{"id":"ecbe0bac-93c4-4f36-81ae-0e982139a14f","arxiv_id":"2607.27156","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"Iterative ghost-Gutzwiller embedding fails for symmetry-invariant Mott insulators under Zeeman field, while direct minimization of the variational functional yields a stable partially polarized paramagnet.","lead":"Two formally equivalent ways of solving the ghost-Gutzwiller method disagree badly in the paramagnetic Mott phase: iterative embedding produces a spurious fully polarized insulator in a magnetic field, while direct energy minimization does not. The result tells practitioners when the cheap DMFT-like loop can be trusted and when a slower variational search is required.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified that undermines the central comparative claim.","rationale":"The strongest claim is a concrete methodological comparison inside ghost-GA, supported by explicit energy and magnetization curves. The iterative scheme’s failure under field is demonstrated variationally: a feasible restricted ansatz already beats it. That fact renders ansatz incompleteness non-decisive for the headline result. AF-phase agreement with DMFT supplies an independent sanity check that the embedding loop is correctly implemented when symmetry breaking is allowed. No internal inconsistency, circularity, or unsupported leap is present. The reader’s ACCEPT / HIGH / low-correctness-risk assessment is therefore unchanged; the residual caveat on Eqs. 18–20 is worth a follow-up calculation but does not move the verdict.","tokens_in":14136,"tokens_out":553,"duration_ms":38068,"concrete_test":"Re-optimize the ghost-GA functional (12) at U=12, h=0.01 with a less constrained impurity wavefunction that relaxes the natural-basis and unpolarized-Hubbard-band conditions of Eqs. 18–20 (e.g., free complex coefficients in the half-filled Sz=0 sector with one decoupled bath orbital). If the energy falls below the reported non-iterative curve while magnetization remains m<1, the quantitative solution improves but the iterative failure is confirmed; only if the optimizer collapses to m=1 with energy equal to the iterative/GA value would the central claim be threatened.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader’s weakest assumption (completeness of the five-parameter Mott ansatz, Eqs. 18–20) is a real caveat on the quantitative accuracy of the partially polarized solution, but it is not load-bearing for the paper’s central claim. Under finite Zeeman field the restricted direct-minimization wavefunction already yields a strictly lower variational energy than the iterative embedding solution (Fig. 3, left), while remaining only partially polarized. Because that ansatz is a legitimate (constrained) point inside the ghost-GA manifold, the iterative result is variationally worse than a feasible trial state and therefore cannot be the minimum of the original functional (12). Any incompleteness of Eqs. 18–20 can only lower the direct-min energy further; it cannot rehabilitate the discontinuous, fully-polarized iterative solution. The h=0 agreement between the two methods (Fig. 2) further corroborates that the ansatz captures the relevant Mott manifold when the degeneracy is intact. Thus the comparative conclusion—iterative embedding is fragile and produces artifacts in the symmetry-invariant Mott phase, while direct minimization does not—stands independently of residual ansatz restrictions.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript compares two formally equivalent routes to the ghost-Gutzwiller (ghost-GA) saddle point for the half-filled single-band Hubbard model: an iterative quantum-embedding loop (analogous to DMFT) and direct minimization of the variational energy functional. In the paramagnetic sector the iterative scheme requires ad-hoc degeneracy resolution in the Mott phase and, under a small Zeeman field, collapses to a discontinuous energy and a spurious fully polarized insulator identical to ordinary GA. Direct minimization with a constrained five-parameter impurity ansatz yields a continuous energy, a only partially polarized Mott state, and a finite spin susceptibility across the transition. When antiferromagnetic order is allowed, the iterative scheme converges cleanly and agrees closely with DMFT (and with HF/GA at strong coupling). The authors conclude that iterative embedding is reliable once symmetry breaking is permitted, but that direct minimization is required for symmetry-invariant Mott states.","tokens_in":14424,"tokens_out":1105,"duration_ms":28556,"significance":"The work cleanly isolates a practical failure mode of iterative embedding that is already known in zero-temperature DMFT and shows that the same pathology appears inside ghost-GA, while also demonstrating a concrete variational fix. The comparison is computational and falsifiable: energies, magnetization, Z and susceptibility are reported side-by-side for iterative embedding, direct minimization, standard GA, HF and DMFT (AF data from Amaricci). The result is useful for anyone using ghost-GA (or related embedding methods) to study paramagnetic Mott insulators, spin liquids or other symmetry-enforced states, and it delineates when the cheaper iterative loop can be trusted. The variational inequality under finite field—direct minimization already lower than the iterative solution—is a particularly strong point.","major_comments":[{"comment":"Sec. III, Eqs. (18)–(20): the direct-minimization results rest on a five-parameter constrained Lanczos ansatz built from a