{"id":"ab384c9b-8d5a-40c2-9b94-4922fa79e182","arxiv_id":"2412.14945","paper_version":3,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The first excited state of the asymmetric Hubbard dimer is not non-interacting v-representable, but becomes complex-v-representable via analytic continuation; state-specific Kohn-Sham calculations with approximate functionals can yield spurious stationary solutions.","lead":"This paper analyzes whether excited states of a two-site Hubbard model can be described with a non-interacting Kohn-Sham system. It finds that the first excited state density requires a complex-valued potential in the non-interacting limit, and that approximate functionals create spurious stationary solutions.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"At Δv=±2ti (t=1/2) the noninteracting Hamiltonian is nilpotent and its unique eigenvector is self-orthogonal, so the complex-v-representability claim rests on a singular 0/0 limit rather than a genuine KS state.","rationale":"Reader's weakest assumption points to the unexplained status of complex-v-representability. The present check locates a concrete mechanism: the λ=0 endpoint is an exceptional point where the c-product energy and density are singular. This does not overturn the paper's main conclusion (absence of real non-interacting v-representability for the first excited state), but it does mean the advertised positive result—that a complex external potential generates the density in the noninteracting limit—is not a standard representability statement and needs a limiting construction. Eq. (58)'s double roots at ±i are the algebraic fingerprint of this defect. The conditional verdict is therefore unchanged, with the condition being that the authors must either define the endpoint via a precise generalized-eigenvector limit or soften the complex-v-representability claim.","tokens_in":19444,"tokens_out":21111,"duration_ms":150360,"concrete_test":"Set t=1/2 and λ=0 in Eq. (55), take Δv=i, and compute H0^3; verify the triple-degenerate eigenvalue 0 and the self-orthogonal eigenvector. Then evaluate the c-product density ⟨D⟩_c = ⟨Ψ|diag(-1,0,1)|Ψ⟩_c / ⟨Ψ|Ψ⟩_c for that eigenvector and for the λ→0+ limit of the branch eigenvector at fixed ρ=1/4. If the endpoint ratio is 0/0 and the limit is obtained only through the singular generalized-eigenvector sector, the claim of complex-v-representability requires revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section IVB's positive claim that the first-excited-state density is non-interacting complex-v-representable (Eq. 60b: Δv0^(1±)=±2ti) does not survive a check of the λ=0 endpoint. With t=1/2, Δv=±i. The noninteracting matrix from Eq. (55) at λ=0 is then nilpotent: direct computation gives H0^3=0, so all three eigenvalues are 0 and there is a single eigenvector (1,-√2 i,-1), whose c-product norm vanishes (1+(-√2 i)^2+1=0). Both the c-product energy (Eq. 57) and the density expectation are therefore 0/0 at the endpoint; the Hamiltonian is a defective exceptional point. Equations (60b)-(62b) can only be obtained as limits along the λ-branch, not by solving an eigenvalue problem at λ=0. This is consistent with Eq. (58), which after substituting u=Δv/(2t) factors as (u^2+1)^2[(ρ^2-1)u^2+ρ^2]=0, so ±i are double roots; the paper notes but does not explain these extra solutions. Hence 'complex-v-representability' is not established as a representability statement, because the noninteracting endpoint has no well-defined c-product density. The robust negative result (no real non-interacting v-representability) is unaffected, but the complex-potential claim needs qualification.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":19783,"tokens_out":2633,"duration_ms":24459,"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":[{"comment":"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).","section":"Sec. IVB, Eq. (58)"},{"comment":"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.","section":"Sec. IVB, Eqs. (60b), (62b)"},{"comment":"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.","section":"Sec. IVB, paragraph introducing analytic continuation"}],"minor_comments":[{"comment":"The phrase 'or vise versa' should read 'or vice versa'.","section":"Sec. V, introduction"},{"comment":"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.","section":"Sec. IVB, discussion after Eq. (62)"},{"comment":"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.","section":"Sec. IVB, Eq. (58)"},{"comment":"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.","section":"Sec. V, discussion of complex roots"}],"recommendation":"major_revision","confidential_remarks":"The paper's main negative result is sound and the model is well chosen, but the advertised positive claim about complex-v-representability at λ=0 is currently presented too strongly and lacks a direct derivation of Eq. (58). The issues are fixable within the scope of the manuscript: reframe Eqs. (60b)–(62b) as analytic-continuation limits, provide the missing derivation, and discuss the exceptional-point nature of the endpoint. I would also encourage the authors to clarify the status of the two extra purely imaginary roots; even a negative statement about their significance would be more satisfactory than leaving them unexplained."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper earns its keep with the negative result: the first excited state of the asymmetric Hubbard dimer at half-filling is not non-interacting v-representable. That is shown cleanly from the exact 3x3 Hamiltonian via a density-fixed adiabatic connection, and it holds for all densities and U. The same machinery also shows the doubly-excited state is fine, and the comparison of exact state-specific functionals gives a useful practical warning about spurious stationary solutions when ground-state functionals are reused for excited-state OO-DFT. The derivation is analytic and internally consistent, and the central result does not depend on the authors' earlier exact-functionals paper, even though that paper is rightly cited as the starting point.