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Universal momentum tail of identical one-dimensional anyons with two-body interactions

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Pith's one-line read For one-dimensional anyons with zero-range interactions, the large-momentum tail is fixed, through order $k^{-3}$, by the statistical parameter and the two- and three-body Tan contacts, and it distinguishes bosonic from fermionic anyons.

desk verdict New universal momentum-tail formulas for 1D anyons, with a solid derivation and exact few-body checks; the one load-bearing gap is the unproved N-body eigenstate claim. read the letter →

arxiv 2505.17669 v1 pith:5OOJGEKJ submitted 2025-05-23 cond-mat.quant-gas

Raúl Hidalgo-Sacoto , Thomas Busch , D. Blume This is my paper · ORCID
classification cond-mat.quant-gas PACS 05.30.Pr03.75.Hh
keywords one-dimensionalanyonsmomentumdistributiontailTancontactszero-rangeinteractionsfractionalstatisticsBose-Fermimappingbosonic-anyon-fermionic-anyonuniversalrelations
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

For systems of $N$ identical one-dimensional anyons with zero-range two-body interactions, this paper claims that the large-momentum tail of the momentum distribution is universal: through order $k^{-3}$ it is fixed by the statistical parameter $\alpha$, the anyonic scattering length $a_{\mathrm{any}}$, and the two- and three-body Tan contacts $C_2$ and $C_3$. The same contacts appear for ordinary bosons and fermions, so the statistics enter only through $\alpha$-dependent prefactors. The $k^{-3}$ term is chiral and differs between bosonic anyons and fermionic anyons, making the two species distinguishable by momentum-resolved measurements. The payoff, if the claim is right, is a concrete universal observable: a momentum-resolved measurement of an anyonic gas would simultaneously expose the statistical phase and the two- and three-body contacts.

What carries the argument

The load-bearing object is the statistical dressing operator $\hat{S}^\dagger_{\alpha/2}$, the product over all pairs of the two-particle gauge phase $\hat{S}_{\alpha/2}(z_{jk})=\exp[-i(\pi\alpha/2)\operatorname{sign}(z_{jk})]$ times a unit-modulus normalization, applied to bosonic or fermionic wavefunctions $\Psi_{\pm}$ that are themselves connected by the Bose-Fermi mapping. Dressing turns the even-parity or odd-parity short-distance boundary conditions into superpositions of both parities, so the resulting anyonic wavefunctions obey the boundary conditions of the Hamiltonian containing both the delta and the derivative-delta pseudopotentials with $a_+=a_-=a_{\mathrm{any}}$. The tail calculation then exploits standard Fourier facts about jump discontinuities: the $\operatorname{sign}(z)$ and $|z|$ pieces of the two-body short-distance behavior contribute at $k^{-2}$ and in interference at $k^{-3}$, while three-body coincidence discontinuities contribute at $k^{-3}$ through the three-body contact $C_3$.

What would settle it

Compute or measure the exact momentum distribution of four identical anyons with a fixed $a_{\mathrm{any}}$ and compare the coefficient of $k^{-3}$ with $4C_2 a_{\mathrm{any}}^{-1}\sin(\pi\alpha)+8C_3\sin^2(\pi\alpha/2)\sin(\pi\alpha)$, with $C_2$ and $C_3$ obtained from the coincidence densities of the same system. Any mismatch at order $k^{-3}$ for $0<\alpha<1$, or any nonzero $k^{-3}$ tail at $\alpha=0$ or $\alpha=1$, would falsify the universal tail formula.

