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

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Pith's one-line read Two identical one-dimensional anyons with zero-range interactions have a consistent scattering theory, and their bosonic and fermionic variants are linked by a sign-function mapping.

desk verdict New and mostly sound scattering framework for 1D zero-range anyons, but Eqs. (29) and (59) contain real localized errors that must be corrected. read the letter →

arxiv 2505.23127 v1 pith:FGFLQE4V submitted 2025-05-29 quant-ph cond-mat.quant-gas

Raúl Hidalgo-Sacoto , Thomas Busch , D. Blume This is my paper · ORCID
classification quant-phcond-mat.quant-gas MSC 81U0581Q1081V70 PACS 03.65.Nk05.30.Pr
keywords one-dimensionalanyonszero-rangeinteractionsexchangestatisticsbosonic-anyon–fermionic-anyonmappingmomentumdistributiontailtwo-bodycontactharmonicconfinementanyonicscatteringlength
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

This paper tries to establish a consistent scattering framework for two identical one-dimensional anyons interacting through a zero-range contact pseudopotential. The anyonic exchange statistics are encoded by the operator $\hat{S}_\alpha(z)=\exp[-i\pi\alpha\,\mathrm{sign}(z)]$, with bosonic anyons using $+\hat{S}_\alpha$ and fermionic anyons using $-\hat{S}_\alpha$. The central claim is that the zero-range potential supports eigenstates satisfying the mapping $\psi_{\alpha,+}(z)=\mathrm{sign}(z)\psi_{\alpha,-}(z)$, a direct anyonic generalization of the ordinary boson-to-fermion mapping, and that the momentum-distribution tails of these states have universal $k^{-2}$ and $k^{-3}$ coefficients plus a non-universal $k^{-4}$ piece. The whole construction requires the even- and odd-parity parts of the interaction to produce exactly the same scattering phase shift, and the authors are explicit that this is an idealization not known to be realized by any finite-range potential. A sympathetic reader would care because this gives the simplest exactly solvable interacting-anyon setting in the continuum and shows which observables can actually fingerprint anyonic statistics.

What carries the argument

The load-bearing object is the zero-range pseudopotential built from even- and odd-parity projectors: $V_+(z)=g_+\delta(z)$ and $V_-(z)=g_-(1/z)\delta(z)\partial_z$, equivalently a logarithmic-derivative boundary condition $\partial_z\psi_s(z)/\psi_s(z)\big|_{|z|=0^+}=-1/a_{\rm sc}$. The anyonic exchange operator $\hat{S}_\alpha(z)=\exp[-i\pi\alpha\,\mathrm{sign}(z)]$ and the normalization factor $N(\alpha)$ generate the regular and irregular reference functions from the ordinary boson and fermion reference functions, and the central requirement is that the even- and odd-parity interaction channels generate the same scattering phase shift $\delta_{\rm sc}(k)$. This equal-phase-shift condition is what makes the outside solution carry the anyonic statistics and what turns the scattering decomposition into the mapping $\psi_{\alpha,+}(z)=\mathrm{sign}(z)\psi_{\alpha,-}(z)$.

What would settle it

Take any finite-range two-body potential whose even- and odd-parity low-energy scattering lengths are equal and numerically solve the relative Schrödinger equation; if the exact wavefunction inside the interaction region fails to satisfy $\psi(-z)=\pm\exp[-i\pi\alpha\,\mathrm{sign}(z)]\psi(z)$, then the anyonic exchange symmetry is a property of the zero-range limit only, not of finite-range physics.

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Extended reading notes

Core claim

Stated on the paper's own terms, the discovery is that a zero-range pseudopotential $V_{\rm pseudo}(z)=g_+\delta(z)+g_-(1/z)\delta(z)\partial_z$, with $g_+$ and $g_-$ fixed by the same anyonic scattering length $a_{\rm sc}$, supports two-anyon scattering and bound states whose exchange statistics are $\psi_{\alpha,\pm}(-z)=\pm\exp[-i\pi\alpha\,\mathrm{sign}(z)]\psi_{\alpha,\pm}(z)$. Free space admits exactly one bosonic-anyon and one fermionic-anyon dimer, both with binding energy $-\hbar^2/(2\mu a_{\rm sc}^2)$; under harmonic confinement the relative wavefunctions are superpositions of the boson and fermion relative states with coefficients $\cos(\pi\alpha/2)$ and $i\sin(\pi\alpha/2)$. Because $\psi_{\alpha,+}(z)=\mathrm{sign}(z)\psi_{\alpha,-}(z)$, local observables such as the two-body contact are statistics-independent, while the one-body density matrix and momentum distribution are not, and the momentum distributions are skewed for $0<\alpha<1$. The paper confirms the universal $k^{-2}$ and $k^{-3}$ tails and shows the $k^{-4}$ tail contains a state-dependent, non-universal coefficient $K_{2,\epsilon}$ in addition to universal terms.

