{"id":"37ef85a5-ddb8-4d9c-96b9-fd1da63253a2","arxiv_id":"2502.07252","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Finite-temperature NLCE calculations find nearest-neighbor ferromagnetic correlations and Nagaoka polarons in the square-lattice Hubbard model for dopings up to about 33 percent at large U/t.","lead":"Kinetic ferromagnetism, the alignment of spins caused by hole motion, is studied at finite temperature in the square-lattice Hubbard model using numerical linked-cluster expansions. The authors find short-range ferromagnetic correlations and Nagaoka polarons for dopings up to roughly 33 percent, providing a map that could guide cold-atom experiments.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The quantitative δc1/δc2 boundaries and the U→∞ extrapolation rest on an internal Wynn-resummation convergence threshold and an unspecified outlier-removal step; a stricter, reproducible convergence check is needed before the central finite-temperature claim is accepted.","rationale":"The reader's weakest assumption correctly identifies the truncated NLCE site expansion plus Wynn resummation as the load-bearing element, and I agree that the 0.003 difference between successive Wynn cycles is an internal consistency criterion rather than a proof of convergence. My partial disagreement is that the more immediately actionable concern is the combination of that loose threshold with the unspecified 'small number of outliers' removal, since both can shift the zero crossings of a small signal. The central qualitative finding — short-range FM correlations and hole-spin-spin FM bubbles at finite temperature on the square lattice — is plausible and consistent with prior ground-state Nagaoka studies, and the paper's use of NLCE with exact diagonalization on topologically distinct clusters is a credible method. However, the paper's headline quantitative claims are the values of δc1, δc2, and their U→∞ extrapolation, and those values are extracted from curves whose magnitude near the crossings is only a few times the acceptance threshold. A stricter and fully specified convergence analysis would settle whether the reported boundaries are robust. Because the issue is fixable through reanalysis of existing data and does not by itself overturn the qualitative picture, the appropriate disposition remains conditional rather than acceptance or rejection.","tokens_in":12291,"tokens_out":3280,"duration_ms":31866,"concrete_test":"Using the deposited data (DOI 10.5281/zenodo.15495644), recompute Css for U/t=80 and 120 at T/t=0.40 and 0.50 with (i) the raw order-7, order-8, and order-9 series, (ii) Wynn cycles 2, 3, and 4 instead of only 3 and 4, (iii) a stricter convergence threshold of 0.0003, and (iv) no outlier removal. Then re-extract δc1 and δc2 from the interpolated zero crossings and the Fig. 4 extrapolation to U→∞. If the boundaries shift by more than ~0.02 in doping, or if the U→∞ δc2 estimate changes by more than ~0.03, the central quantitative claim is not robust to resummation and data-selection choices.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim — that short-range ferromagnetic correlations and Nagaoka polarons persist in the thermodynamic limit up to interaction-dependent critical dopings, with δc2 approaching ~33% as U→∞ — depends on the sign and zero crossings of Css computed by NLCE at T/t≈0.3–0.5, exactly where the raw site expansion is not converged and Wynn resummation is essential. Section III.A accepts results only when two successive Wynn cycles differ by less than 0.003, but this is an internal consistency check, not an estimate of the systematic error of the resummation. Near the δc1/δc2 crossings, Css has magnitude roughly 0.01–0.02, so the 0.003 threshold is a substantial fraction of the signal; a modest resummation bias could move the zero crossings by several percent in doping and thereby shift the boundaries plotted in Fig. 4. Additionally, the text states that 'A small number of outliers at various temperatures are removed' without specifying the number, the temperatures, or whether removal was sign-blind. If outlier removal is correlated with the sign of Css, the extracted δc1 and δc2 could be systematically biased. The abstract's 'temperatures as low as 0.2' also conflicts with the text's T/t∼0.3 and with the displayed data, which begin near 0.3–0.4. These issues do not invalidate the qualitative observation of short-range FM correlations, but they leave the quantitative boundaries — the paper's main deliverable — insufficiently secured.