Watch

Extended reading notes

Core claim

The paper's central claim is that the wavefunctions $\Psi_{\alpha,\pm}=\hat{S}^\dagger_{\alpha/2}\Psi_{\pm}$, obtained by dressing Bose-Fermi-mapped wavefunctions with a statistical gauge phase, are eigenstates of a Hamiltonian with additive even- and odd-parity zero-range interactions when $a_+=a_-=a_{\mathrm{any}}$, and that their momentum distributions satisfy $n_{\alpha,+}(k)=\frac{4C_2}{k^2}\sin^2(\pi\alpha/2)+\frac{4C_2}{a_{\mathrm{any}}k^3}\sin(\pi\alpha)+\frac{8C_3}{k^3}\sin^2(\pi\alpha/2)\sin(\pi\alpha)+O(k^{-4})$ and the companion result $n_{\alpha,-}(k)=\frac{4C_2}{k^2}\cos^2(\pi\alpha/2)-\frac{4C_2}{a_{\mathrm{any}}k^3}\sin(\pi\alpha)-\frac{8C_3}{k^3}\cos^2(\pi\alpha/2)\sin(\pi\alpha)+O(k^{-4})$. Here $C_2$ and $C_3$ are the statistics-independent two- and three-body Tan contacts, defined through coincidence densities. The $k^{-3}$ term is nonzero only for $0<\alpha<1$ and finite $a_{\mathrm{any}}$, and it changes sign or angular structure between bosonic and fermionic anyons. The paper also establishes a bosonic-anyon-fermionic-anyon mapping that reduces to the Bose-Fermi mapping at $\alpha=0$ and to its dual complement at $\alpha=1$.

Load-bearing premise

The load-bearing premise is that the gauge-dressed wavefunctions $\Psi_{\alpha,\pm}=\hat{S}^\dagger_{\alpha/2}\Psi_{\pm}$ are genuine eigenstates of the two-body-interaction Hamiltonian for every $N$; the paper states this follows by the same logic as the Bose-Fermi mapping but does not explicitly demonstrate the cancellation of the kinetic energy acting on the statistical sign discontinuities.

Editorial extensions

If this is right

  • If the eigenstate construction holds for all $N$, then the momentum tail of any one-dimensional anyonic gas, trapped or free and few- or many-body, is fixed through order $k^{-3}$ by just $\alpha$, $a_{\mathrm{any}}$, and $C_2$ and $C_3$.
  • A measurement resolving the $k^{-3}$ tail would identify the anyon species and the statistical parameter, because the chiral prefactor differs between bosonic and fermionic anyons and vanishes at $\alpha=0$ and $\alpha=1$.
  • The formulas reduce to the known bosonic and fermionic tails at the endpoints, and the anyonic duality exchanges the two species, so the result unifies the previously separate boson and fermion universal relations.
  • The presence of a $k^{-3}$ tail is a direct fingerprint of fractional statistics combined with finite zero-range interactions, rather than of either ingredient alone.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the tail formula survives checks beyond the two- and three-body examples in the paper, the $k^{-3}$ coefficient would give a practical route to extracting the three-body contact $C_3$ from momentum-resolved images, without relying on three-body loss measurements.
  • The scaling table in the supplemental material hints at a general hierarchy in which $M$-body coincidence discontinuities enter at $k^{-M}$; one testable extension is that the $k^{-4}$ term mixes a four-body contact with subleading two- and three-body information for $N\ge4$.
  • Because the $k^{-3}$ contribution is chiral, the asymmetry of $n_{\alpha,\pm}(k)$ under $k\to -k$ is a direct observable for the statistics-induced chiral symmetry breaking the paper invokes, for instance in expansion or time-of-flight experiments.
  • The paper notes that its $\alpha=1/2$ case connects to spin-orbit-coupled Fermi gases, so existing spin-orbit-coupled systems could serve as testbeds for the predicted $k^{-3}$ tail before dedicated anyon experiments become routine.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper constructs two classes of one-dimensional anyonic wavefunctions by multiplying ordinary bosonic or fermionic wavefunctions by a statistical gauge factor built from sign functions, proposes a many-body Hamiltonian with additive even- and odd-parity zero-range two-body interactions that is claimed to support these anyonic eigenstates, and derives universal large-momentum tails of the momentum distributions in terms of the statistics-independent two- and three-body Tan contacts C2 and C3. The central results are Eqs. (21) and (22), which give k^-2 and k^-3 prefactors for bosonic anyons and fermionic anyons that differ in a characteristic way. The Supplemental Material contains a step-by-step Fourier derivation of the tails and validates the formulas against exact two- and three-anyon momentum distributions in free space.

