Higgs and Nambu-Goldstone modes in a spin-1 $XY$ model with long-range interactions

Daiki Kawasaki; Ippei Danshita

REVIEW 3 major objections 4 minor 47 references

Higgs and Nambu-Goldstone modes in a spin-1 $XY$ model with long-range interactions

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read In a two-dimensional spin-1 XY model with dipolar (1/r^3) interactions, the Higgs mode acquires a linear dispersion and its Beliaev damping is strongly suppressed, so the amplitude mode can be long-lived at finite temperatures near the quan

desk verdict A clean extension of the known HP/Bogoliubov machinery to long-range interactions with a concrete, checkable prediction for Rydberg arrays; the one-loop truncation and a missing derivation keep the central claim from being fully established. read the letter →

arxiv 2512.24557 v2 pith:BEALATNW submitted 2025-12-31 cond-mat.quant-gas

Daiki Kawasaki , Ippei Danshita This is my paper · ORCID

classification cond-mat.quant-gas

keywords spin-1XYmodellong-rangeinteractionsHiggsmodeNambu-GoldstoneBeliaevdampingRydbergatomarraysquantumphasetransitionfinite-temperatureGreen'sfunction

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 analyzes collective excitations in a spin-1 XY model on a square lattice with long-range interactions decaying as 1/r^3, as realized in Rydberg-atom arrays. For two dimensions, it finds that the Higgs mode—a gapped amplitude fluctuation of the magnetic order—has a linear dispersion, while the Nambu-Goldstone mode has a square-root dispersion. The central result is that the damping rate of the zero-momentum Higgs mode, evaluated via the Beliaev process of decaying into two NG modes, is strongly suppressed near the quantum critical point, so the damping-to-gap ratio stays finite at low temperatures. This suggests the Higgs mode can be observed as a long-lived coherent oscillation in Rydberg experiments, in contrast to short-range interacting systems where it is overdamped.

What carries the argument

The Holstein-Primakoff expansion truncated at cubic order, combined with a Bogoliubov transformation and one-loop finite-temperature Green's functions. The cubic term captures the Beliaev decay of a Higgs mode into two NG modes. The crucial small-momentum expansion γ_k ≈ 1 - A|k|^{α-d} (valid for d<α<d+2) converts the lattice dispersions into anomalous power laws, and its exponent α-d controls the momentum-space integrals in the damping formula.

What would settle it

A direct measurement of the zero-momentum Higgs mode's damping rate as a function of temperature and u in a Rydberg-atom array with three states, using the proposed quench protocol, would test the predicted finite Γ/Δ_h as u→1. If the damping-to-gap ratio diverges or the Higgs oscillation is not resolved, the one-loop truncation would be falsified. Alternatively, a numerically exact quantum Monte Carlo calculation of the spectral function near the critical point could check whether corrections beyond one loop alter the suppression.

Watch

Extended reading notes

Core claim

The paper claims that in the XY-ferromagnetic ordered phase of a spin-1 XY model with a quadratic Zeeman term and long-range interactions, the dispersion relations of the Higgs and Nambu-Goldstone modes are dramatically modified when the interaction decays as 1/r^3 in two dimensions: the Higgs mode becomes linear in momentum, E_A,k ≈ Δ_h + (g_h^2/2Δ_h)|k|, and the NG mode becomes square-root-like, E_Φ,k ∝ |k|^{1/2}. Using a finite-temperature Green's-function formalism with a one-loop self-energy, the Beliaev damping of the k=0 Higgs mode is derived analytically, yielding Γ_A,0 ∝ (1-u^2) coth(βΔ_h/4) for d=2, α=3. Consequently, Γ_A,0/Δ_h converges to a finite value proportional to temperatur

Load-bearing premise

The calculation assumes that the Holstein-Primakoff expansion truncated at cubic order and the one-loop self-energy (retaining only the A→ΦΦ decay channel) remain accurate even close to the quantum critical point, where quantum fluctuations grow and higher-order terms or alternative decay channels could become significant.

Editorial extensions

If this is right

The Higgs mode in Rydberg-atom arrays with dipole-dipole interactions should be observable as a long-lived oscillation after a sudden quench of the quadratic Zeeman field, with a finite damping-to-gap ratio at low temperatures.
The linear Higgs dispersion and square-root NG dispersion are distinctive signatures that can be probed via momentum-resolved spectroscopy or Bragg scattering, distinguishing long-range from short-range interactions.
The damping rate vanishes at the critical point u→1 for all temperatures within the one-loop approximation, implying undamped amplitude oscillations exactly at the transition.
The results interpolate continuously to the nearest-neighbor interaction limit as α→d+2, recovering the standard quadratic Higgs and linear NG dispersions.
The proposed excitation protocol (adiabatic preparation followed by a sudden quench) provides a concrete experimental route to test the theory.

