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Anyonization of bosons in one dimension: an effective swap model

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Pith's one-line read This paper claims that putting a complex phase on the swapping of two bosonic species in a lattice makes the impurity's one-body correlation function exactly match the anyonic correlations of the one-dimensional anyon-Hubbard model.

desk verdict Swap model gives a clean new route to anyonic correlations in 1D spinor gases; the formal mapping is conditional on spin-charge separation, but the numerics support it in the tested regime. read the letter →

arxiv 2504.21208 v1 pith:S35XNHFI submitted 2025-04-29 cond-mat.quant-gas cond-mat.mes-hallquant-ph

Botao Wang , Amit Vashisht , Yanliang Guo , Sudipta Dhar , Manuele Landini , Hanns-Christoph Nägerl , Nathan Goldman This is my paper · ORCID
classification cond-mat.quant-gascond-mat.mes-hallquant-ph
keywords anyonsfractionalstatisticsspin-chargeseparationswapmodelanyon-Hubbardhard-corebosonsspinorquantumgasone-dimensionalgases
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

The paper argues that a two-species gas of hard-core bosons on a lattice can display one-dimensional anyonic correlations even though the microscopic particles are ordinary bosons. The key is a "swap" Hamiltonian that assigns a complex phase $e^{i\theta}$ to the exchange of a host particle and an impurity on neighboring sites. Using a spin-charge separation analysis, the authors show that the impurity's one-body correlation function is formally identical to the one-body correlation of the anyon-Hubbard model, with the swap processes playing the role of the Jordan-Wigner string operators. If correct, this gives a simpler and directly measurable route to anyonic statistics in ultracold gases, because the asymmetric momentum distribution belongs to the impurity itself rather than having to be reconstructed from the bare bosons.

What carries the argument

The load-bearing object is the swap Hamiltonian $\hat{H}_{\text{swap}}$ of Eq. (4), a two-component hard-core boson model on a lattice in which nearest-neighbor spin-swapping terms carry the complex phase $e^{i\theta}$ while ordinary hopping terms do not. The argument is carried by a spin-charge separation analysis: each Fock state is decomposed into a charge sector (which particle is where) and a spin sector (which spin label sits on each occupied site), and the impurity one-body correlator is built from charge hopping combined with a sequence of spin-swap operators. In the low-energy sector of the $\theta=0$ model, the spin part is a spin wave with momentum $p$ and eigenvalue $e^{i2\pi p/N}$; identifying $\theta=2\pi p/N$ turns the accumulated swap phase into exactly the string phase of the anyon-Hubbard model. This spin-wave structure is what converts ordinary two-body exchange into effective anyonic statistics in the single-particle observable of the impurity.

What would settle it

Perform exact diagonalization of the swap Hamiltonian for a small system (for example, $L=12$, $N_{\uparrow}=11$, $N_{\downarrow}=1$) over a range of $J_{\text{ex}}$ from $0.01J$ to $J$, and compute the overlap between the exact ground state at finite $\theta$ and the spin-charge separated spin-wave state with $p=N\theta/(2\pi)$; if that overlap drops well below one away from $J_{\text{ex}}\approx0.1J$, or if the impurity one-body correlator deviates from the anyon-Hubbard prediction beyond numerical precision, the claimed formal equivalence would be refuted.

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Extended reading notes

Core claim

The central discovery is a formal equivalence: in the swap model, the one-body correlation function of a single impurity, derived through spin-charge separation, equals the anyonic one-body correlation of the anyon-Hubbard model up to a normalization factor. The phase accumulated in the impurity correlator, $e^{i\theta(N_{\ell'}-N_{\ell})}$, emerges from a series of spin-swap processes between the impurity and the host particles, mirroring the string operators $e^{i\theta(\hat{N}_{\ell'}-\hat{N}_{\ell})}$ that appear in the fractional Jordan-Wigner transformation of the anyon-Hubbard model. The paper further shows that the ground state of the swap model at finite $\theta$ can be approximated by exciting a spin wave in the $\theta=0$ model, with the statistical angle related to the spin-wave momentum by $\theta=2\pi p/N$, and that a weak constant force applied to the impurity adiabatically prepares such states with fidelity approaching unity. The equivalence is verified numerically for $J_{\text{ex}}\approx 0.1J$ and extends qualitatively to multiple impurities, where the momentum-distribution peak sits at $k_{\text{peak}}=\theta N_{\uparrow}/L$.

