{"id":"cb7c32fd-154d-42a1-9943-825afa973f90","arxiv_id":"2411.19938","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Neural-network quantum Monte Carlo reproduces the Tomonaga-Luttinger liquid state in one-dimensional hydrogen chains and indicates a transition to a Fermi-liquid-like phase at short interatomic distances.","lead":"This paper applies neural-network quantum Monte Carlo to hydrogen chains and reports that the method reproduces the Tomonaga-Luttinger liquid behavior at long bond lengths and detects a transition to a Fermi-liquid-like phase at short bond lengths. A generalist might read it to see whether deep-learning wavefunctions can map quantum phase diagrams of real materials.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Fermi-liquid identification rests on single-size N=24 PBC data with no finite-size scaling or error bars; a finite-size TLL rounding could mimic the reported n(k) jump and q-peak shift.","rationale":"Good-faith reading: the paper's methodological core is to use real-space NNQMC to measure RDMs, structure factors, and n(k) for H chains, with the central new physics claim being a TLL-to-Fermi-liquid transition at R<1.6 a_B driven by self-doping. The TLL identification at large R has independent support: the energy agrees with high-level benchmarks within 1 mEh, the exponent eta=1.16(5) at R=2.8 is consistent with prior DMRG, and the linear small-q behavior of S_sigma(q) gives K_sigma~1.067. These make the methodological part credible. The load-bearing weakness is the Fermi-liquid part: all three signatures in Fig. 3(d-f) are shown for a single PBC size, N=24, and the paper itself notes in the SI only that n(k) is not a finite-size artifact (SI Sec. I C), but that SI is not included in the arXiv text. Without N-scaling and error bars, the reported 'abrupt' changes could be the finite-size rounding of a Luttinger liquid with a small alpha, a discrete-k mesh effect (2pi/24 ~ 15 degrees), or an incommensurate oscillation; none of these is excluded by the presented evidence. The proposed check--running N=36/48/72 and extrapolating the jump height Z_N, q_max, and n(k)>1 overshoot--would directly distinguish a true Fermi-surface discontinuity from finite-size effects. If the signatures vanish, the central claim fails; if they persist, the CONDITIONAL verdict should be upgraded. A secondary concern: the self-doping mechanism is supported by orbital occupations reported only below ~1.0 a_B, while the transition is claimed at 1.6 a_B, so the causal link is not quantitatively established. Because this is an addressable evidentiary gap rather than an internal contradiction, the reader's CONDITIONAL verdict is unchanged.","tokens_in":10119,"tokens_out":6050,"duration_ms":59395,"concrete_test":"Perform DeepSolid PBC calculations at N=36 and N=48 (and N=72 if feasible) for R=1.0, 1.3, 1.6, and 2.0 a_B. Compute S_sigma(q) and n(k) from block-averaged MCMC samples with error bars. Define the observables: q_max = argmax S_sigma(q), Z_N = n(k_F^+)-n(k_F^-) excluding the nearest k point to k_F, and O_N = max_k[n(k)-1] near k=0. Compare q_max, Z_N, O_N across N: if they extrapolate to pi, 0, 0, respectively, the FL signatures are finite-size artifacts and the TLL phase extends across the nominal transition; if q_max < pi and Z_N, O_N stay nonzero at large N, the FL claim is supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing step is the identification of a Fermi-liquid phase from Fig. 3(d-f), which is computed only for N=24 PBC, with no finite-size scaling and no error bars on S_sigma(q) or n(k). In a 1D TLL with Luttinger parameter near 1, n(k) is continuous but can be very sharp in a small finite system; the apparent peak shift away from q=pi may also be a discrete-k mesh artifact (q spacing 2pi/24) or an incommensurate finite-size oscillation. The claim therefore requires that the n(k) jump height Z and the q_max shift persist in the thermodynamic limit, and that n(k)>1 near k=0 is not a basis-set/projection artifact. A second, independent inconsistency: the self-doping orbital-occupation evidence in Fig. 4(d) is only reported to become non-negligible below ~1.0 a_B, whereas the claimed transition starts at R<1.6 a_B, leaving the mechanism unquantified across the transition.