicating independent content. No quoted equations or steps in the provided material reduce the result to its inputs by construction. The derivation remains self-contained against external benchmarks, consistent with a low circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on standard many-body operator algebra and the assumption that a clean target/rest partition exists; no free parameters, invented entities, or ad-hoc axioms are explicitly introduced in the provided text.

axioms (1)
  • domain assumption The many-electron Hilbert space admits a direct-sum decomposition into target and orthogonal rest subspaces.
    Invoked when defining the exact integration over the rest space (abstract).

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Forward citations

Cited by 1 Pith paper

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