Load-bearing premise

The equivalence holds only if the true state of the swap model is, or is very close to, a spin-charge separated state in which the spin part is a single spin wave whose momentum is locked to the statistical angle by $\theta = 2\pi p/N$; if the ground state leaves that spin-wave sector, the match to the anyon-Hubbard model breaks.

Editorial extensions

If this is right

  • The impurity's quasi-momentum distribution in the swap model directly mirrors the anyonic momentum distribution of the anyon-Hubbard model, so measurements on the impurity provide a direct probe of fractional statistics without reconstructing bosonic correlations from the host gas.
  • A weak constant force applied to the impurity in the $\theta=0$ swap model prepares the ground state at finite $\theta$ with high fidelity, giving an explicit adiabatic protocol for generating anyonic correlations in a strongly interacting spinor gas.
  • The equivalence links the swap model to the spinful Bose-Hubbard model in the strongly interacting regime, connecting the proposed mechanism to the recent experimental observation of anyonization in an array of one-dimensional Bose gases.
  • In quench dynamics after releasing a harmonic trap, the impurity's momentum distribution for all initial $\theta$ converges to the same asymptotic distribution, a signature of dynamical fermionization of anyonic correlations.
  • The extension to multiple impurities shows that anyonic momentum distributions survive for $N_{\downarrow}>1$, with the peak still located at $k_{\text{peak}}=\theta N_{\uparrow}/L$, indicating the effect is not limited to a single mobile particle.

Reading between the lines

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

  • If the equivalence holds generally, the internal spin degree of freedom becomes a practical resource for simulating fractional statistics without engineering density-dependent hopping; a Floquet-driven protocol for the complex swapping phase could make the statistical angle continuously tunable in existing ultracold-gas setups.
  • The spin-wave preparation picture suggests a geometric interpretation of the statistical angle as momentum-space displacement of a spin excitation; this could be tested by measuring the impurity momentum distribution as a function of host density to verify the linear scaling $k_{\text{peak}}=\theta N_{\uparrow}/L$.
  • The multi-impurity results raise the question of whether impurity-impurity interactions inherit anyonic statistics as well, which the paper does not address; a natural extension would compute the two-impurity correlation function in the swap model and compare it with two-anyon predictions.
  • Since the derivation assumes a sharply defined spin-wave sector, the same machinery might be adapted to simulate non-Abelian statistics by replacing the single spin species with multiple internal states whose swaps carry matrix-valued phases, though the paper only lists this as an open possibility.
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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

3 major / 6 minor

Summary. The paper proposes a one-dimensional two-component hard-core boson model, the 'swap model' (Eq. 4), in which the spin-swap term carries a complex phase e^{iθ}. Using a spin-charge separation analysis, the authors argue that the one-body correlation function of a single impurity in this model reproduces the anyonic correlation function of the anyon-Hubbard model (AHM), with the swap processes playing the role of the Jordan-Wigner string operator. Numerical matrix-product-state results show good agreement between the impurity correlator of the swap model and the AHM correlator for an open chain with J_ex = 0.1J. The paper also proposes an adiabatic preparation protocol based on a weak force, studies quench dynamics after trap release, and presents qualitative results for multiple impurities. The central claim is that this model provides a new, experimentally accessible route to 1D anyonic correlations in spinor quantum gases.

Significance. If the central claim holds, this is a valuable contribution: it offers a simple lattice model that realizes anyonic correlations through spin exchange rather than density-dependent hopping or scattering, and it connects directly to recent experiments on impurity motion in strongly interacting bosonic gases (including the companion experimental work of Dhar et al.). The numerical evidence in Figs. 1(c), 2, and S3 is supportive and the proposed preparation protocol via a spin wave is concrete and testable. A particular strength is that the impurity's momentum distribution is directly measurable, avoiding the reconstruction step needed in conventional anyon-Hubbard implementations. However, the central formal claim is conditional on a spin-wave/sector assumption that is numerically plausible but not proven, and the quantitative match relies on tuning J_ex to about 0.1J. These points need to be clarified before the claim of formal equivalence can be accepted.