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript uses numerical linked-cluster expansions (NLCE) up to 9 sites for the Hubbard model and 11 sites for the t-J model to compute finite-temperature nearest-neighbor spin correlations Css and hole-spin-spin correlations Chss on the square lattice. The authors identify a doping window, bounded by critical dopings delta_c1 and delta_c2, within which Css is ferromagnetic, and they study its temperature and interaction dependence for U/t between 60 and 240 (and the J=0 t-J limit). They report evidence for finite-temperature Nagaoka polarons and conclude that ferromagnetic nearest-neighbor correlations survive in the thermodynamic limit up to dopings of about 33% in the U to infinity limit. The paper includes a comparison with recent cold-atom data on the triangular lattice and provides a public data repository.","tokens_in":12578,"tokens_out":5179,"duration_ms":48727,"significance":"If the quantitative boundaries are correct, this is a useful finite-temperature extension of the Nagaoka problem and provides concrete doping and temperature targets for cold-atom experiments in square optical lattices. The study is largely parameter-free: the correlation functions are computed directly from the Hubbard and t-J Hamiltonians, and no fitted parameters enter the central claim. The use of an established cluster method, the comparison with t-J results, the connection to existing variational and DMRG ground-state literature, and the deposition of data in Zenodo are all strengths. However, the quantitative deliverables—delta_c1, delta_c2, and the U to infinity extrapolation—rest on an internal Wynn-resummation consistency threshold and an incompletely specified outlier-removal step, which need to be addressed before the central finite-temperature claim is fully secured.","major_comments":[{"comment":"The convergence criterion that successive Wynn resummation cycles differ by less than 0.003 is an internal consistency check, not a systematic error estimate. Near the delta_c1 and delta_c2 crossings in Figs. 1-4, Css has magnitude roughly 0.01-0.02, so the 0.003 threshold is a substantial fraction of the signal; a modest resummation bias could move the zero crossings by several percent in doping and shift the phase boundaries plotted in Fig. 4. The manuscript should provide a quantitative convergence study, for example by comparing site-expansion orders, varying the acceptance threshold, or cross-checking against an independent method such as determinant quantum Monte Carlo at T/t around 0.4-0.5 for U/t = 80-120.","section":"Section III.A, Figs. 1-4"},{"comment":"The caption states that 'A small number of outliers at various temperatures are removed' without specifying how many outliers, at which temperatures, or whether the removal was sign-blind. If outlier removal is correlated with the sign or magnitude of Css, the extracted delta_c1 and delta_c2 values in Fig. 4 could be systematically biased. Please report the exact removal criteria or show the raw data alongside the cleaned data.","section":"Fig. 1 caption"},{"comment":"The quantitative claims that delta_c2 approaches about 33% in the U to infinity limit and that there is a lower bound of U/t around 20 are read off from the gray crosses and the dashed linear-regression line in Fig. 4, but no uncertainties or regression parameters are given for this extrapolation. The only error bars mentioned are from the mu-T interpolation and are smaller than the symbols; they do not include NLCE truncation or resummation uncertainty. Please propagate the convergence uncertainty into delta_c1 and delta_c2 and into the extrapolated quantities, or state explicitly that the gray crosses and linear regression are only qualitative guides.","section":"Fig. 4, Conclusions"}],"minor_comments":[{"comment":"The abstract says temperatures as low as 0.2 of the hopping energy, but Section V states T/t ~ 0.3 and the data in Fig. 1 appear to begin near 0.3-0.4; please reconcile this inconsistency.","section":"Abstract vs. Conclusions"},{"comment":"The sentence 'Here, we do not show data for which the difference between the two resummations is smaller than 0.003' appears to reverse the criterion used elsewhere in the paper; it should presumably read 'larger than 0.003'.","section":"Fig. 6 caption"},{"comment":"There is a typo: 'neaerest-neighbor' should be 'nearest-neighbor'.","section":"Fig. 7 caption"},{"comment":"The caption refers to yellow bonds placed where 'the agreement between Wynn orders exceeds our error thresholds,' but the threshold value is not defined in the text; please specify it or refer explicitly to Section III.A.","section":"Fig. 10 caption"},{"comment":"Reference [37] is listed as '(In preparation)'; if the cluster enumeration algorithm is important for reproducibility, please provide a preprint identifier or DOI when available.","section":"Reference [37]"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely publishable after revision. The main risk is that the quantitative phase boundaries are insufficiently secured by the current convergence and outlier-handling documentation. The Zenodo data deposit is a strength and should be preserved. I have no concerns about integrity; the issues are technical and presentation-level, but one of them is load-bearing for the central quantitative claim."