{"id":"0f2af023-2a40-4d90-9eda-3ec4b2bde31e","arxiv_id":"2510.25056","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Long-time momentum distributions after trap release are identical for 1D bosons, fermions, and anyons with the same scattering length, and equal the initial quasi-momentum distribution.","lead":"This paper proves that one-dimensional identical particles with any statistics (bosons, fermions, or anyons) and the same scattering length share the same momentum distribution long after being released from a trap. This universal distribution is set by the initial state's quasi-momenta, generalizing the known dynamical fermionization of hard-core bosons.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the derivation is internally consistent and the completeness input is standard, so I would not change the ACCEPT verdict.","rationale":"The reader's verdict is ACCEPT with high confidence and low correctness risk. My stress-test focused on the single most delicate assumption, the completeness of the free-space Bethe-ansatz basis for arbitrary statistics at fixed negative scattering length. That assumption is indeed the natural place to probe, and the reader correctly identifies it as the weakest point. On inspection, however, the assumption is well supported: the phase-factor mapping (3) is a bijection between bosonic and α-symmetric Hilbert spaces, so the completeness of the α BA basis is inherited from the standard Lieb-Liniger completeness for α=0. The derivation of Eq. (16) is internally consistent: the A coefficients are pure phases, the theta functions select exactly one ordering sector in momentum space, and the α-dependent phase in Eq. (15) becomes a global phase that cancels in the momentum distribution. The SPA steps are the same as in the established dynamical-fermionization literature. The numerical confirmation for two and three particles, although restricted to one coupling value, is consistent with the claim. No correction to the reader's verdict is needed.","tokens_in":9720,"tokens_out":29980,"duration_ms":300251,"concrete_test":"Run an independent completeness check for N=3, α=π/2: instead of using the BA expansion, solve the trapped initial state exactly and time-evolve the released wavefunction on a real-space grid with a regularized zero-range potential implementing the contact condition (2), then compare the long-time n(k) with the predicted n_q(k). Agreement to within SPA numerics would confirm that no missing BA states contribute; systematic disagreement would indicate that the assumed completeness of the anyonic BA basis fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I find no load-bearing objection. The central claim rests on three steps: (i) expansion of the trapped α-statistics state in free-space BA states (Eq. 10); (ii) α-independence of the coefficients c(k) (Eq. 11); and (iii) the stationary-phase reduction to n_q(k) (Eqs. 14-16). The most delicate input is the completeness of the BA basis for fractional α at fixed l, which the reader also identifies. Although the paper does not re-prove completeness, it follows from the standard bosonic Lieb-Liniger completeness via the unitary phase mapping in Eq. (3): the phase factor is constant on each ordering sector, so it maps symmetric functions bijectively onto α-symmetric functions and maps bosonic BA eigenstates to the α BA states of Eq. (6) with the same quasi-momenta. The α-dependent periodic condition (8) is exactly the braiding phase that cancels when deriving Eq. (9), so the spectrum and basis are α-independent. The remaining subtleties—the pure-phase factor A in Eq. (7) and the ordering phase in Eq. (15)—drop out of |Ψ|², giving Eq. (16). The numerical results for N=2 and N=3 support the derivation. I therefore see no internal inconsistency or unstated assumption beyond those already standard in the cited dynamical-fermionization literature.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims a generalized dynamical duality for 1D identical particles with arbitrary statistics α (bosons, fermions, anyons) and a common negative scattering length l. It proves that after a sudden release from a trap, the long-time one-body momentum distribution is independent of α and equals the quasi-momentum distribution of the initial trapped state: n^(α)(k,t→∞)=n_q(k,t=0). The proof expands the initial state in free-space Bethe ansatz states, shows that the expansion coefficients are α-independent because the initial state and the basis share the same ordering-phase structure, and then applies stationary-phase approximation to obtain the asymptotic momentum distribution. Exact numerics for N=2 and N=3 particles at l=-l_T support the prediction, and the effect of a finite p-wave effective range is discussed.","tokens_in":10107,"tokens_out":15895,"duration_ms":153763,"significance":"If the result holds, it substantially extends the known phenomenon of dynamical fermionization to arbitrary statistics and arbitrary (negative) scattering length, unifying equilibrium and dynamical dualities in 1D. The derivation is transparent and parameter-free, and the prediction is falsifiable in quasi-1D ultracold gases with tunable s- and p-wave interactions. The