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REVIEW 3 major objections 4 minor 107 references

Excited-State-Specific Kohn-Sham Formalism for the Asymmetric Hubbard Dimer

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The first excited state of the asymmetric Hubbard dimer at half-filling has a density that no real local potential can produce in a non-interacting system; only a purely imaginary external potential, $\Delta v = \pm 2ti$, reproduces it.

desk verdict Solid no-go result for excited-state KS on the Hubbard dimer, but the complex-v-representability add-on rests on a singular endpoint and needs qualification. read the letter →

arxiv 2412.14945 v3 pith:4YJMKCTL submitted 2024-12-19 physics.chem-ph cond-mat.mtrl-scicond-mat.str-elnucl-th

Pierre-François Loos , Sara Giarrusso This is my paper · ORCID
classification physics.chem-phcond-mat.mtrl-scicond-mat.str-elnucl-th PACS 31.15.E71.10.Fd
keywords excited-statedensityfunctionaltheoryKohn-Shamformalismnon-interactingv-representabilityHubbarddimeradiabaticconnectionanalyticcontinuationcomplexexternalpotentialstate-specificcorrelationfunctionals
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Kohn–Sham density-functional theory assumes that the density of an interacting system can be reproduced by non-interacting electrons moving in a local potential. This paper asks whether that assumption survives for excited states, using the asymmetric Hubbard dimer at half-filling, where every quantity can be written exactly. It finds that the doubly-excited singlet state fits the Kohn–Sham framework, but the first excited state does not: no real local potential generates its density in the non-interacting limit. The paper then shows that if the external potential is allowed to be complex, a purely imaginary potential $\Delta v = \pm 2ti$ does reproduce the density, and it demonstrates that approximate correlation functionals can create spurious stationary solutions when used in state-specific calculations.

What carries the argument

The load-bearing object is the density-fixed adiabatic connection for state $m$, $F_\lambda^{(m)}(\rho) = \operatorname{stat}_{\Delta v}\,[E_\lambda^{(m)} - \Delta v\,\rho]$, where $E_\lambda^{(m)}$ is an eigenvalue of the dimer Hamiltonian with the interaction scaled by $\lambda$. For the first excited state the stationarity equation has two branches that merge at a critical $\lambda_c$; analytic continuation past this point, using the c-product $\langle f|g\rangle_c = \langle f^*|g\rangle$ and the complex-stationary principle for the energy, turns the two real stationary points into a complex-conjugate pair that reaches $\Delta v = \pm 2ti$ at $\lambda=0$. This mechanism converts a failure of real v-representability into complex v-representability.

What would settle it

Compute the stationarity condition $\partial f_\lambda^{(1)}(\rho,\Delta v)/\partial \Delta v = 0$ for $U=1$, $t=1/2$, and a fixed density such as $\rho=1/4$, and take the limit $\lambda \to 0$. The paper predicts the only optimizers are the purely imaginary $\Delta v = \pm 2ti$ (plus two additional purely imaginary roots); discovering a real $\Delta v$ that satisfies the condition, or any real non-interacting potential whose density equals $1/4$, would falsify the claim that the first excited state lacks non-interacting v-representability.

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Extended reading notes

Core claim

The paper's central claim is that the density of the first excited singlet state of the asymmetric Hubbard dimer at half-filling is not non-interacting v-representable: along a density-fixed adiabatic connection in which the electron–electron interaction is scaled by $\lambda$, the two stationary branches of the first excited state merge at a critical coupling $\lambda_c$ and no real external potential survives to $\lambda=0$. Analytic continuation of the same connection into the complex plane yields, at $\lambda=0$, purely imaginary optimizers $\Delta v = \pm 2ti$, making the density complex-v-representable in the non-interacting limit. For the ground state and the doubly-excited state, real Kohn–Sham potentials $v_s^{(0)}(\rho) = -2t\rho/\sqrt{1-\rho^2}$ and $v_s^{(2)}(\rho) = +2t\rho/\sqrt{1-\rho^2}$ exist for every density. State-specific correlation functionals are computed for each state, and self-consistent Kohn–Sham calculations with these functionals show that using an approximate functional for the wrong state (notably the ground-state functional in a doubly-excited-state calculation) can produce spurious stationary solutions, most severely for small $|\Delta v|$ and large $U$.

Load-bearing premise

The load-bearing premise is that complex-valued external potentials and the associated complex-stationary states are legitimate representations of the real density of the first excited state once no real Kohn–Sham potential exists.