\n\nThe soft spot is the positive complex-v-representability claim. At lambda=0 with t=1/2 and Delta v = ±i, the noninteracting Hamiltonian is nilpotent, its only eigenvector is self-orthogonal under the c-product, and both the c-product energy and the density expectation become 0/0. So Eq. (60b) describes a limit along the analytically continued branch, not a well-defined noninteracting KS state at the endpoint. The paper's own Eq. (58) shows the ±i roots are double roots, and the two extra purely imaginary solutions are left unexplained. That does not break the no-real-v-representability result, but it means 'complex-v-representable' is oversold. What is established is an analytic continuation of the adiabatic connection, not representability by a genuine noninteracting state in the Hilbert space.\n\nOther issues are minor by comparison: Eq. (58) is stated without derivation, the assignment of the two complex branches to the convex/concave branches is arbitrary, and the spurious-solution analysis in Sec. V is graphical rather than exhaustive. The citation pattern is fine; reusing Ref. 52 is legitimate here.\n\nThis paper is for the DFT-foundations subfield and for practitioners doing orbital-optimized DFT on model systems. I would send it to referees, but with the expectation of a major revision: qualify the complex-v-representability language, confront the nilpotent endpoint head-on, and either justify the limit interpretation or soften the claim.","headline":"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.","tokens_in":20270,"tokens_out":4544,"would_cite":true,"duration_ms":38232,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["31.15.E-","71.10.Fd"],"model":"deepseek-v4-flash","headline":"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.","keywords":["excited-state density functional theory","Kohn-Sham formalism","non-interacting v-representability","Hubbard dimer","adiabatic connection","analytic continuation","complex external potential","state-specific correlation functionals"],"falsifier":"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.","tokens_in":19267,"feed_emoji":"⚛️","tokens_out":11190,"duration_ms":76064,"temperature":0.7,"pith_summary":"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.","feed_headline":"Excited state needs an imaginary Kohn-Sham potential","feed_subtitle":"In the Hubbard dimer, the first excited state's density is unreachable by any real potential, only by Δv = ±2ti.","key_machinery":"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.","core_discovery":"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 $U$.","pith_inferences":["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."],"forward_implications":["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 $\\Delta v$."],"supporting_citations":[{"why":"Provides the exact excited-state universal functionals of the asymmetric Hubbard dimer that this work extends to the Kohn–Sham adiabatic connection.","marker":"Ref. 52"},{"why":"Introduces the stationarity principle for excited states that replaces the constrained search in the construction of state-specific functionals.","marker":"10"},{"why":"Lieb's convex formulation underlies the density-fixed adiabatic connection $F_\\lambda^{(m)}(\\rho) = \\operatorname{stat}_v [E_\\lambda^{(m)} - v\\rho]$ used throughout.","marker":"57"},{"why":"Levy's constrained search defines the universal functional and frames the v-representability question the paper tests.","marker":"56"},{"why":"Supplies the c-product and complex-stationary principle used to continue the adiabatic connection into the complex plane.","marker":"91"},{"why":"Establishes the complex adiabatic connection as a hidden non-Hermitian path from ground to excited states, the template for the analytic continuation performed here.","marker":"101"},{"why":"Gives the ground-state Hubbard dimer Kohn–Sham potential and reference results that the paper's non-interacting limit recovers.","marker":"Ref.76"},{"why":"Documents violations of v-representability in real atoms, motivating the study of densities that lack non-interacting v-representability.","marker":"7"}],"fun_headline_variants":["Imaginary potential unlocks excited state's density","Excited state density needs complex potential","Hubbard dimer's excited state defies real potentials","Spurious solutions in state-specific DFT","Complex Kohn-Sham potential for excited states"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Imaginary potential unlocks excited state's density","Excited state density needs complex potential","Hubbard dimer's excited state defies real potentials","Spurious solutions in state-specific DFT","Complex Kohn-Sham potential for excited states"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000499,"raw_usage":{"total_tokens":2488,"prompt_tokens":1032,"completion_tokens":1456,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":648,"completion_tokens_details":{"reasoning_tokens":1387}},"tokens_in":648,"tokens_out":1456,"duration_ms":9392,"temperature":1.0,"reasoning_tokens":1387,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T11:46:21.480189+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the complex adiabatic connection as a hidden non-Hermitian path from ground to excited states, the template for the analytic continuation performed here."}],"review_version":1}