manuscript includes exact small-cluster numerical verification and a discussion of experimental feasibility, which strengthens the claim.","major_comments":[],"minor_comments":[{"comment":"The expansion of the initial state in the free-space Bethe ansatz basis Φ_k^(α) presupposes that these states form a complete basis for arbitrary α at fixed l<0. This is the only non-elementary input of the proof and is not stated explicitly. Please add a sentence (or a citation, e.g., to Ref. [17] together with the unitary phase mapping in Eq. (3)) justifying completeness; for α=0 this is the standard Lieb-Liniger completeness, and the phase mapping extends it to other α.","section":"Main text, Eq. (10)"},{"comment":"The stationary-phase expressions omit overall normalization and phase prefactors that depend on t. Since the final momentum distribution is obtained from |Ψ|^2, the omission does not affect the result, but the paper should state explicitly that Eqs. (14) and (15) hold up to an overall t-dependent normalization factor.","section":"Main text, Eqs. (14) and (15)"},{"comment":"There is a typo in the abstract: \"uniqued given\" should be \"uniquely given\". The same typo appears in the closing paragraph of the main text.","section":"Abstract"},{"comment":"The caption would be clearer if it explicitly stated that the gray curve is the same for all α (as predicted by Eq. (16)) and that the colored momentum-distribution curves at t=10 overlap with it, rather than relying on visual inspection.","section":"Figure 1 caption"},{"comment":"The statement \"Eqs.(10-16) remain unchanged\" for finite r_p could be misunderstood: the Bethe ansatz states themselves are modified by the r_p-dependent equation (S15). What remains unchanged is the structure of the stationary-phase argument leading from the expansion to the asymptotic momentum distribution. Please clarify this wording.","section":"Main text, finite p-wave effective range paragraph"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Chen and Cui have proven a clean generalization of dynamical fermionization: in 1D, any particles with the same short-range scattering length, regardless of exchange statistics, expand from a trap to the same long-time momentum distribution, equal to the initial quasi-momentum distribution. That is a real step beyond the known hard-core boson and anyon cases, and it unifies a lot of previous results under one mechanism. The derivation is simple in the best sense: expand in Bethe ansatz states, note the overlap coefficients are statistics-independent because the phase factors cancel, then stationary phase does the rest. The numerics for N=2 and 3 at l=-l_T agree with the prediction, and the finite-p-wave-range discussion is a useful practical addition.\n\nSoft spots are minor. The main input is completeness of the BA basis for fractional statistics at fixed scattering length; the paper takes this from the equilibrium duality of Ref [3] rather than proving it. That is fine — the phase mapping is unitary, so completeness transfers from the bosonic Lieb-Liniger case — but a referee should make sure the citation actually covers the continuum case. Also, the boundary condition in Eq. (2) is written for Ψ(α) itself, while the construction in Eq. (3) makes clear it applies to the common spatial-ordering function; this could confuse readers. The numerics are small clusters at one scattering length, but they are exact and the figure shows the convergence clearly. I do not see a load-bearing flaw.\n\nThe paper is for people working on 1D gases, exact integrability, and dynamical fermionization. It will likely be a standard reference for the generalized duality. I would accept it after minor revisions; the main thing is to clarify the boundary condition and maybe add one sentence on completeness.","headline":"Clean generalization of dynamical fermionization to arbitrary statistics; the central argument holds and the paper deserves refereeing.","tokens_in":10473,"tokens_out":4608,"would_cite":true,"duration_ms":43295,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The late-time momentum distribution of one-dimensional particles is independent of exchange statistics, and equals the initial quasi-momentum distribution.","keywords":["dynamical duality","dynamical fermionization","Bethe ansatz","one-dimensional anyons","momentum distribution","quasi-momentum distribution","scattering length","ultracold atomic gases"],"falsifier":"Take two or three particles in a harmonic trap at fixed $l=-l_T$, prepare the same trapped ground state as bosons ($\\alpha=0$) and as anyons ($\\alpha=\\pi/2$), and measure the momentum distribution after an expansion time $t\\sim 10/\\omega$. If the two late-time curves differ by more than the numerical convergence error, or if either fails to match the quasi-momentum distribution $n_q(k)$ computed from Eq. (9), the generalized dynamical duality is false. A sharper test is to check completeness directly: expand the trapped ground state in the Bethe-ansatz basis (Eq. 6) with a large cutoff and verify that the squared overlaps sum to 1 independently of $\\alpha$; any residual $\\alpha$-dependence in the overlap sum would break