Load-bearing premise

The load-bearing premise is that the even- and odd-parity parts of the two-body interaction generate exactly the same scattering phase shift across the relevant energy range, together with the zero-range idealization that confines the interaction to a point; the paper states plainly that no finite-range potential satisfying this condition is known, so a real cold-atom implementation would violate the anyonic statistics in the inner $|z|<z_0$ region.

Editorial extensions

If this is right

  • For a harmonically trapped two-anyon state, the $k^{-2}$ and $k^{-3}$ coefficients of the momentum tail are fixed by the two-body contact, the scattering length, and $\alpha$, so a high-momentum measurement can test the anyonic prediction without knowing the short-range interaction details.
  • The $k^{-4}$ coefficient is not universal: for trapped anyons the non-universal part is $K_{2,\epsilon}=((2\epsilon+3)/4)(a_{\rm sc}/a_{\rm HO})^2$, meaning the same zero-range Hamiltonian realized in different trap states produces visibly different $k^{-4}$ tails.
  • Free-space dimers exist for both bosonic and fermionic anyons with identical binding energy, and their wavefunctions are related by the sign factor, extending the boson-fermion mapping beyond $\alpha=0$.
  • The momentum distributions of both anyon species are asymmetric about $k=0$ for $0<\alpha<1$, and the locations and heights of their extrema provide a quantitative measure of the anyonic chirality.
  • The identities $n_{\alpha,+}(k)=n_{1-\alpha,-}(-k)$ connect physical observables at complementary statistics and imply that off-diagonal correlations distinguish anyonic statistics even when the two-body contact is identical.

Reading between the lines

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

  • If the zero-range model is taken as the effective low-energy theory, the non-universal $k^{-4}$ terms imply that any lattice or cold-atom emulation must fix the short-range physics before its momentum tail can be compared with the model; only the $k^{-2}$ and $k^{-3}$ pieces are protected.
  • The equal-phase-shift requirement gives a practical search criterion for finite-range approximations to one-dimensional anyons: tune a two-channel potential until the even- and odd-parity phase shifts coincide over the relevant momentum window, and quantify the symmetry violation by the mismatch of the inside wavefunction.
  • The skewed momentum distribution suggests a braiding-free diagnostic: measuring the asymmetry of the momentum distribution about $k=0$ in an anyonic simulator would directly probe the chiral $\alpha$-dependent phase without needing to perform an exchange or braid.
  • For more than two particles, the same construction points to the three-body contact $C_3$ entering the $k^{-3}$ tail, so momentum-tail measurements in few-anyon systems could serve as a route to extract $C_3$ experimentally.
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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

1 major / 3 minor

Summary. This paper develops a two-particle scattering theory for identical 1D bosonic anyons and fermionic anyons with zero-range interactions. The authors define regular and irregular reference functions with anyonic exchange symmetry, construct a pseudopotential whose even- and odd-parity parts produce the same scattering phase shift, and derive short-distance boundary conditions. From these ingredients they obtain the free-space dimer, the harmonically confined two-body eigenstates, the one-body density matrix, and the momentum distribution. The central claims are that the two-particle states obey the anyonic exchange statistics of Eqs. (7)-(8), that the mapping psi_{alpha,+}=sign(z) psi_{alpha,-} of Eq. (24) holds, that the k^{-2} and k^{-3} momentum-tail coefficients are universal, and that the k^{-4} coefficient contains a non-universal contribution K_{2,epsilon} in the trapped case. The analytic tail results are supported by numerical calculations.