singly-occupied decoupled bath at half-filling and Sz=0. While the h=0 agreement with the embedding solution (Fig. 2) and the strict variational superiority under h=0.01 (Fig. 3, left) already prove that the iterative fully-polarized state is not the ghost-GA minimum, the manuscript should state more explicitly that the ansatz is a restricted submanifold and that any incompleteness can only lower the direct-min energy further. A short remark on whether an unconstrained optimization of the full impurity coefficients in (5) was attempted (or is feasible) would strengthen the claim that the partially polarized solution is the true ghost-GA minimum rather than an artifact of the parametrization.","section":"Sec. III, Eqs. (18)–(20)"},{"comment":"Sec. II, Lagrangian (13) and the surrounding discussion of representability: the argument that the Mott minimum can lie on the boundary of the admissible region for R and Delta is central to why the embedding scheme fails. The paper would benefit from a sharper statement of which matrix elements of R vanish at Uc and how that maps onto the loss of interior critical points of L. A brief diagnostic (e.g., monitoring the smallest singular value of R across the transition) would make the boundary-minimum claim more concrete and transferable to multi-orbital or crystal-field cases mentioned in the text.","section":"Sec. II, Eq. (13)"}],"minor_comments":[{"comment":"Abstract, first sentence: typographical error “long-standing ing challenge” should be “long-standing challenge”.","section":"Abstract"},{"comment":"Fig. 3 right panel: the susceptibility is assembled from embedding (metal) and direct minimization (Mott). Please state explicitly in the caption how the metallic chi is extracted and whether the two branches join continuously at Uc within numerical resolution.","section":"Fig. 3"},{"comment":"Sec. IV: the AF direct-minimization ansatz (24) is less accurate than iterative embedding (Fig. 5). A one-sentence remark on why the constrained form is still useful (fixed-m scans, mutual information in Fig. 6) would help the reader.","section":"Sec. IV"},{"comment":"Notation: N_ghosts is fixed to 2 throughout; a brief comment on whether the iterative failure under Zeeman field persists for larger bath sizes would be welcome, even if only as a qualitative expectation.","section":"Sec. III"},{"comment":"References: the unifying ghost-GA/DMFT work cited as arXiv:2603.20559 is central to the embedding formulation; ensure the citation is updated if a journal version appears before publication.","section":"References"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is methodologically solid and the central comparative claim is well supported. The two major comments are clarifications rather than blockers; either could be handled in revision without new calculations. Fit for a specialized condensed-matter journal is good. No concerns about novelty disclosure or citation pattern."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The useful takeaway is practical. For the half-filled Hubbard model, the iterative embedding form of ghost-GA and a direct minimization of the same variational functional agree in the metal and in the AF insulator, and both track DMFT in the AF case. They diverge once you stay paramagnetic through the Mott transition and turn on a small Zeeman field: the iterative loop jumps onto a fully polarized band insulator with a discontinuous energy, while direct min stays partially polarized, continuous, and lower in energy, with finite spin susceptibility.\n\nThat comparison is new and cleanly shown. The failure mode is not hand-waved; Figs. 2–3 and the AF checks against Amaricci’s DMFT data make the point. The paper also explains why the embedding reformulation can land on the boundary of the representable region when a bath decouples, so the formal equivalence of the two routes is not automatic in the Mott phase. For people who actually run ghost-GA (or who care about zero-T iterative embedding more generally), this is usable guidance: trust the loop when symmetry breaking is allowed; do not trust it for a forced paramagnetic Mott state under perturbations that split the atomic degeneracy.\n\nSoft spots are real but secondary. N_ghosts is fixed at 2, and the direct-min Mott wavefunction is a five-parameter constrained ansatz. That limits how quantitative the partially polarized solution is. It does not rescue the iterative result: under field the restricted ansatz already beats the iterative energy, so the loop cannot be sitting at the minimum of the original functional. No code is shipped, which is a minor annoyance for a methods paper. Citations are appropriate; self-cites are to the formalism they are stress-testing.\n\nThis is for the embedding / Mott-methods crowd, not a broad condensed-matter audience. It deserves a serious referee. I would engage with it if I were writing or reviewing ghost-GA or related zero-T embedding work.","headline":"Clear operational result: iterative ghost-GA embedding fails under Zeeman field in the paramagnetic Mott phase while direct minimization of the same functional does not.","tokens_in":15011,"tokens_out":492,"would_cite":true,"duration_ms":10477,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["71.27.+a","71.30.