{"id":"1de2cb21-5007-4c97-8756-b6c07157198d","arxiv_id":"2607.08176","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"NGS-MPS simulations of Hubbard–Holstein models show soft phonons promote 1D phase separation and stabilize 2D stripe phases, including a novel bipolaronic stripe with enlarged unit cell.","lead":"A hybrid non-Gaussian matrix-product-state method scales simulations of strongly correlated electrons coupled to phonons to large 1D chains and 2D cylinders. It finds soft phonons drive phase separation in 1D and stabilize stripe orders in 2D via local and non-local retardation, including a novel bipolaronic stripe.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The central claim rests on two concrete observations (enhanced charge / diminished spin response under soft phonons in the doped AFM; spontaneous 16\times2 bipolaronic stripe that preempts PS in the doped CDW) that are directly tied to the analytically derived effective interaction (Eq. 12) and are cross-checked by independent diagnostics (double-occupancy perturbation theory, local compressibility, energy comparison). The soft-phonon regime is the hardest for any wave-function method, yet the paper documents the exact failure modes (divergence of Γ_pp, local minima) and the practical controls used to avoid them. Because those controls are already in place and the physical narrative does not require an untested extrapolation, the concern does not move the verdict. The reader's ACCEPT / HIGH assessment is therefore left unchanged.","tokens_in":33184,"tokens_out":521,"duration_ms":7004,"concrete_test":"Re-optimize the (u=2, λ=2, ω=t) bipolaronic stripe of Fig. 6a on the same 48\times4 cylinder with bond dimension doubled (D \to 4\times10^4) and two additional random seeds; if the 16\times2 period, energy ordering versus macroscopic PS, and long-range V profile remain unchanged within the reported 10^{-6} energy precision, the soft-phonon fidelity concern is closed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest assumption correctly flags soft-phonon fidelity of the NGS-MPS ansatz (Eq. 3) and workflow on 48\times4 cylinders, but the manuscript already supplies the concrete safeguards that keep this from becoming load-bearing against the central claim. Benchmarks against NGSED on 4\times4 tori (Appendix B.2) recover pairing, structure factors and local moments; the 1D soft-phonon PS is independently confirmed by Maxwell construction under the restricted ansatz (Appendix B.1); the effective Hamiltonian (Eqs. 6, 12) is derived analytically and used to interpret both the local pinning (Eqs. 18a–c) and the non-local V profile (Fig. 6b); truncation error is held at 10^{-5} and energies compared across distinct initializations (Sec. II B). No internal inconsistency or untested leap is required for the local/non-local retardation narrative of Sec. III B 2.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript introduces a hybrid non-Gaussian matrix-product-state (NGS-MPS) variational method for strongly correlated electron-phonon systems. The ansatz (Eq. 3) applies a non-Gaussian transformation that encodes e-ph entanglement, after which residual phonons are treated as a bosonic Gaussian state and the electrons by an MPS; a self-consistent workflow of NGS flows and MPS sweeps is used to minimize the energy. The method is applied to generalized Hubbard-Holstein models. In 1D it recovers the phonon-mediated nearest-neighbor attraction of prior NGSED work and, on larger systems, finds a pronounced tendency to phase separation. In 2D it maps the half-filled phase diagram (metallic intermediate phase between AFM and CDW) and, at 1/8 doping, shows that soft phonons stabilize fully filled stripes via a local retardation effect in the doped AFM and a novel bipolaronic stripe (16\times2 unit cell) via non-local phonon-mediated interactions that suppress phase separation in the doped CDW regime. Analytic effective electronic Hamiltonians (Eqs. 6, 12) are used to interpret both mechanisms.","tokens_in":33445,"tokens_out":804,"duration_ms":8468,"significance":"If the numerical results hold, the work supplies a scalable, systematically benchmarked tool that reaches system sizes previously inaccessible to NGSED while remaining competitive with pure fermionic MPS. The concrete physical claims—soft-phonon stabilization of stripes by local versus non-local retardation, and the spontaneous appearance of a large-unit-cell bipolaronic stripe—are of direct interest for cuprate phenomenology and for the broader theory of intertwined orders. Strengths include quantitative