Significance. If the two printed formulas identified below are corrected, the paper provides a clean, analytically solvable illustration of the bosonic-anyon/fermionic-anyon mapping and gives useful explicit benchmarks for off-diagonal correlations and momentum tails. Its strengths are the explicit wavefunction constructions, the analytic tail expansions, and the numerical verification in Figs. 3-4. The model's physical realizability is limited by the admitted absence of any known finite-range potential whose even- and odd-parity sectors produce the same phase shift, and by the authors' own statement that a cold-atom realization would violate the anyonic statistics for |z|<z0; this is a disclosed limitation rather than an internal inconsistency. The overall framework is coherent, but Eqs. (29) and (59) are load-bearing printed errors that must be fixed before the results can be used as stated.

major comments (1)
  1. [Sec. II.D, Eq. (29)] The printed fermionic-anyon boundary condition is dimensionally inconsistent and violates the mapping Eq. (24). The sine bracket 'z - |z|/a_sc' mixes a coordinate with a dimensionless ratio and does not satisfy the logarithmic-derivative condition Eq. (25). Multiplying Eq. (29) by sign(z) gives cos(pi alpha/2)[sign(z)-z/a_sc] + i sin(pi alpha/2)[|z|-z/a_sc], whereas Eq. (28) has i sin(pi alpha/2)[sign(z)-z/a_sc]. The corrected form, obtained from Table I and Eq. (15), should read cos(pi alpha/2)[sign(z)-z/a_sc] + i sin(pi alpha/2)[1-|z|/a_sc]. Because this boundary condition underlies the fermionic-anyon scattering solutions and tail analysis, the authors should correct Eq. (29) and re-check the fermionic-anyon expressions; the tabulated tails in Tables V/VI and Eq. (61) appear to correspond to the corrected condition, suggesting a typo rather than a systematic error.
minor comments (3)
  1. [Sec. IV.A after Eq. (63)] The statements about the alpha=0 and alpha=1 limits of Eq. (63) are interchanged. For alpha=0, Eq. (63) gives a leading k^{-2} tail (two identical fermions), while for alpha=1 it gives a k^{-4} tail (two identical bosons); the printed sentences say the opposite.
  2. [Sec. II.D, Eq. (25) and surrounding text] The sentence stating that the pseudopotential imposes no constraints on z^2, z^3, etc. terms is slightly misleading in the harmonically trapped case, where the external potential fixes the higher-order Taylor coefficients through the Schrodinger equation; consider clarifying that this refers to the free-space zero-range boundary condition only.
  3. [Abstract and Sec. V] The abstract says the previously derived k^{-2} and k^{-3} coefficients are 'confirmed' for two harmonically confined anyons; since the k^{-3} term is absent in the special cases epsilon=1/2 and 3/2, a more precise wording would state that the coefficients are confirmed for generic finite a_sc and 0<alpha<1.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the tail coefficients and the anyon-anyon mapping are re-derived from explicit wavefunctions and boundary conditions, with the companion paper [39] used for comparison rather than as the source of the computed results.

full rationale

The paper's central results are self-contained derivations from the zero-range model. The scattering reference functions in Table I and the pseudopotential in Eqs. (16)-(20) are constructed so that the even- and odd-parity channels share a common phase shift; consequently, the mapping psi_{alpha,+} = sign(z) psi_{alpha,-} of Eq. (24) and the anyonic exchange-statistics properties of the eigenstates are consistency properties of the model, not empirical predictions fitted to data. The universal k^{-2} and k^{-3} tail coefficients are not merely quoted from the earlier companion paper [39]: they are re-derived in Sec. IV by expanding the explicit free-space and harmonically trapped eigenstates (Eqs. (62)-(63), Tables V-VI) and checked against numerical momentum distributions. The non-universal k^{-4} coefficient K_{2,epsilon} is an analytic function of the state energy and scattering length, Eq. (72), and is not fitted to the momentum distribution it is used to predict. The self-citation to [39] is therefore present but not load-bearing. I flag two non-circular correctness concerns in the manuscript: Eq. (29) appears dimensionally inconsistent and, when derived from Eq. (28) via the paper's own mapping, violates Eq. (24); and Eq. (59) has a normalization that conflicts with Eq. (41). These are internal consistency or accuracy issues, not evidence that a derived result is equivalent to its input. The Sec. II.D statement that no finite-range potential realizing the equal-phase-shift condition is known is a disclosed physical-realizability limitation, not a circular step.