Significance. If the central eigenstate claim is correct, the paper offers a clean unification of the 1D Bose-Fermi mapping with anyonic statistics and makes falsifiable predictions: the momentum tails of bosonic and fermionic anyons differ at order k^-3, while the Tan contacts are statistics-independent. The derivation is essentially parameter-free, and the exact few-body checks in Appendix D are a genuine strength. The main weakness is that the N-body eigenstate statement is asserted rather than proved; this is load-bearing for the claimed universality of Eqs. (21) and (22) for arbitrary N.

major comments (2)
  1. [Appendix D] The proof that the dressed states Ψ_{α,±}=S†_{α/2}Ψ± are eigenstates of H is incomplete. Equation (16) verifies the two-body boundary condition for one pair at a time with all other coordinates fixed, but it does not show that (H-E)Ψ_{α,±} vanishes as a distribution on R^N. In particular, the kinetic energy acting on the product of sign functions in S† generates codimension-two terms supported at triple coincidences, such as δ(z_jk)δ(z_jl), which are not ruled out by the displayed argument and which cannot be canceled by the two-body pseudopotentials in Eq. (15). The text says this follows 'using the same logic' as the Bose-Fermi mapping, but no such N-body distributional identity is provided. The authors should either supply this proof or reformulate the claim, e.g., by defining the Hamiltonian through boundary conditions on the cut configuration space and demonstrating that the dressed wavefunctions lie in the corresponding domain.
  2. [Section C] The two- and three-body validations confirm the Fourier algebra and the contact definitions, but they do not independently confirm the eigenstate property for N≥3. The three-body check computes the momentum distribution of the dressed wavefunction and expands its tail; if the dressed wavefunction were not an eigenstate of H, that comparison would still be internally consistent. Thus the exact N=3 result does not supply the missing distributional proof for the Hamiltonian in Eq. (15), and the universality claim for arbitrary N rests on the unproven Section C assertion.
minor comments (4)
  1. [Section C / Appendix B] The operator defined in Eq. (6) includes a factor Nα, but Eq. (16) and Eq. (B4) write S†_{α/2}(z) simply as cos(πα/2)+i sin(πα/2) sign(z). This is not an identity as written (for example, at α=1 the factor Nα contributes a factor -i). Since |Nα|=1, the final momentum tails are unaffected, but the notation should be clarified and the equations corrected.
  2. [Fig. 1] The caption says 'three identical free-space bosonic anyons and fermionic anyons', but the distributions shown are for three bound anyons (Appendix D). Please correct the caption to say 'bound' rather than 'free-space'.
  3. [Abstract] The abstract mentions 'statistics-induced chiral symmetry breaking', but this concept is not defined or discussed in the main text. Please define it explicitly or remove the phrase.
  4. [Section A / Appendix A] The sign function is said to be undefined at z=0, yet products such as sign²(z)=1 are used at z=0 and densities are evaluated at coincident particles. Please state the convention used for the value of sign at zero when evaluating such expressions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the momentum-tail formulas are derived from the stated short-distance boundary conditions and gauge phase, not assumed or fitted; the only concern is an omitted N-body eigenstate proof in Section C, which is a correctness gap rather than circularity.

full rationale

The paper's central results, Eqs. (21) and (22), are not circular. The anyonic wavefunctions Ψα,± are defined by applying the gauge operator S†_{α/2} to Ψ±, and the tails are obtained by Fourier-transforming the resulting short-distance factors h^{(2)}_{α,±} and h^{(3)}_{α,±}. The coefficients are expressed in terms of C2 and C3, which are defined independently via g2 and g3, not fitted to the momentum distribution. Because |S†_{α/2}|=1, C2 and C3 coincide with the underlying boson/fermion contacts, a fact the paper uses consistently. No parameter is fitted to data and no load-bearing self-citation closes the argument; the Bose-Fermi mapping [19] and previous Tan-contact results are external, standard results. The N=2 and N=3 free-space examples are genuine checks: the full momentum distributions are computed and their large-k expansions reproduce the general formulas with the separately computed C2 and C3. The only notable weakness is Section C's assertion that Ψα,± are eigenstates of H = T + ΣV+ + ΣV− 'using the same logic as was used when deriving the Bose-Fermi mapping'; the paper verifies only the two-particle boundary condition (Eq. (16)) and does not display a distributional proof that kinetic-energy terms at triple-coincidence surfaces are canceled. This is an omitted proof, not evidence of circularity; it affects rigor, not the self-containedness of the tail derivation conditional on the eigenstate assumption.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim relies on standard Fourier asymptotics, on the model assumption that zero-range interactions with equal even- and odd-parity scattering lengths support both types of anyonic eigenstates, and on the leading-order smoothness of the three-body short-distance function. No numerical parameters are fitted; a_any and the Tan contacts are inputs or outputs of the model, and alpha is the statistical parameter.