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper applies real-space neural-network quantum Monte Carlo (FermiNet for open chains, DeepSolid for periodic chains) to equispaced hydrogen chains over a range of interatomic distances R, and develops Monte Carlo sampling of 1- and 2-RDMs, spin and charge structure factors, and momentum distributions. After validating against published AFQMC/MRCI+Q energies (within ~1 mEh) and demonstrating Peierls dimerization and natural-orbital correlation signatures, the paper reports that for R>1.6 a_B the chains exhibit Tomonaga-Luttinger-liquid behavior: spin-spin correlations decaying as eta=1.16(5) close to the Heisenberg value, K_sigma=1.067 from the small-q slope of S_sigma(q), a gapped charge structure factor, and continuous n(k). For R<1.6 a_B the authors claim a transition to a Fermi-liquid-like phase, marked by a shift of the S_sigma(q) peak away from q=pi, a kink in S_rho(q), a sharpened n(k) at k=pi/2, and n(k)>1 near k=0, which they attribute to a self-doping mechanism involving orbitals above 1s.","tokens_in":10329,"tokens_out":19549,"duration_ms":164475,"significance":"If the Fermi-liquid claim holds, this is a significant result: it would identify a strictly one-dimensional ab initio system that escapes TLL universality and exhibits a quasi-particle-like Fermi surface, with implications for 1D materials and for the validity of one-band models. The paper's methodological contribution is also substantial: the importance-sampled RDM and structure-factor estimators provide a general route to extract second-quantized observables from continuous-space neural-network wavefunctions, and the energy benchmarks against AFQMC/MRCI+Q (within 1 mEh) plus the quantitative agreement of eta with the DMRG value 1.11(1) are concrete, reproducible checks. The TLL part of the paper is solid, and its falsifiable predictions (transition near 1.6 a_B, incommensurate spin peak, multi-band occupancy below the transition) make it a useful anchor for future work. The central new physical claim, however, is currently supported only by single-size, qualitative evidence, and the reported transition distance is not yet connected to the proposed self-doping mechanism.","major_comments":[{"comment":"The Fermi-liquid identification rests entirely on N=24 PBC data, with no finite-size scaling and no error bars reported for S_sigma(q) or n(k), and the paper does not state the set of interatomic distances computed or the procedure by which R_c=1.6 a_B is bracketed. The claim in §III.B that 'In SI Sec. I C we verify that this is not an artifact of finite size' is not checkable from the present version, since no supplementary material accompanies it. This matters because in a TLL with K_sigma near 1 and weak charge coupling, n(k) is continuous yet can be extremely steep at N=24, and the apparent S_sigma(q) peak shift could be a discrete-k mesh artifact (Delta q = 2 pi/24) or an incommensurate finite-size oscillation; similarly, n(k)>1 near k=0 needs a band-resolved or normalization check, since a single-band momentum distribution is bounded by unity. Please (i) plot S_sigma(q) and n(k) with statistical error bars; (ii) repeat for N=16, 24, 32, and 48, stating whether twisted boundary conditions are averaged over theta and including the average if so; (iii) extract the apparent n(k) jump height Z at k=pi/2 and the S_sigma(q) peak position q_max as functions of 1/N, since a Z(N) decaying as a power law in N is the TLL expectation whereas a Z(N) extrapolating to a nonzero constant would support the Fermi-liquid reading; and (iv) state the R values used and how R_c=1.6 a_B is determined. Without these, the abstract's claim of 'abrupt changes' and the transition itself are not established.","section":"§III.C, Fig. 3(d-f)"},{"comment":"The self-doping mechanism is claimed to drive the transition at R<1.6 a_B, but the orbital-occupation evidence is only reported to depart from the one-band description below roughly 1.0 a_B: the text states that a non-negligible amount of electrons occupies orbitals above 1s, 2s, and 2p 'for separation less than 1.0 a_B' and that the lowest localized orbital contains less than 90% of the electrons only 'for atomic distances smaller than 1.0 a_B.' The 1.0 to 1.6 a_B window, in which the TLL-to-Fermi-liquid transition is claimed, is therefore not covered by the mechanism evidence. Please report the orbital occupations at R=1.2, 1.4, and 1.6 a_B and correlate the onset of multi-band occupation with the changes in S_sigma(q) and n(k), or revise the statement of the mechanism.","section":"§III.C vs. Fig. 4(d)"},{"comment":"The theoretical identification of the small-R phase as a Fermi liquid needs an argument or direct evidence beyond the sharpening of n(k). In