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The useful new thing here is a finite-temperature, thermodynamic-limit map of short-range ferromagnetic correlations in the square-lattice Hubbard and t-J models, using NLCE to order 9 (Hubbard) and 11 (t-J). The qualitative result—nearest-neighbor spin correlations turn FM in a doping window at low T, with Nagaoka polarons visible in hole-spin-spin correlations—looks right. The paper also does a good job cross-checking Hubbard and t-J results against each other and against experimental data from Lebrat et al., and the data are deposited. No fitted parameters enter the central claim; self-citations are for method validation and are appropriate.\n\nThe soft spots are real but mostly fixable. First, the Wynn resummation is accepted when two cycles agree to 0.003, but that is an internal consistency check, not an error estimate. Near the delta_c1/delta_c2 crossings, Css is only about 0.01–0.02 in magnitude, so the threshold is a substantial fraction of the signal. A biased resummation could shift the boundaries by a few percent in doping. The authors should provide some sense of systematic error—for example, by comparing with DQMC where feasible or by checking sensitivity to the threshold.\n\nSecond, the Fig. 1 caption says \"a small number of outliers at various temperatures are removed\" without saying how many, at which temperatures, or whether removal was sign-blind. That is too vague for a result whose main deliverable is the zero crossings. I would want this specified, or better, the raw data made available so readers can judge.\n\nThird, the abstract says \"temperatures as low as 0.2\" while the conclusions say \"T/t~0.3,\" and the actual FM-window analysis is mostly at T/t >= 0.4. That inconsistency needs fixing. Also, calling the FM region a \"phase\" is an overstatement—the paper itself notes the correlations remain short-ranged. \"Region with FM nearest-neighbor correlations\" is the honest wording.\n\nNone of this undermines the main qualitative finding, which is consistent with prior ground-state results and with the triangular-lattice experiments. The quantitative boundaries should be treated as indicative rather than exact until the convergence question is addressed.\n\nWho is this for? People working on Nagaoka physics, cold atoms in square lattices, and NLCE practitioners. It deserves a serious referee: the method is appropriate, the topic is timely, and the issues are addressable in revision. I would send it to peer review.","headline":"A solid finite-temperature NLCE study of short-range kinetic ferromagnetism on the square lattice; the qualitative picture is convincing, but the quantitative boundaries are less secure than the abstract suggests.","tokens_in":13108,"tokens_out":3253,"would_cite":true,"duration_ms":31479,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The square-lattice Hubbard model hosts finite-temperature kinetic ferromagnetism in a doping window that widens with interaction strength, reaching about 33% in the infinite-U limit.","keywords":["Nagaoka ferromagnetism","Hubbard model","t-J model","numerical linked-cluster expansion","finite temperature","spin correlations","Nagaoka polaron","square lattice"],"falsifier":"A determinant quantum Monte Carlo calculation or a square-lattice ultracold-atom experiment at $U/t \\approx 120$, $T/t \\approx 0.4$, and hole doping near 18% should measure a positive nearest-neighbor spin correlation $C_{ss}$ if the claim is right; a negative value would falsify the ferromagnetic window. The same measurement at 35% doping in the $U\\to\\infty$ limit would test the predicted $\\delta_{c2} \\sim 33\\%$ boundary, since a positive $C_{ss}$ there would contradict it.","tokens_in":12050,"feed_emoji":"🧲","tokens_out":8765,"duration_ms":71558,"temperature":0.7,"pith_summary":"The paper is trying to establish that Nagaoka's kinetic ferromagnetism, usually proved for a single hole in an infinite-$U$ Hubbard band at zero temperature, survives as a finite-temperature, finite-doping phenomenon on the square lattice. Using numerical linked-cluster expansions (NLCE), the authors find that for $U/t$ between 80 and 240 and temperatures down to $T/t \\sim 0.3$, the nearest-neighbor spin correlations turn ferromagnetic in a doping interval $\\delta_{c1} < \\delta < \\delta_{c2}$, with $\\delta_{c2}$ growing with $U/t$ toward about 33% and $\\delta_{c1}$ shrinking toward zero. They also identify Nagaoka polarons, ferromagnetic bubbles bound to holes, persisting to roughly 20% doping. A reader should care because this maps where cold-atom square-lattice quantum simulators could look for kinetic magnetism, and it