assumptions (3)
  • standard math The tail of the momentum distribution is controlled by jump discontinuities of the wavefunction and its derivatives, with separated discontinuities contributing additively.
    Used throughout Section D and SM Appendices A and B; based on Fourier transform asymptotics of Ref. [55].
  • domain assumption The gauge-dressed wavefunctions Psi_{alpha,pm} = S-dagger_{alpha/2} Psi_pm solve H with a+ = a- = a_any.
    Stated as readily shown in Section C, Eq. (15); no explicit N-body proof is given for cancellation of kinetic terms acting on sign-function discontinuities.
  • domain assumption The leading-order three-body short-distance function h^(3)_+ equals 1, with no jump discontinuity.
    SM Appendix B, used to evaluate T^(3); justified by exchange symmetry and the absence of explicit three-body forces.

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Cite this review

Pith. "Pith review of Universal momentum tail of identical one-dimensional anyons with two-body interactions." pith.science (2026). https://pith.science/paper/5OOJGEKJ

@misc{pith2026250517669,
  author       = {Pith},
  title        = {Pith review of: Universal momentum tail of identical one-dimensional anyons with two-body interactions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5OOJGEKJ}},
  note         = {Machine review of arXiv:2505.17669}
}
abstract

Non-relativistic anyons in 1D possess generalized exchange statistics in which the exchange of two identical anyons generates a non-local phase that is governed by the spatial ordering of the particles and the statistical parameter $\alpha$. Working in the continuum, we demonstrate the existence of two distinct types of 1D anyons, namely bosonic anyons and fermionic anyons. We identify a many-body Hamiltonian with additive two-body zero-range interactions that supports bosonic and fermionic anyon eigenstates, which are, for arbitrary interaction strength, related through a generalized bosonic-anyon--fermionic-anyon mapping, an extension of the celebrated Bose-Fermi mapping for zero-range interacting 1D systems. The momentum distributions of bosonic and fermionic anyons are distinct: while both feature $k^{-2}$ and $k^{-3}$ tails, the associated prefactors differ. Our work reveals intricate connections between the generalized exchange statistics, the universal two- and three-body Tan contacts of systems consisting of $N$ identical particles, and the emergence of statistics-induced chiral symmetry breaking.

Figures

Figures reproduced from arXiv: 2505.17669 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Bosonic-anyon — fermionic-anyon mapping that [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Analysis of momentum distribution [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

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Forward citations

Cited by 2 Pith papers

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  1. Efficient determination of eigenenergies and eigenstates of $N$ ($N=3$--$4$) identical 1D bosons and fermions under external harmonic confinement

    cond-mat.quant-gas 2025-09 accept novelty 7.0 of 10

    A Lippmann-Schwinger based scheme with Hermite recursion efficiently computes converged spectra for three/four 1D bosons and three 1D fermions with zero-range interactions, validated by Bose-Fermi mapping and free-spa...

  2. Two identical 1D anyons with zero-range interactions: Exchange statistics, scattering theory, and anyon-anyon mapping

    quant-ph 2025-05 conditional novelty 6.0 of 10

    Develops a scattering framework for two 1D anyons with zero-range interactions, derives their momentum distribution tails to order k^{-4}, and confirms a bosonic-anyon to fermionic-anyon mapping.

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