Editorial extensions

If this is right

  • Excited-state Kohn–Sham theory does not automatically inherit ground-state v-representability: the first excited state of the dimer has no real local Kohn–Sham potential, so state-specific calculations for this state must confront non-v-representability directly.
  • Admitting complex-valued external potentials gives the first excited state a non-interacting reference system, suggesting that non-Hermitian or complex formulations can serve as a fallback when real v-representability fails.
  • Correlation functionals are genuinely state-specific in this model, and reusing the ground-state functional in an excited-state calculation can generate spurious stationary solutions, making convergence to the wrong density a practical risk in orbital-optimized DFT.
  • The doubly-excited singlet state is a well-behaved Kohn–Sham state with a real potential opposite in sign to the ground-state potential, so the Kohn–Sham framework extends cleanly to the highest singlet of the dimer.
  • The concave branch of the first-excited-state functional corresponds to a charge-transfer-like configuration and is a poor approximation for ground- or doubly-excited-state calculations, while the convex branch performs well at small $\Delta v$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The signature used here — two real stationary branches of the density-fixed adiabatic connection merging at $\lambda_c > 0$ before reaching $\lambda=0$, with complex-conjugate potentials appearing below $\lambda_c$ — could serve as a diagnostic for non-v-representability in other finite lattice or few-electron systems.
  • If complex-v-representability is accepted as meaningful, excited-state Kohn–Sham systems might generally be non-Hermitian single-particle problems whose real densities come from complex potentials; whether such systems have a foundation beyond this dimer model is an open question.
  • The observed spurious stationary solutions suggest a practical test for orbital-optimized DFT: repeat a state-specific calculation from several initial densities and check whether the self-consistent equation has a unique root, since multiple roots indicate functional mismatch rather than physical multistability.
  • The two additional purely imaginary solutions of Eq. (58), whose meaning the paper leaves unexplained, may correspond to states outside the singlet manifold or to resonant solutions; identifying them could sharpen or delimit the complex-v-representability claim.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This paper extends ground-state Kohn-Sham density-functional theory to the singlet excited states of the asymmetric Hubbard dimer at half-filling, using exact diagonalization of the 3×3 model Hamiltonian. For the doubly-excited state the authors find that a conventional KS description exists via the density-fixed adiabatic connection. For the first excited state they show that the density-potential map is non-invertible and that no real non-interacting potential reproduces the density; they then propose to continue the adiabatic connection into the complex plane, obtaining non-interacting potentials Δv = ±2ti at λ=0 and complex kinetic-energy functionals. The paper also performs state-specific KS calculations with exact and approximate correlation functionals and identifies spurious stationary solutions that arise when the ground-state functional is used for the doubly-excited state.

Significance. If the central claims hold, the paper provides an instructive exactly solvable model in which the first excited-state density fails non-interacting v-representability, and it offers a concrete demonstration of how approximate state-specific functionals produce spurious KS solutions. The negative result on real non-interacting v-representability is derived directly from the model Hamiltonian and is robust. The paper is also valuable for clearly separating the m=0 and m=2 cases, where the adiabatic connection ties smoothly to a KS system, from the m=1 case, where it does not. The practical section illustrates a caveat relevant to orbital-optimized DFT: using a ground-state functional for an excited state can produce multiple stationary solutions. However, the paper's positive claim that the first excited-state density is 'complex-v-representable' at λ=0 requires qualification, because the non-interacting endpoint is singular (see major comments).

major comments (3)
  1. [Sec. IVB, Eq. (58)] Eq. (58) is stated without derivation. The text says it is obtained as the limit of ∂f_λ^(m)/∂Δv = 0 as λ→0, but the actual limit is nontrivial: the stationarity condition may develop additional roots or degenerate behavior. Moreover, for the first excited state, substituting u=Δv/(2t) factors Eq. (58) as (u^2+1)^2[(ρ^2−1)u^2+ρ^2]=0, so ±i are double roots. The paper acknowledges 'two additional solutions that are purely imaginary for all values of ρ' but does not explain them. Please provide the derivation of Eq. (58) and clarify the multiplicity and selection of the branches leading to Eq. (60b).
  2. [Sec. IVB, Eqs. (60b), (62b)] The claim that the first excited-state density is 'complex-v-representable' at the non-interacting endpoint is not established. At λ=0 and Δv=±2ti (with t=1/2), the non-interacting Hamiltonian in Eq. (55) is nilpotent (H_0^3=0), all three eigenvalues are zero, and there is a single eigenvector whose c-product norm vanishes. Consequently the c-product energy in Eq. (57) and the density expectation are 0/0 at the endpoint; this is an exceptional point, not a well-defined KS state. Equations (60b)–(62b) should be presented as limits of the λ-branch analytic continuation rather than as solutions of an eigenvalue problem at λ=0. The robust negative statement (no real non-interacting v-representability) is unaffected, but the positive complex-potential interpretation needs this qualification.
  3. [Sec. IVB, paragraph introducing analytic continuation] The analytic continuation of the adiabatic connection across λ_c is assumed to be unique and physically meaningful, but no criterion is given for selecting the continuation, and the paper notes that the c-product is not a valid metric and can make eigenfunctions self-orthogonal. Please state explicitly which property (e.g., continuity of the branch at λ_c, agreement of the real part with the known real branch, or a resonance-theoretic interpretation) defines the chosen continuation, and show that the resulting λ→0 limit is independent of the path of continuation.
minor comments (4)
  1. [Sec. V, introduction] The phrase 'or vise versa' should read 'or vice versa'.
  2. [Sec. IVB, discussion after Eq. (62)] The assignment of the positive complex branch to the concave branch and the negative complex branch to the convex branch is stated to be arbitrary. Since later sections use this assignment in the KS calculations, please note whether any physically motivated convention is possible or whether the results in Sec. V are independent of the choice.
  3. [Sec. IVB, Eq. (58)] The polynomial in Eq. (58) is of sixth order, but the text introduces it as a way to obtain optimizers; a brief derivation or a reference to supporting material would help readers reproduce the factorization and the root structure.
  4. [Sec. V, discussion of complex roots] In Sec. VC it is stated that complex-valued roots of Eq. (68) 'are not considered here due to their unphysical nature.' Given that the preceding section accepts complex-valued potentials as meaningful, this restriction deserves a sentence of justification or a reference to the convention adopted.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the central v-representability result is derived from exact diagonalization of the model Hamiltonian, and the self-citations are contextual or illustrative rather than definitionally load-bearing.