major comments (3)
  1. [Swap model, Eq. (7), and Supplementary Sec. II] The statement that the anyonic correlations 'equivalent to Eq. (3) are formally established in the one-body correlation of the impurity' overstates what has been shown. The derivation of Eq. (7) in the main text assigns phase factors e^{±iθ} to the swap operators by hand; it does not derive them from the Hamiltonian. The only derivation is the spin-wave calculation in Supplementary Sec. II, which assumes that the state factorizes as |Ψ⟩ = |φ⟩⊗|χ⟩ and that the spin sector is exactly the spin-wave eigenstate |χ⟩ ∝ Σ_s e^{iθ̃s}|χ_s⟩ with θ̃ = 2πp/N. It is not proven that the ground state of H_swap(θ) for arbitrary θ lies in this sector, nor that spin-charge factorization holds for the parameters considered. For an open chain and a value such as θ = 0.75π in Fig. 2, the commensuration condition θ = 2πp/N is not satisfied for N = 51, so the spin-wave argument does not directly apply. The numerical agreement is excellent for the cases shown, but the formal claim should be softened or the conditions under which it holds must be stated explicitly.
  2. [Supplementary Sec. I (J_ex dependence)] Supplementary Sec. I states that quantitative agreement with the AHM requires treating J_ex as a free parameter and that values J_ex ≈ 0.1J match the AHM results, while J_ex = 0.01J and 0.4J produce different distributions (Fig. S1(b)). This means the mapping is not parameter-free: the equivalence between the swap model and the AHM holds only in a particular regime of J_ex relative to J. The paper's central claim, however, is phrased as a general formal equivalence. The authors should either demonstrate analytically why J_ex ≈ 0.1J is special or explicitly frame the result as an approximate, regime-specific correspondence. As written, a reader could reasonably conclude that the anyonic correlations are a generic property of the swap model, which the authors' own data contradict.
  3. [Effective realization, Fig. 3] The adiabatic preparation protocol relies on the fidelity F(θ,t) approaching unity for small forces, as shown numerically for system sizes L = 20 and N↑ = 10. This is a finite-size statement. The text argues that the ground state |Ψ_swap(θ)⟩ can be 'effectively realized' via a slow force, but no scaling analysis in system size or adiabatic gap estimate is provided. Since the spin-wave argument indicates that the target state is not the ground state of H_swap(θ) but rather a specific excited state of H_swap(0), it is not obvious that a weak force will prepare it with high fidelity in the thermodynamic limit. The authors should either supply a gap argument, a relevant scaling analysis, or explicitly present this as a finite-size numerical observation.
minor comments (6)
  1. [Supplementary Sec. II, Eq. (S3)] Equation (S3) contains a typographical error: the state |χ(θ̃=0)⟩ is written as 1/√2 (|↑↓⟩ + |↑↓⟩), with both terms identical. It should presumably read (|↑↓⟩ + |↓↑⟩). The same applies to the θ̃=π state.
  2. [Supplementary Sec. II] The text says 'choosing a particular spin-wave excited state with a finite value of θ̃ ... is similar to considering the ground state of the swap Hamiltonian H_swap with a finite swapping phase (θ = θ̃).' This is an important identification and should be made more precise, since the spin-wave state is an eigenstate of H_swap(0), not of H_swap(θ). The relationship between these two settings is the crux of the derivation and deserves a clear statement of the logic.
  3. [Main text, Effective realization] The mapping θ = 2πp/N connecting the statistical angle to the spin-wave momentum is introduced only in the supplementary material. Since it is central to the physical interpretation, it would help to state it explicitly in the main text (with the quantization caveat) rather than referring the reader to the supplement.
  4. [Fig. 2 and Fig. 1(c)] The comparison between the swap model and the AHM is made after rescaling the peak of the momentum distribution 'according to a normalization factor determined at θ = 0'. It would be helpful to state in the main text whether the same rescaling also applies to the real-space correlator in Fig. 2, and what physical normalization is being used.
  5. [Supplementary Sec. I, Eq. (S1)] In Eq. (S1), the gauge transformation is applied to a Hamiltonian on an open chain, but the interpretation in terms of a magnetic flux is presented for a ring. The gauge transformation changes boundary conditions; the authors should clarify that the flux interpretation is valid for the closed chain and that the open-chain results in the main text are not affected by this subtlety.
  6. [Supplementary Sec. III] There are a few typographical errors: 'storngly-interacting' and 'particuluar' should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the swap-model mapping is a self-contained calculation; theta is a bare model parameter and Jex is openly calibrated, not a hidden fit.