one dimension, interacting electronic systems generically form (possibly multi-component) TLLs, and neither multiple Fermi points (refs. 24-25) nor the presence of extended interactions (ref. 2) by itself restores a nonzero quasiparticle weight; the natural alternative reading of the data is a weakly interacting TLL whose continuous but steep n(k) and incommensurate S_sigma(q) peak are finite-size effects (subject to the test in the first major comment), or an incommensurate spin-density-wave state. The paper itself labels the R=1.0 chain a 'band conductor' in Fig. 4(a), which is weaker than 'Fermi liquid.' Please either (i) soften the terminology to 'Fermi-liquid-like' or 'band conductor' throughout, or (ii) provide an additional diagnostic that directly probes quasiparticles, e.g., a momentum-resolved spectral function using the RDM/photoemission route cited as ref. 28, or momentum distributions at larger L from an independent method such as DMRG for R=1.2-1.4 a_B.","section":"§III.C"}],"minor_comments":[{"comment":"The text reads 'the well-known Peiers instability'; this should be 'Peierls instability.'","section":"§III.A"},{"comment":"The sentence 'For the momentum distribution (Fig. 3(d))' should refer to Fig. 3(f), which is the panel showing n(k).","section":"§III.C, second paragraph"},{"comment":"The spin-sector parameter K_sigma is a Luttinger parameter, not a central charge; the sentence 'the slope of the spin structure factor is related to the central charge K_sigma' should be reworded, since the central charge of this TLL is c=1.","section":"§III.B"},{"comment":"Only the critical exponent at 2.8 a_B is quoted (eta=1.16(5)); to support the claim that eta and K_sigma are unchanged for medium-to-large R, please report the fitted values, uncertainties, and fitting ranges for every computed interatomic distance, either in the text or in a table.","section":"§III.B"},{"comment":"The twisted-boundary-condition k-point formula k=(2m pi + theta)/N is dimensionally inconsistent; the lattice constant a should appear in the denominator, i.e., k=(2m pi + theta)/(N a).","section":"§II.C"},{"comment":"The manuscript cites 'SI Sec. I C' and 'SI Sec. II' for the finite-size verification and the magnetic-perturbation test, but no supplementary material accompanies this version, so those supporting checks are not verifiable as submitted.","section":"Supplementary material"}],"recommendation":"major_revision","confidential_remarks":"The paper has two parts of very different strength: the TLL demonstration (energies within 1 mEh of AFQMC/MRCI+Q, eta and K_sigma consistent with prior DMRG) is solid and publishable as a methodological result, while the central new physics claim (a Fermi liquid in a 1D system at R<1.6 a_B) rests on single-system-size qualitative evidence. The finite-size scaling and spectral-function checks requested in the major comments are feasible within the paper's scope, so this is a major revision rather than a rejection. If the Z(N) scaling turns out to follow the TLL expectation, the main phase-transition claim would need to be withdrawn, but the TLL demonstration and the RDM/structure-factor sampling methodology would remain a useful contribution. The authors should also be asked to upload the SI, since the finite-size verification is delegated to it."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper up front. First, the methodological core is real: the RDM sampling in real-space NNQMC is a concrete, useful addition, and the demonstration that neural network wavefunctions capture TLL critical behavior in an ab initio 1D system is genuinely new. Second, the headline physical claim—a TLL-to-Fermi-liquid transition at R < 1.6 a_B—is under-supported as presented.\n\nWhat the paper does well: energy benchmarks are solid, within 1 mEh of AFQMC/MRCI+Q, and the TLL signatures at large R are consistent with prior DMRG and VMC work. The critical exponent η ≈ 1.16(5) at 2.8 a_B and the Kσ ≈ 1.07 value are sensible. The self-citation of method papers (refs. 13–15, 27) is not a problem; those methods are published and independently used.