Assumptions & free parameters 1 free parameters · 6 assumptions · 1 invented entities

The model's inputs are the statistical parameter alpha and the scattering length a_sc, plus standard contact-interaction machinery. No parameters are fitted to data. The main load-bearing assumptions are the definition of 1D anyonic exchange and the requirement of equal even/odd phase shifts, which the paper acknowledges is not known to be realizable with finite-range potentials.

free parameters (1)
  • statistical parameter alpha = 0 < alpha < 1 (continuously variable, not fitted)
    Chosen by hand to interpolate between bosons (alpha=0) and fermions (alpha=1); all results depend on it, but it is not determined by data.
assumptions (6)
  • domain assumption Anyonic exchange statistics in 1D are defined by psi_{alpha,pm}(-z)=pm exp(-i pi alpha sign(z)) psi_{alpha,pm}(z) with alpha in [0,1].
    This is the paper's model of exchange statistics, distinguishing bosonic and fermionic anyons; it is a definition rather than a derived fact (Sec. II.B).
  • ad hoc to paper The even and odd parity parts of the two-body interaction must generate the same scattering phase shift to support anyonic eigenstates.
    Required for the outside solution Eq. (15) to be consistent; if violated, the model does not produce anyonic wavefunctions (Sec. II.D).
  • domain assumption Zero-range interactions are represented by the pseudopotential V_pseudo=V_+ + V_- with V_+=g_+ delta(z) and V_-=g_- (1/z) delta(z) partial_z.
    Standard 1D contact interaction formalism (Cheon-Shigehara); the paper relies on it to define the anyonic scattering problem (Sec. II.D).
  • domain assumption In the low-energy limit the k-dependent couplings can be replaced by energy-independent scattering-length parameters g_+ = -h_bar^2/(mu a_sc) and g_- = h_bar^2 a_sc/mu.
    This is the usual low-energy parametrization; it makes the scattering length the only interaction input (Sec. II.D).
  • domain assumption The Taylor expansion of the confined two-anyon wavefunction to third order around the contact point determines the large-|k| tail through order k^{-4}.
    The paper uses this to derive the non-universal k^{-4} coefficient; higher-order wavefunction terms are not fixed by the pseudopotential (Sec. IV.B).
  • standard math The harmonically confined relative eigenstates are built from confluent hypergeometric functions with quantum numbers fixed by transcendental equations.
    Standard solution of the two-body harmonic trap problem; the paper invokes these in Appendix B and Eq. (69).
invented entities (1)
  • Bosonic anyons and fermionic anyons as distinct 1D particle statistics
    purpose: Define particles whose exchange phase interpolates continuously between bosons and fermions, with two separate species for the two endpoint statistics.
    The statistics are introduced as a mathematical model. The paper offers no experimental realization or independent falsifiable prediction for the entity itself, only observable consequences (momentum distributions) within the model.

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Cite this review

Pith. "Pith review of Two identical 1D anyons with zero-range interactions: Exchange statistics, scattering theory, and anyon-anyon mapping." pith.science (2026). https://pith.science/paper/FGFLQE4V

@misc{pith2026250523127,
  author       = {Pith},
  title        = {Pith review of: Two identical 1D anyons with zero-range interactions: Exchange statistics, scattering theory, and anyon-anyon mapping},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FGFLQE4V}},
  note         = {Machine review of arXiv:2505.23127}
}
abstract

While elementary particles obey either bosonic or fermionic exchange statistics, generalized exchange statistics that interpolate between bosons and fermions -- applicable to quasi-particles -- constitute an intriguing topic, both from the fundamental and practical points of view. This work develops a scattering framework for two identical 1D bosonic anyons and two identical 1D fermionic anyons with zero-range contact interactions. The two-body system with zero-range interactions, both in free space and under external confinement, is used to illustrate the recently proposed bosonic-anyon -- fermionic-anyon mapping~(R. Hidalgo-Sacoto {\em{et al.}}, arXiv:2505.17669), which connects the eigenstates of bosonic anyons to those of fermionic anyons and vice versa. Performing explicit calculations for two-particle systems, the momentum distributions and the off-diagonal correlations of the single-particle density matrix for bosonic anyons and fermionic anyons are confirmed to be distinct. We also confirm the previously derived asymptotic coefficients of the momentum distribution tail at orders $k^{-2}$ and $k^{-3}$ for two harmonically confined anyons. Non-universal contributions at order $k^{-4}$ are discussed.

Figures

Figures reproduced from arXiv: 2505.23127 by the authors.

Figure 1
Figure 1. FIG. 1. Illustration of exchange operator [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Properties of off-diagonal elements of anyonic bound state in free space. (a) The first and second columns show, [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Analysis of the tail of the momentum distribution [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Analysis of the quantity lim [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

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