complements recent triangular-lattice observations by showing that a bipartite geometry still hosts the effect, just over a finite doping window.","feed_headline":"Kinetic ferromagnetism survives on square lattice to 33% doping","feed_subtitle":"Cluster-expansion calculations find ferromagnetic Nagaoka polarons up to ~30 percent hole doping at temperatures as low as T/t = 0.3.","key_machinery":"The numerical linked-cluster expansion (NLCE) is the central tool: it expresses an extensive property of the infinite lattice as a sum over all topologically distinct connected clusters, with inclusion-exclusion weights, and computes each cluster exactly by diagonalization, up to 9 sites for the Hubbard model and 11 sites for the $t$-$J$ model. Wynn's epsilon algorithm resums the site series to reach lower temperatures, and results are accepted when successive resummation cycles differ by less than 0.003. The order parameters are normalized correlation functions: $C_{ss}$, the two-point nearest-neighbor spin correlation normalized by the squared fraction of singly occupied sites, and $C_{hss}$, the three-point hole-spin-spin correlation normalized by the hole fraction times the squared singly occupied fraction. The zero crossings of $C_{ss}$ as a function of doping define $\\delta_{c1}$ and $\\delta_{c2}$, the boundaries of the ferromagnetic window.","core_discovery":"The paper argues that, at interaction strengths $U/t \\gtrsim 80$ and temperatures as low as $T/t \\sim 0.3$, the square-lattice Hubbard model and its strong-coupling $t$-$J$ limit develop a window of hole doping, $\\delta_{c1} < \\delta < \\delta_{c2}$, in which the nearest-neighbor spin correlation $C_{ss}$ is ferromagnetic. The window widens with increasing $U/t$; in the $U \\to \\infty$ ($J \\to 0$) limit $C_{ss}$ is ferromagnetic for any infinitesimal doping and the upper boundary approaches $\\delta_{c2} \\sim 33\\%$. The three-point hole-spin-spin correlation $C_{hss}$ shows ferromagnetic bubbles around dopants, Nagaoka polarons, that persist to about 20% doping, and the polarons grow as temperature drops. The central claim is that kinetic ferromagnetism on a bipartite lattice is not confined to the single-hole, zero-temperature Nagaoka limit, but appears as a finite-temperature, finite-doping short-range correlated phase.","pith_inferences":["The 0.003 threshold between Wynn cycles is an internal consistency check; if it were tightened, the accessible temperature range would shrink and the reported $\\delta_{c1}$ values at low temperature might shift upward, so the extrapolation $\\delta_{c1}\\to 0$ at zero temperature is the most fragile part of the picture.","A direct extension would be to compute the same correlators on a tilted or weakly anisotropic square lattice to test whether the finite ferromagnetic window is specific to the perfect bipartite square geometry or survives mild asymmetry.","The $J=0$ $t$-$J$ results can be read as a parameter-free prediction: at $U\\to\\infty$, $C_{ss}$ should be positive for any infinitesimal doping at the lowest temperatures, a statement an optical-lattice experiment with tunable interaction could test directly by measuring the sign of nearest-neighbor correlations at very low doping.","The apparent lower bound $U/t \\sim 20$ suggests the kinetic-ferromagnetism window is an intermediate-to-strong-coupling phenomenon; a natural follow-up is to search for how the window depends on temperature just above that threshold."],"forward_implications":["A square optical lattice filled with ultracold fermions at $U/t \\sim 100$-$240$ and $T/t \\sim 0.4$ should show positive nearest-neighbor $C_{ss}$ for hole dopings roughly between a few percent and 20-30%, with the window shrinking as temperature rises.","The $t$-$J$ model with $J=0$ predicts $\\delta_{c1}=0$ and $\\delta_{c2}\\sim 1/3$ in the $U\\to\\infty$ ground state, giving a concrete target for ground-state methods to confirm or refute.","Extrapolating the finite-temperature boundary $\\delta_{c2}$ to zero yields a lower bound $U/t \\sim 20$ for nearest-neighbor ferromagnetic correlations, so experiments below that interaction should see none.","Because the ferromagnetic correlations remain short-ranged at these temperatures, the predicted phase is a short-range correlated state rather than a fully polarized ferromagnet; detecting it requires nearest-neighbor correlation measurements, not bulk magnetization alone.","The close agreement between Hubbard and $t$-$J$ results at small doping validates using the $t$-$J$ model as a computational shortcut for strong-coupling kinetic-magnetism physics."],"supporting_citations":[{"why":"Proves the single-hole ferromagnetic Nagaoka state in the infinite-U Hubbard model that this paper extends to finite temperature and doping.","marker":"[6]"},{"why":"Recent ultracold-atom observation of Nagaoka polarons on