\n\nWhere the soft spots are, in proportion: the Fermi-liquid identification rests almost entirely on Fig. 3(d-f), computed for a single system size (N=24, PBC), with no finite-size scaling and no error bars on Sσ(q) or n(k). The stress-test concern is on target: in a small 1D system, a TLL with a large Luttinger parameter can have a sharply peaked n(k) that is still continuous, and the q-peak shift away from π could be a discrete-k mesh artifact (spacing 2π/24). The claim needs at least a second system size, ideally a finite-size scaling of the apparent jump and peak position, plus statistical error bars. Separately, the self-doping mechanism is quantified in Fig. 4(d) only below about 1.0 a_B, while the transition is claimed at 1.6 a_B; that leaves the mechanism unquantified across the transition. The referenced SI is not included, which limits reproducibility. These are addressable gaps, not fatal flaws. The TLL results and the method developments are likely to survive scrutiny.\n\nWho gets value from this: anyone working on neural network quantum Monte Carlo, or on 1D correlated solids, will find the RDM sampling and the TLL demonstration worth engaging with. The Fermi-liquid claim is the part to be skeptical about.\n\nMy recommendation: send it to peer review. The method advance and the TLL results deserve referee time, and the Fermi-liquid claim can be sorted out with requested revisions. A editor should not desk-reject this, but a referee should push hard for finite-size and error-bar evidence.","headline":"A solid NNQMC methods paper with credible TLL results, but the Fermi-liquid transition claim rests on single-size evidence and needs harder support before it carries weight.","tokens_in":810,"tokens_out":727,"would_cite":true,"duration_ms":23899,"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":"Neural-network wavefunctions capture a quantum phase flip in hydrogen chains.","keywords":["neural network quantum Monte Carlo","hydrogen chain","Tomonaga-Luttinger liquid","Fermi liquid transition","self-doping","reduced density matrices","spin structure factor","momentum distribution"],"falsifier":"Recompute the periodic-chain observables for $N = 48$ and $N = 96$: if the spin-structure-factor peak returns to $q = \\pi$ and the momentum-distribution step at $k = \\pi/2$ flattens as the system grows, the claimed Fermi-liquid transition would be a finite-size artifact rather than a bulk phase. The decisive check is whether the discontinuity in $n(k)$ at $k = \\pi/2$, extrapolated to the thermodynamic limit, is nonzero.","tokens_in":9901,"feed_emoji":"🧠","tokens_out":9660,"duration_ms":80091,"temperature":0.7,"pith_summary":"The paper sets out to show that real-space neural network quantum Monte Carlo can probe quantum critical phases, not just ground-state energies, in a realistic one-dimensional system: the hydrogen chain. Using neural-network trial wavefunctions trained by the variational principle, the authors compute reduced density matrices, spin and charge structure factors, and momentum distributions across a wide range of interatomic distances. They report that chains longer than about 1.6 Bohr radii behave as a Tomonaga-Luttinger liquid, with the predicted gapless spin mode and algebraic spin correlations, and that below that distance the system crosses over to a Fermi-liquid-like phase characterized by a shifted spin-structure-factor peak and a sharper momentum distribution. The driving mechanism they identify is self-doping: electrons spill out of hydrogen 1s orbitals into higher bands, so minimal one-band descriptions break down. If this picture holds, neural-network wavefunctions become a practical tool for locating quantum phase transitions in real materials without basis-set extrapolations.","feed_headline":"Hydrogen chains switch quantum phase below 1.6 Bohr, neural net shows","feed_subtitle":"Fully correlated neural-network wavefunctions reveal when one-dimensional quantum liquids break down in hydrogen chains.","key_machinery":"The load-bearing object is the real-space neural-network wavefunction — a Slater determinant of neural-network orbitals (FermiNet form for open chains, DeepSolid's generalized Bloch form for periodic chains) — optimized by variational Monte Carlo. The argument then runs through one- and two-particle reduced density matrices sampled from that wavefunction, which yield natural-orbital occupations, spin-spin correlations, the spin structure factor $S_\\sigma(q)$, and the momentum distribution $n(k)$. The quantitative diagnostic is the low-$q$ slope of $S_\\sigma(q)$, which fixes the Luttinger parameter $K_\\sigma$ through $\\lim_{q\\to 0} S_\\sigma(q)/q = K_\\sigma/\\pi$, together with the peak position of $S_\\sigma(q)$ and the sharpness of $n(k)$ near $k = \\pi/2$. The mechanism offered for the transition is self-doping, read off from orbital-occupation data and Wannier orbitals.","core_discovery":"The