full rationale

The central derivation starts from the swap Hamiltonian (Eq. 4), which contains the complex phase e^{i theta} on nearest-neighbor spin-swap processes, and computes the impurity one-body correlator by spin-charge decomposition. The density-dependent phase e^{i theta (N_{l'}-N_l)} in Eq. (7) is not inserted by hand; it emerges from counting the number of swap processes between the initial and final impurity positions, each carrying the phase already present in H_swap. This is a genuine calculation: the string operator is not a term in the Hamiltonian, and its appearance in the correlator is a nontrivial consequence of the model. The equivalence with the anyon-Hubbard correlator, Eq. (3), is therefore a structural mapping rather than a restatement of the model definition. The spin-wave discussion in Supplementary Sec. II is an ansatz about which sector of H_swap(theta=0) reproduces the same correlator; it is an assumption about the ground state, not a circular import, and the main numerical comparison with the AHM in Fig. 2 uses the full H_swap(theta) directly. The parameter Jex is explicitly described as a free parameter used to match the AHM (Supplementary Sec. I: 'We treat Jex as a free parameter... Jex ~ 0.1J match the results obtained from the AHM'), so the numerical agreement is an honest calibration rather than a disguised prediction. The self-citation to the experimental companion [72] provides implementation context and experimental support, but the formal mapping and numerical benchmarks are independent of it. No circular step can be exhibited.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the spin-charge separated spin-wave ansatz for the ground state, the hard-core approximation, and the mapping from the spinful Bose-Hubbard model to the effective swap Hamiltonian. The only explicit free parameter is the exchange strength J_ex, set to 0.1J to reproduce the anyon-Hubbard correlations. No new physical entities are introduced.

free parameters (1)
  • J_ex = 0.1 J
    Supplementary Sec. I: 'We treat Jex as a free parameter... the values of Jex ≈ 0.1J match the results obtained from the AHM.' Used for quantitative agreement of momentum distributions and correlations.
assumptions (4)
  • domain assumption The ground state of the swap model factorizes into charge and spin sectors (spin-charge separation).
    Invoked in the main text around Eq. (5)-(7) and the supplementary; standard for strongly interacting 1D gases but not exact for finite J_ex.
  • ad hoc to paper The spin sector is a spin-wave eigenstate whose momentum p maps to the statistical angle θ = 2πp/N.
    Supplementary Eq. (S2) and the 'Effective realization' section; this identification is used to derive the anyonic correlator and to justify the tilt preparation protocol.
  • domain assumption Hard-core constraint with no double occupancy.
    Stated after Eq. (4); the swap model applies in the strongly interacting regime.
  • domain assumption The swap Hamiltonian approximates the spinful Bose-Hubbard model in the strongly interacting regime (neglect of nearest-neighbor interactions).
    Supplementary Sec. III; verified numerically for U_↑↓ = 40 in Fig. S3.

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Pith. "Pith review of Anyonization of bosons in one dimension: an effective swap model." pith.science (2026). https://pith.science/paper/S35XNHFI

@misc{pith2026250421208,
  author       = {Pith},
  title        = {Pith review of: Anyonization of bosons in one dimension: an effective swap model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S35XNHFI}},
  note         = {Machine review of arXiv:2504.21208}
}
read the original abstract

Anyons emerge as elementary excitations in low-dimensional quantum systems and exhibit behavior distinct from bosons or fermions. Previous models of anyons in one dimension (1D) are mainly categorized into two types: those that rely on nontrivial scattering behavior, and those based on density-dependent hopping processes in discrete lattices. Here, we introduce a novel framework for realizing anyonic correlations using the internal degrees of freedom of a spinor quantum gas. We propose a "swap" model, which assigns a complex phase factor to the swapping processes between two different species, referred to as "host particles" and "impurities". The anyonic characteristics are demonstrated through the one-body correlator of the impurity, using a spin-charge separation analysis. For a single impurity, our swap model can be effectively implemented by applying tilt potentials in a strongly interacting quantum gas [Dhar et al., arXiv:2412.21131]. We further explore the dynamical properties of anyonic correlations and extend our analysis to the case of multiple impurities. Our work provides new avenues for engineering many-body anyonic behavior in quantum simulation platforms.

Figures

Figures reproduced from arXiv: 2504.21208 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Sketch of one-body correlations [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The real (a) and the imaginary (b) part of the one [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) The fidelity [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Spatial-temporal density distribution of the single impurity after abruptly releasing the trap for [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

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