the triangular lattice, used as the experimental comparison in the insets.","marker":"[11]"},{"why":"DMRG analysis of the Nagaoka polaron in the two-dimensional t-J model, grounding the polaron picture used here.","marker":"[14]"},{"why":"Variational quantum Monte Carlo study finding ferromagnetism at up to 40% doping, providing a ground-state benchmark.","marker":"[24]"},{"why":"Introduces the numerical linked-cluster expansion method that is the paper's central computational tool.","marker":"[32]"},{"why":"Describes the site-expansion formulation and Wynn resummation used to reach low temperatures.","marker":"[33]"},{"why":"Supplies the epsilon algorithm used for the Wynn resummation.","marker":"[38]"},{"why":"Earlier variational result giving a 25% upper doping bound and critical U/t near 77.7, against which the new boundaries are compared.","marker":"[25]"},{"why":"Earlier variational estimate of a 29% upper doping bound and critical U/t near 42, another comparison point.","marker":"[23]"}],"fun_headline_variants":["Square-lattice kinetic ferromagnetism at finite temperature","Nagaoka polarons in square lattice Hubbard model up to 30% doping","Kinetic ferromagnetism extends to finite doping on square lattice","Ferromagnetic polarons at up to 30% doping in Hubbard model","Square-lattice Hubbard model shows kinetic ferromagnetism at finite T"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that the truncated cluster expansion, clusters of up to 9 sites for the Hubbard model and 11 for the t-J model resummed with Wynn's algorithm, correctly represents the infinite lattice at the reported temperatures, with the only evidence being agreement between successive resummation cycles to within 0.003.","fun_headline_variants_meta":{"raw":{"variants":["Square-lattice kinetic ferromagnetism at finite temperature","Nagaoka polarons in square lattice Hubbard model up to 30% doping","Kinetic ferromagnetism extends to finite doping on square lattice","Ferromagnetic polarons at up to 30% doping in Hubbard model","Square-lattice Hubbard model shows kinetic ferromagnetism at finite T"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000681,"raw_usage":{"total_tokens":3088,"prompt_tokens":938,"completion_tokens":2150,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":554,"completion_tokens_details":{"reasoning_tokens":2059}},"tokens_in":554,"tokens_out":2150,"duration_ms":15841,"temperature":1.0,"reasoning_tokens":2059,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T13:19:22.359363+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A determinant quantum Monte Carlo calculation or a square-lattice ultracold-atom experiment at $U/t \\approx 120$, $T/t \\approx 0.4$, and hole doping near 18% should measure a positive nearest-neighbor spin correlation $C_{ss}$ if the claim is right; a negative value would falsify the ferromagnetic window. The same measurement at 35% doping in the $U\\to\\infty$ limit would test the predicted $\\delta_{c2} \\sim 33\\%$ boundary, since a positive $C_{ss}$ there would contradict it.","supporting_citations":[{"cited_title":"Nagaoka, Ferromagnetism in a narrow, almost half- filled s band, Physical Review 147, 392 (1966)","cited_arxiv_id":null,"evidence_quote":"Proves the single-hole ferromagnetic Nagaoka state in the infinite-U Hubbard model that this paper extends to finite temperature and doping."},{"cited_title":"Lebrat, M","cited_arxiv_id":null,"evidence_quote":"Recent ultracold-atom observation of Nagaoka polarons on the triangular lattice, used as the experimental comparison in the insets."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"DMRG analysis of the Nagaoka polaron in the two-dimensional t-J model, grounding the polaron picture used here."},{"cited_title":"Becca and S","cited_arxiv_id":null,"evidence_quote":"Variational quantum Monte Carlo study finding ferromagnetism at up to 40% doping, providing a ground-state benchmark."},{"cited_title":"Rigol, T","cited_arxiv_id":null,"evidence_quote":"Introduces the numerical linked-cluster expansion method that is the paper's central computational tool."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Describes the site-expansion formulation and Wynn resummation used to reach low temperatures."},{"cited_title":"Wynn, On the Convergence and Stability of the Ep- silon Algorithm, SIAM Journal on Numerical Analysis 3, 91 (1966)","cited_arxiv_id":null,"evidence_quote":"Supplies the epsilon algorithm used for the Wynn resummation."},{"cited_title":"Wurth, G","cited_arxiv_id":null,"evidence_quote":"Earlier variational result giving a 25% upper doping bound and critical U/t near 77.7, against which the new boundaries are compared."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier variational estimate of a 29% upper doping bound and critical U/t near 42, another comparison point."}],"review_version":1}