authors claim that a fully correlated real-space neural-network wavefunction supplies the correct quantum-critical physics of the one-dimensional hydrogen chain, and that this is the first time neural networks capture Tomonaga-Luttinger liquid behavior. At interatomic distances $R > 1.6 a_{\\rm B}$, the spin-spin correlation decays algebraically with exponent $\\eta \\approx 1.16$, the spin structure factor peaks at $q = \\pi$ and disperses linearly near $q = 0$ with a Luttinger parameter $K_\\sigma \\approx 1.07$, and the momentum distribution is continuous at $k = \\pi/2$ — the signatures of a Tomonaga-Luttinger liquid. At $R < 1.6 a_{\\rm B}$, the spin-structure-factor peak shifts away from $q = \\pi$, a kink appears at the same wave vector in the charge structure factor, and $n(k)$ sharpens at $k = \\pi/2$ while building a peak above unity near $k = 0$; the authors interpret these abrupt changes as a breakdown of the Tomonaga-Luttinger liquid and the emergence of a Fermi-liquid-like phase. They connect this transition to self-doping, in which electrons occupy orbitals above 1s when atoms approach within about one Bohr radius, and support it with occupation numbers and Wannier orbitals sampled from the wavefunction.","pith_inferences":["A natural next calculation is to track the momentum-distribution step at $k = \\pi/2$ and the $S_\\sigma(q)$ peak shift with increasing cell size, which would separate the bulk Fermi-liquid transition from finite-size effects in the one-size ($N = 24$) periodic calculation.","The same reduced density matrix protocol could be applied to other quasi-one-dimensional materials, such as doped chains or nanotubes, where Tomonaga-Luttinger liquid behavior coexists with multi-band physics.","The authors do not directly resolve spin-charge separation; extracting the spectral function, for example through transition reduced density matrices as they suggest, would give a sharper test of whether the low-energy excitations are spinons and holons in the Tomonaga-Luttinger liquid regime."],"forward_implications":["For interatomic distances above about 1.6 Bohr radii, hydrogen chains exhibit Tomonaga-Luttinger liquid behavior: algebraic spin correlations with $\\eta \\approx 1.16$, a linear spin structure factor at small $q$ with $K_\\sigma \\approx 1.07$, and a gapped charge mode.","Below about 1.6 Bohr radii, the same chains show Fermi-liquid-like signatures—the spin structure factor peak leaving $q = \\pi$, a kink in the charge structure factor, and a sharper momentum distribution at $k = \\pi/2$—marking a quantum phase transition driven by self-doping.","One-band lattice models based only on the hydrogen 1s orbital are inadequate for the small-distance regime, where electrons occupy higher orbitals and interactions extend beyond nearest neighbors.","The reduced density matrix sampling framework can be carried over to other observables, including spectral densities and transition reduced density matrices relevant to photoemission, within the same real-space neural-network ansatz.","Neural-network quantum Monte Carlo can therefore serve as a probe of quantum critical phases and transitions in realistic correlated systems without relying on basis-set extrapolation."],"supporting_citations":[{"why":"Supplies the state-of-the-art benchmark energies against which the neural-network hydrogen-chain energies are validated.","marker":"[17]"},{"why":"Gives reference DMRG results for hydrogen-chain dimerization, insulator-to-metal transition, magnetic phases, and the $\\eta \\approx 1.11$ exponent this work compares with.","marker":"[4]"},{"why":"Introduces the FermiNet neural-network ansatz used for open chains in this work.","marker":"[11]"},{"why":"Introduces the DeepSolid periodic neural-network ansatz used for the periodic calculations.","marker":"[13]"},{"why":"Supplies the relation between the small-$q$ spin structure factor slope and the Luttinger parameter $K_\\sigma$ used to identify the gapless spin mode.","marker":"[21]"},{"why":"Provides the conformal-field-theory prediction $\\eta = 1$ for spin correlations of the Heisenberg chain, the comparison point for the fitted exponent.","marker":"[20]"},{"why":"Establishes the critical exponents of the one-band Hubbard model, the Tomonaga-Luttinger liquid reference that large-distance hydrogen chains are expected to match.","marker":"[22]"},{"why":"Reports a previous variational Monte Carlo study of strong correlation in hydrogen chains whose conclusions about small-distance behavior