full rationale

The central derivation is self-contained. The lack of non-interacting v-representability of the first excited state is obtained from the 3x3 Hubbard Hamiltonian in Eq. (55), the stationary principle in Eq. (53), and the stationarity condition Eq. (59); the merging of the two branches at lambda_c is computed, not assumed. The complex continuation is also derived in the paper: the c-product and complex-stationary principle are introduced in Eq. (57), and the non-interacting optimizers Eq. (60b) follow from the explicit polynomial Eq. (58). No fitted parameter is renamed as a prediction, and the conclusion does not reduce to the definition of the optimizer. Self-citations appear (Ref. 52 provides the exact state-specific functionals used in Sec. V, and Ref. 101, with overlapping authorship, is cited for the complex adiabatic-connection idea), but they are not load-bearing for the v-representability claim, which can be checked directly from the model Hamiltonian. The paper also flags its own limitations: it states that a general proof of the stationarity duality 'has not yet been established,' that the two extra purely imaginary solutions of Eq. (58) have 'physical and mathematical significance remains unclear,' and that the c-norm can vanish through self-orthogonality. The lambda=0 endpoint of the complex branch is indeed delicate because the c-norm and density expectation can become 0/0, but this is a mathematical singularity or correctness concern rather than a circular reduction of the paper's derivation to its inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the Hubbard dimer model, the stationarity principle for excited states, the complex-stationary principle, and the analytic continuation of the adiabatic connection. No free parameters are fitted; all results follow from exact diagonalization. The only hand-chosen convention is the branch assignment for the two complex branches of the first excited state.

assumptions (4)
  • domain assumption Görling's stationarity principle: excited-state universal functionals are obtained as stationary points of <Ψ|T+W|Ψ> under density constraint.
    Used in Sec. IIB to define F^(m)[ρ] and the KS decomposition; relies on Refs 10, 47-50.
  • domain assumption Complex-stationary principle with c-product inner product for non-Hermitian (complex-symmetric) Hamiltonians.
    Used in Sec. IVB to analytically continue the adiabatic connection for complex Δv, citing Refs 91-96.
  • ad hoc to paper The analytic continuation of the adiabatic path across λc is unique and physically meaningful.
    The paper continues the stationary solutions into the complex plane beyond the merging point; no theorem guarantees the continuation corresponds to a physical density or that the branch choices are correct.
  • domain assumption Hubbard dimer at half-filling with singlet states is a representative model for studying v-representability in DFT.
    Used throughout; the paper notes applicability to atoms with partially filled p-shells in conclusion, but extrapolation is not proven.

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Pith. "Pith review of Excited-State-Specific Kohn-Sham Formalism for the Asymmetric Hubbard Dimer." pith.science (2026). https://pith.science/paper/4YJMKCTL

@misc{pith2026241214945,
  author       = {Pith},
  title        = {Pith review of: Excited-State-Specific Kohn-Sham Formalism for the Asymmetric Hubbard Dimer},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4YJMKCTL}},
  note         = {Machine review of arXiv:2412.14945}
}
abstract

Building on our recent study [https://doi.org/10.1021/acs.jpclett.3c02052, J. Phys. Chem. Lett. 14, 8780 (2023)], we explore the generalization of the ground-state Kohn-Sham (KS) formalism of density-functional theory (DFT) to the (singlet) excited states of the asymmetric Hubbard dimer at half-filling. While we found that the KS-DFT framework can be straightforwardly generalized to the highest-lying doubly-excited state, the treatment of the first excited state presents significant challenges. Specifically, using a density-fixed adiabatic connection, we show that the density of the first excited state lacks non-interacting $v$-representability. However, by employing an analytic continuation of the adiabatic path, we demonstrate that the density of the first excited state can be generated by a complex-valued external potential in the non-interacting case. More practically, by performing state-specific KS calculations with exact and approximate correlation functionals -- each state possessing a distinct correlation functional -- we observe that spurious stationary solutions of the KS equations may arise due to the approximate nature of the functional.

Figures

Figures reproduced from arXiv: 2412.14945 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic representation of the asymmetric Hubbard [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Exact functional [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (6 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Analytic continuation of [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 6
Figure 6. Figure 6: depicts the correlation functional (top) and cor￾relation potential (bottom) for each singlet state of the asymmetric Hubbard dimer. As discussed below, these results reveal notable differences as well as intriguing similarities. The correlation energy functionals are …
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]

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