benchmarks against NGSED (Fig. 1, Appendix B.2), independent Maxwell-construction confirmation of 1D phase separation (Appendix B.1), analytic derivation of the effective interactions used for interpretation, and explicit control of truncation error and multi-seed validation. These features make the central narrative falsifiable and reusable by other groups.","major_comments":[],"minor_comments":[{"comment":"In Sec. II B the switch from imaginary-time evolution to DMRG is described only qualitatively; a short statement of the practical criterion (e.g., energy change or truncation-error threshold) used to decide the switch would improve reproducibility.","section":null},{"comment":"Fig. 1 caption and panels (b),(d) note the absence of data points at large λ as the onset of phase separation; a brief quantitative criterion (energy divergence, density variance, or compressibility sign) would make the identification unambiguous.","section":null},{"comment":"Table II reports renormalized hoppings along the u=2λ line; the corresponding bare values or the explicit definition of the central bulk site would help the reader verify the polaron-mass trend discussed in the text.","section":null},{"comment":"Appendix A.4, Eq. (A25): the approximation 〈PS〉_{0} ≈ 〈PS〉 is stated to leading order; a one-sentence estimate of the neglected O((t̃/ΔE)^{4}) correction for the parameters of Fig. 5 would strengthen the comparison with the full NGS-MPS double occupancy.","section":null},{"comment":"A few typographical inconsistencies appear (e.g., “NGS/uni2010ED”, “L = 80 system,,” double commas). A final proofreading pass would remove them.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is a solid, self-contained contribution that meets the standards of a high-quality condensed-matter theory journal. The soft-phonon fidelity concern raised in the reader’s weakest-assumption note is already addressed by the paper’s own benchmarks and safeguards; I do not regard it as load-bearing against acceptance. No novelty or citation issues warrant editorial attention."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The new piece is the NGS-MPS hybrid: non-Gaussian transformation for e-ph entanglement, Gaussian phonons, MPS for the electrons, plus a self-consistent loop that starts from pure NGS seeds. That combination actually reaches 1D chains of length 120 and 48\times4 cylinders at soft frequencies, which pure NGSED and standard DMRG with phonon cutoffs cannot. The physics payoff is concrete: macroscopic PS in soft-phonon 1D HH (confirmed by Maxwell construction under the restricted ansatz), a half-filled metallic window on 48\times4 between AFM and CDW, and—most useful—the distinction between local retardation that pins the fully filled stripe when doping the AFM and non-local V that arrests PS into a spontaneous 16\times2 bipolaronic stripe when doping the CDW.\n\nWhat works: the effective electronic Hamiltonian (Eqs. 6, 12) is derived analytically, so the local pinning terms (18a–c) and the long-range V profile in Fig. 6b are not post-hoc stories. Benchmarks against NGSED on V_nn and 4\times4 observables, energy consistency with LBO, and phase-diagram features that line up with existing QMC/VMC are all present. Truncation is held at 10^{-5}, multiple seeds are compared, and the paper itself flags when the bosonic sector can lose positive-definiteness.\n\nSoft spots are real but proportional. Soft phonons (ω ≤ t) on 48\times4 still sit near dense near-degeneracies and extended clouds; the ansatz can diverge or trap if D or the seed is poor. Finite-width effects (even-odd spin gap, plaquette pairing) mean the metallic window will shrink toward the 2D limit. Neither issue is hidden, and neither overturns the local/non-local narrative that is the paper’s strongest claim.\n\nThis is for people who actually run or need large-scale e-ph numerics on cuprate-relevant models. The math and citation pattern look solid; self-cites are to the prior NGS machinery that is being extended, not circular. I would send it to referees. Worth reading and, for anyone working on stripes or Holstein physics, worth citing.","headline":"Solid hybrid solver plus large-scale HH results that cleanly separate local vs non-local phonon retardation; soft-phonon caveats are real but already stress-tested in the paper.","tokens_in":34093,"tokens_out":557,"would_cite":true,"duration_ms":7632,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Soft phonons pin stripes in doped cuprate-like models by two distinct retardation effects.","keywords":["electron-phonon","Hubbard-Holstein","non-Gaussian