are contrasted here.","marker":"[8]"},{"why":"Documents a known breakdown of Luttinger liquid behavior in an extended one-dimensional model, cited as precedent that Tomonaga-Luttinger liquid universality is not guaranteed.","marker":"[2]"}],"fun_headline_variants":["Neural wavefunction captures hydrogen chain quantum criticality","Hydrogen chain quantum phase shift seen by neural network","First neural-net capture of Tomonaga-Luttinger critical behavior","Neural network wavefunction maps hydrogen chain phase transition"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The Fermi-liquid identification rests on the assumption that the shifted spin-structure-factor peak and the sharpened momentum distribution seen at one periodic system size ($N = 24$, no finite-size extrapolation) are genuine Fermi-surface signatures rather than finite-size effects, an incommensurate spin-density wave, or a Tomonaga-Luttinger liquid with a large Luttinger parameter.","fun_headline_variants_meta":{"raw":{"variants":["Neural wavefunction captures hydrogen chain quantum criticality","Hydrogen chain quantum phase shift seen by neural network","First neural-net capture of Tomonaga-Luttinger critical behavior","Neural network wavefunction maps hydrogen chain phase transition"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000781,"raw_usage":{"total_tokens":3479,"prompt_tokens":1004,"completion_tokens":2475,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":620,"completion_tokens_details":{"reasoning_tokens":2409}},"tokens_in":620,"tokens_out":2475,"duration_ms":17512,"temperature":1.0,"reasoning_tokens":2409,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T05:40:45.382013+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the periodic-chain observables for $N = 48$ and $N = 96$: if the spin-structure-factor peak returns to $q = \\pi$ and the momentum-distribution step at $k = \\pi/2$ flattens as the system grows, the claimed Fermi-liquid transition would be a finite-size artifact rather than a bulk phase. The decisive check is whether the discontinuity in $n(k)$ at $k = \\pi/2$, extrapolated to the thermodynamic limit, is nonzero.","supporting_citations":[{"cited_title":"Electric polariza- tion from a many-body neural network ansatz","cited_arxiv_id":null,"evidence_quote":"Supplies the state-of-the-art benchmark energies against which the neural-network hydrogen-chain energies are validated."},{"cited_title":"The conventional understanding of the hy- drogenchainisbasedontheone-bandmodels, whereonly the hydrogen 1s orbital is considered [20, 22, 23]","cited_arxiv_id":null,"evidence_quote":"Gives reference DMRG results for hydrogen-chain dimerization, insulator-to-metal transition, magnetic phases, and the $\\eta \\approx 1.11$ exponent this work compares with."},{"cited_title":"Sinitskiy, Loren Greenman, and David A","cited_arxiv_id":null,"evidence_quote":"Introduces the FermiNet neural-network ansatz used for open chains in this work."},{"cited_title":"Spencer, Alexander G","cited_arxiv_id":null,"evidence_quote":"Introduces the DeepSolid periodic neural-network ansatz used for the periodic calculations."},{"cited_title":"Kądzielawa, and Józef Spałek","cited_arxiv_id":null,"evidence_quote":"Supplies the relation between the small-$q$ spin structure factor slope and the Luttinger parameter $K_\\sigma$ used to identify the gapless spin mode."},{"cited_title":"Kądzielawa, Andrzej Biborski, and Józef Spałek","cited_arxiv_id":null,"evidence_quote":"Provides the conformal-field-theory prediction $\\eta = 1$ for spin correlations of the Heisenberg chain, the comparison point for the fitted exponent."},{"cited_title":"Critical behaviour of spin-s Heisenberg antiferromagnetic chains: Analytic and numerical results","cited_arxiv_id":null,"evidence_quote":"Establishes the critical exponents of the one-band Hubbard model, the Tomonaga-Luttinger liquid reference that large-distance hydrogen chains are expected to match."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reports a previous variational Monte Carlo study of strong correlation in hydrogen chains whose conclusions about small-distance behavior are contrasted here."},{"cited_title":"BreakdownofLut- tinger liquid state in a one-dimensional frustrated spin- less fermion model","cited_arxiv_id":null,"evidence_quote":"Documents a known breakdown of Luttinger liquid behavior in an extended one-dimensional model, cited as precedent that Tomonaga-Luttinger liquid universality is not guaranteed."}],"review_version":1}