Eq. (16).","tokens_in":9534,"feed_emoji":"⚛️","tokens_out":10211,"duration_ms":78482,"temperature":0.7,"pith_summary":"This paper proves that in one dimension, identical particles with different exchange statistics—bosons, fermions, and anyons—end up with the same one-particle momentum distribution after a long free expansion from a trap, as long as their short-range interactions share the same scattering length $l$. The asymptotic distribution is not a thermal or interaction-renormalized shape; it is exactly the quasi-momentum distribution of the initial trapped state, the conserved rapidity content of the integrable system. This turns the known phenomenon of dynamical fermionization (hard-core bosons becoming indistinguishable from free fermions in momentum space) into one special case, $l=0$, of a general duality that holds for any coupling and any statistics. The proof works by expanding the initial state in a statistics-independent Bethe-ansatz basis and applying stationary phase to the long-time wavefunction. A sympathetic reader should care because it identifies a direct observable—the late-time momentum distribution—that can certify the generalized equilibrium duality in real ultracold-gas experiments.","feed_headline":"All statistics converge to one momentum curve after expansion","feed_subtitle":"Bose, Fermi, and anyonic gases share the same late-time spread, set by the initial quasi-momenta.","key_machinery":"The machinery is the Bethe-ansatz (BA) basis for free 1D particles with a contact interaction, Eq. (6): each eigenstate is a sum over spatial orderings with a statistics phase $e^{i\\alpha\\Lambda/2}$, and the same function $\\phi_{\\vec k}$ in every ordering. The BA equation (9), $e^{ik_jL}=\\prod_{j\\ne l}(k_j-k_l-2i/l)/(k_j-k_l+2i/l)$, is universal in $\\alpha$, so the quasi-momenta and energies are shared by bosons, fermions, and anyons at fixed $l$. The initial trapped state is expanded in this basis (Eq. 10), and because initial state and basis have the same ordering-sector phase structure, the coefficients $c(\\vec k)$ are $\\alpha$-independent. The long-time limit is taken by stationary phase (Eq. 13), which selects $k_{Pj}=mx_{Qj}/t$ and converts the asymptotic real-space wavefunction into a momentum-space wavefunction with the same sector structure. That structure is what makes the traced one-body distribution collapse to $n_q(k,t=0)$.","core_discovery":"The central claim is Eq. (16): for $N$ identical 1D particles with any statistics $\\alpha$ and common negative scattering length $l$, the one-body momentum distribution after a long free expansion satisfies $n^{(\\alpha)}(k,t\\to\\infty)=n_q(k,t=0)$, where $n_q$ is the quasi-momentum distribution of the initial state, built from overlaps $|c(k,k_2,\\dots,k_N)|^2$ with free-space Bethe-ansatz states labeled by quasi-momenta. Because the contact boundary condition determines the same real-space wavefunction in each ordering sector for all $\\alpha$, and because the Bethe-ansatz equation is the same for all $\\alpha$, the expansion coefficients $c(\\vec k)$ do not depend on statistics. At long times, stationary phase maps each spatial sector onto a momentum sector with $\\vec{x}=\\vec{k}t/m$, so the momentum-space wavefunction inherits the same ordering structure as the real-space one; tracing out all but one momentum then leaves only the $\\alpha$-independent coefficients. The result generalizes dynamical fermionization from hard-core bosons at $l=0$ to arbitrary coupling and to fractional statistics.","pith_inferences":["The ordering-sector phase argument should carry over to lattice anyons realized in cold atoms, where a lattice counterpart of dynamical fermionization is already known; the paper mentions this as future work, but the mechanism suggests a direct construction.","By analogy with the known extensions of dynamical fermionization to spinor gases and finite temperature, the generalized duality likely holds for spin mixtures and at finite temperature; this is a conjecture beyond the present proof.","A quantitative prediction worth testing is that the approach to the asymptotic curve is governed by the spacing of the quasi-momenta: deeper traps or stronger interactions, which change that spacing, should speed up or slow down convergence to $n_q(k)$.","The same stationary-phase mechanism might extend the duality to any 1D integrable gas whose eigenstates factor into ordering sectors, including spin chains after an interaction quench, not just particles with contact interactions."],"forward_implications":["After a long expansion, the one-body momentum distribution of any 1D gas with a fixed scattering length becomes a direct readout of the initial state's quasi-momentum (rapidity) content, not of its initial momentum content.","Dynamical fermionization is the $l\\to0$ limit of this duality, so the previously separate bosonic and anyonic fermionization results are unified in one derivation.","In quasi-1D ultracold gases with tunable s- and p-wave interactions, releasing trapped bosons, fermions, and anyons with matched scattering length should produce identical asymptotic $n(k)$ curves; the paper's small-cluster numerics confirm this for two and three particles.","A finite p-wave