+h","75.10.Lp"],"model":"grok-4.5","headline":"Iterative embedding and direct minimization of ghost-Gutzwiller, though formally equivalent, disagree on paramagnetic Mott insulators under a magnetic field.","keywords":["ghost-Gutzwiller","Mott insulator","quantum embedding","Hubbard model","paramagnetism","antiferromagnetism","variational methods","spin susceptibility"],"falsifier":"Unrestricted direct minimization of the full ghost-Gutzwiller functional in a finite Zeeman field: if that calculation recovers the iterative scheme's energy jump and fully polarized insulator, the claim that direct minimization is required for a physical paramagnetic Mott solution fails.","tokens_in":14977,"feed_emoji":"🧲","tokens_out":894,"duration_ms":37621,"temperature":0.7,"pith_summary":"Quantum embedding methods have long struggled to describe a Mott insulator that keeps full spin symmetry, because iterative solvers either break that symmetry or need hand-built fixes. This paper compares two ways to solve the ghost-Gutzwiller approximation for the single-band Hubbard model: a cheap iterative embedding loop, and direct minimization of the variational energy. In the paramagnetic Mott phase the iterative route needs ad-hoc recipes; in a Zeeman field those recipes fail, producing an energy jump and a spurious fully polarized insulator. Direct minimization avoids the artifacts, giving a smooth, only partly polarized paramagnetic solution with finite spin susceptibility. When antiferromagnetic order is allowed, the iterative scheme works and matches dynamical mean-field theory. The practical message is when the fast loop can be trusted and when the harder minimization is required.","feed_headline":"Iterative ghost-Gutzwiller fails for paramagnetic Motts","feed_subtitle":"In a magnetic field it jumps to a fake fully polarized insulator; direct minimization does not.","key_machinery":"The ghost-Gutzwiller variational energy functional, solved either by iterative embedding (self-consistent impurity ground state, analogous to dynamical mean-field theory) or by direct minimization of a constrained impurity wavefunction that keeps a decoupled Fermi-level bath and enforces half-filling and zero total Sz.","core_discovery":"Across the Mott transition of the single-band Hubbard model, the iterative embedding scheme and direct minimization of the ghost-Gutzwiller functional behave very differently despite formal equivalence. The iterative scheme is efficient but fragile in the symmetry-invariant Mott phase: ad-hoc degeneracy recipes fail under a Zeeman field, yielding a discontinuous energy and a spurious fully polarized insulator. Direct minimization stabilizes a genuinely paramagnetic, only partially polarized Mott solution with continuous energy and finite zero-field spin susceptibility. When antiferromagnetism is allowed, the iterative scheme recovers the correct solution and aligns with dynamical mean-field","pith_inferences":["The same iterative fragility likely appears in other zero-temperature embedding methods whenever fields or multiplet structure split the atomic manifold.","Constrained impurity ansatzes that stay inside the representable density-matrix domain may become a practical fix whenever Lagrange-multiplier embedding hits a boundary minimum.","Direct variational control of the impurity wavefunction is a natural route to forced spin-charge fractionalization and multi-orbital spin-liquid trial states."],"forward_implications":["Iterative embedding can be trusted when symmetry breaking such as antiferromagnetism is allowed.","Direct minimization is required for symmetry-invariant Mott states and under perturbations that split atomic degeneracy.","Ghost-Gutzwiller with direct minimization yields finite zero-field spin susceptibility continuously across the Mott transition, unlike standard Gutzwiller.","The direct route lets one force fixed order parameters and explore metastable paramagnetic or spin-liquid-like trial states without artificial spin symmetrization."],"fun_headline_variants":["Iterative ghost-Gutzwiller fails for paramagnetic Motts","Direct minimization saves ghost-Gutzwiller paramagnetic Mott phase","Iterative embedding gives spurious polarized Mott insulator","Ghost-Gutzwiller iterative scheme breaks under Zeeman field","Direct min needed for symmetry-invariant Motts in ghost-Gutzwiller"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The compact five-parameter impurity wavefunction built from a singly-occupied decoupled bath is assumed to span the true ghost-Gutzwiller minimum in the Mott phase, including under a magnetic field.","fun_headline_variants_meta":{"raw":{"variants":["Iterative ghost-Gutzwiller fails for paramagnetic Motts","Direct minimization saves ghost-Gutzwiller paramagnetic Mott phase","Iterative embedding gives spurious polarized Mott insulator","Ghost-Gutzwiller iterative scheme breaks under Zeeman field","Direct min needed for symmetry-invariant Motts in ghost-Gutzwiller"]},"model":"grok-4.5","effort":"low","cost_usd":0.00394,"raw_usage":{"total_tokens":1205,"prompt_tokens":768,"num_sources_used":0,"completion_tokens":73,"cost_in_usd_ticks":39404000,"prompt_tokens_details":{"text_tokens":768,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":364,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":768,"tokens_out":73,"duration_ms":7395,"temperature":1.0,"reasoning_tokens":364,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-30T10:46:47.924744+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Unrestricted direct minimization of the full ghost-Gutzwiller functional in a finite Zeeman field: if that calculation recovers the iterative scheme's energy jump and fully polarized insulator, the claim that direct minimization is required for a physical paramagnetic Mott solution fails.","supporting_citations":[],"review_version":1}