states","matrix product states","stripe order","phase separation","retardation","soft phonons"],"falsifier":"On a 48\times4 cylinder at 1/8 doping in the doped CDW regime, an independent method that can resolve large unit cells either finds a lower-energy macroscopic phase-separated state than the claimed bipolaronic stripe, or finds that the 16-site period collapses once bond dimension is pushed higher.","tokens_in":34050,"feed_emoji":"🧲","tokens_out":643,"duration_ms":6322,"temperature":0.7,"pith_summary":"The paper introduces a hybrid non-Gaussian matrix product state method that can treat strongly correlated electrons and phonons of unbounded Hilbert space at the large scales needed to resolve competing orders. Using that tool on generalized Hubbard–Holstein models, the authors show that soft phonons drive a strong tendency to phase separation in one dimension and, in two dimensions, open an intermediate metallic window at half filling. Upon doping they find that the same soft phonons stabilize stripe order in two different ways: a local retardation that pins charge while weakening the spin response in the doped antiferromagnet, and a non-local retardation that uses long-range phonon-mediated interactions to suppress phase separation and lock a novel bipolaronic stripe with a 16-site period in the doped charge-density-wave regime. The concrete claim is that these retardation mechanisms are the microscopic reason soft phonons favor stripes over phase separation, giving a systematic route to the electron–phonon interplay that is thought to matter for superconductivity.","feed_headline":"Soft phonons pin stripes two ways in doped models","feed_subtitle":"Local and non-local retardation stabilize stripes over phase separation at 1/8 doping","key_machinery":"The non-Gaussian matrix product state (NGS-MPS) ansatz: a non-Gaussian transformation that entangles a bosonic Gaussian phonon state with an electronic matrix product state, followed by a self-consistent alternating optimization that yields an analytic effective electronic Hamiltonian containing both instantaneous and retarded phonon-mediated interactions.","core_discovery":"Soft phonons stabilize stripe phases in doped two-dimensional Hubbard–Holstein models through two distinct retardation effects: a local one that pins charge order and diminishes the spin response in the doped antiferromagnet, and a non-local one in which long-range phonon-mediated interactions suppress phase separation and stabilize a bipolaronic stripe of 16-site period in the doped charge-density-wave regime.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Soft phonons pin stripes two ways via local and non-local retardation","Phonons stabilize stripes: local pin in AF, non-local bipolaronic in CDW","Soft phonons curb phase separation to lock in doped stripe phases","Local retardation pins charge order while non-local interactions kill PS","Two retardation paths let soft phonons stabilize 1/8 stripes"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The hybrid variational ansatz plus its self-consistent optimization remains faithful for soft phonons on large cylinders, where near-degenerate states and extended phonon clouds make local minima and insufficient bond dimension especially dangerous.","fun_headline_variants_meta":{"raw":{"variants":["Soft phonons pin stripes two ways via local and non-local retardation","Phonons stabilize stripes: local pin in AF, non-local bipolaronic in CDW","Soft phonons curb phase separation to lock in doped stripe phases","Local retardation pins charge order while non-local interactions kill PS","Two retardation paths let soft phonons stabilize 1/8 stripes"]},"model":"grok-4.5","effort":"low","cost_usd":0.003504,"raw_usage":{"total_tokens":1176,"prompt_tokens":792,"num_sources_used":0,"completion_tokens":78,"cost_in_usd_ticks":35040000,"prompt_tokens_details":{"text_tokens":792,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":306,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":792,"tokens_out":78,"duration_ms":3819,"temperature":1.0,"reasoning_tokens":306,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-10T11:50:44.485069+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"On a 48\times4 cylinder at 1/8 doping in the doped CDW regime, an independent method that can resolve large unit cells either finds a lower-energy macroscopic phase-separated state than the claimed bipolaronic stripe, or finds that the 16-site period collapses once bond dimension is pushed higher.","supporting_citations":[],"review_version":1}