effective range modifies the Bethe-ansatz equation and the initial quasi-momentum distribution, but leaves the identity $n(k,t\\to\\infty)=n_q(k,t=0)$ intact, so the duality remains observable under realistic experimental parameters."],"supporting_citations":[{"why":"Supplies the original hard-core-boson to noninteracting-fermion duality in one dimension, the seed case of the generalized duality.","marker":"[1]"},{"why":"Extends the duality to bosons and fermions with general contact interactions, fixing the scattering-length parametrization used here.","marker":"[2]"},{"why":"Establishes the equilibrium boson-anyon-fermion mapping with a universal Bethe-ansatz equation, the completeness input for the present construction.","marker":"[3]"},{"why":"Provides the experimental observation of dynamical fermionization that the paper generalizes to arbitrary statistics and coupling.","marker":"[6]"},{"why":"Gives the exact coherent-state description of a harmonically confined Tonks gas, the forerunner of the stationary-phase momentum mapping.","marker":"[7]"},{"why":"Shows for bosons that the long-time momentum distribution is the initial rapidity distribution, the precise statement extended here.","marker":"[8]"},{"why":"Supplies the Bethe-ansatz solution of one-dimensional gases and the form of the universal Bethe-ansatz equation.","marker":"[17]"},{"why":"Provides the stationary-phase approximation used to obtain the asymptotic wavefunction and the momentum-space limit.","marker":"[19]"}],"fun_headline_variants":["Any statistics, one momentum curve after 1D expansion","1D anyons join bosons and fermions in expansion universality","Statistics fade away in long-time 1D expansion","Momentum distribution loses statistical memory at late times","All 1D statistics share same late-time momentum profile"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The derivation assumes that the free-space Bethe-ansatz states with the universal Bethe-ansatz equation form a complete basis for every statistics $\\alpha$ at a fixed scattering length $l$; that completeness is inherited from integrability and from the equilibrium duality, but it is not re-proved in this paper.","fun_headline_variants_meta":{"raw":{"variants":["Any statistics, one momentum curve after 1D expansion","1D anyons join bosons and fermions in expansion universality","Statistics fade away in long-time 1D expansion","Momentum distribution loses statistical memory at late times","All 1D statistics share same late-time momentum profile"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000652,"raw_usage":{"total_tokens":2952,"prompt_tokens":870,"completion_tokens":2082,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":486,"completion_tokens_details":{"reasoning_tokens":2000}},"tokens_in":486,"tokens_out":2082,"duration_ms":12543,"temperature":1.0,"reasoning_tokens":2000,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:39:53.531477+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take two or three particles in a harmonic trap at fixed $l=-l_T$, prepare the same trapped ground state as bosons ($\\alpha=0$) and as anyons ($\\alpha=\\pi/2$), and measure the momentum distribution after an expansion time $t\\sim 10/\\omega$. If the two late-time curves differ by more than the numerical convergence error, or if either fails to match the quasi-momentum distribution $n_q(k)$ computed from Eq. (9), the generalized dynamical duality is false. A sharper test is to check completeness directly: expand the trapped ground state in the Bethe-ansatz basis (Eq. 6) with a large cutoff and verify that the squared overlaps sum to 1 independently of $\\alpha$; any residual $\\alpha$-dependence in the overlap sum would break Eq. (16).","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the original hard-core-boson to noninteracting-fermion duality in one dimension, the seed case of the generalized duality."},{"cited_title":"Cheon, T","cited_arxiv_id":null,"evidence_quote":"Extends the duality to bosons and fermions with general contact interactions, fixing the scattering-length parametrization used here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the equilibrium boson-anyon-fermion mapping with a universal Bethe-ansatz equation, the completeness input for the present construction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the experimental observation of dynamical fermionization that the paper generalizes to arbitrary statistics and coupling."},{"cited_title":"Minguzzi and D","cited_arxiv_id":null,"evidence_quote":"Gives the exact coherent-state description of a harmonically confined Tonks gas, the forerunner of the stationary-phase momentum mapping."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows for bosons that the long-time momentum distribution is the initial rapidity distribution, the precise statement extended here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Bethe-ansatz solution of one-dimensional gases and the form of the universal Bethe-ansatz equation."},{"cited_title":"Juki´ c, R","cited_arxiv_id":null,"evidence_quote":"Provides the stationary-phase approximation used to obtain the asymptotic wavefunction and the momentum-space limit."}],"review_version":2}