Reading between the lines

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

The suppression of damping likely persists beyond one loop as long as the A→ΦΦ channel remains the dominant decay process, but near the critical point higher-order corrections could alter the quantitative prediction; a two-loop calculation or a numerical simulation of the spectral function would test this.
The square-root NG dispersion implies a modified density of states at low energies, which could affect spin transport or thermalization dynamics in a measurable way, perhaps through spin-diffusion experiments.
A natural extension is to include spatial inhomogeneity or finite system sizes, where the Higgs mode may hybridize with edge states or form bound states; the paper mentions this direction in its conclusion.
Because the ratio Γ/Δ_h becomes temperature-independent near u_c, a single experimental measurement of the oscillation decay over a range of temperatures could extract both the gap and the damping strength simultaneously.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

The paper does exactly what the abstract says: it takes the known HP/Bogoliubov/one-loop treatment of the spin-1 XY model and swaps the nearest-neighbor dispersion for a long-range one, γ_k ≈ 1 - A|k|^{α-d}. For d=2, α=3 the resulting dispersions are new: linear Higgs, sqrt NG. The Beliaev damping formula, Eq. (39), with the concrete d=2, α=3 limit Eq. (41) is also new. The consistency check with α→5, d=3 reproduces the previous nearest-neighbor result, which is a good sign that the algebra is right.

The physics point is genuinely interesting. In the short-range 2D case the Higgs mode is overdamped because Γ/Δ_h diverges as u→u_c. Here the one-loop calculation gives Γ/Δ_h → finite ∝ T, so the ratio stays below unity at low T. That is a concrete, falsifiable prediction for Rydberg atom arrays, and the experimental protocol in Sec. IV B is sensible.

Now the soft spots, in order of size. First, the paper does not show the integration from Eq. (33) to Eq. (39). The prefactor of Eq. (41) is therefore not checkable from the text. That should have been an appendix line at minimum. Second, the Holstein-Primakoff expansion is truncated at cubic order and the self-energy is one-loop. The paper does not address the validity of this truncation near the 2D critical point. The stress-test concern about soft NG modes carrying large thermal occupation is real: on-shell k0 ~ Δ_h^2, u_Φ(k0) ~ (1-u^2)^{-1/4}, and n_B ~ T/Δ_h. Higher-order diagrams with additional NG lines could in principle contribute powers of T/Δ_h and change the ratio Γ/Δ_h. The paper doesn't estimate those. This is not a fatal flaw — the approximation is standard and the short-range limit checks out — but it means the 'long-lived' claim is established only within the one-loop approximation. Third, the mean-field treatment of the critical point ignores the known deviation of the critical exponent from 1/2, though for long-range interactions that may be less severe.

Who should read this: anyone working on collective modes in long-range spin systems or Rydberg arrays. It's a solid contribution, but the missing derivation and the unquantified truncation mean it deserves a serious referee rather than immediate acceptance. I would send it to peer review.

Referee Report

3 major / 4 minor

Summary. The manuscript studies the spin-1 XY model with a quadratic Zeeman term and power-law interactions, Eq. (1), in the XY-ferromagnetic phase near the transition to the disordered phase. Using a mean-field product state, a Holstein-Primakoff expansion truncated at cubic order, and finite-temperature Green's functions, the authors derive analytic dispersions for the Higgs and NG modes, Eqs. (24)-(25), and a one-loop Beliaev damping rate for the k=0 Higgs mode, Eqs. (33), (39), and (41). For d=2 and alpha=3, the parameter set relevant to Rydberg-atom arrays, they find a linear Higgs dispersion, a square-root NG dispersion, and a damping-to-gap ratio Gamma/Delta_h that remains finite as u->u_c, in contrast to the short-range case where it diverges. The alpha->5, d=3 limit reproduces the previously known short-range result, Eq. (42). The paper also proposes a concrete experimental protocol for exciting and detecting the Higgs mode.

Significance. If the one-loop calculation is quantitatively reliable, the paper gives an experimentally relevant prediction for Rydberg-atom arrays: the long-range nature of the interaction suppresses Higgs-mode damping near the critical point, making a long-lived amplitude mode observable. The analytic closed forms are a strength, and the reduction to the known short-range result is a useful consistency check. However, the central conclusion depends on an uncontrolled cubic-order, one-loop truncation near the critical point and on the numerical prefactor of Eq. (41). The significance is therefore contingent on the robustness of that approximation, which the manuscript does not currently establish.

major comments (3)

[Sec. III.C and III.E, Eq. (41)] The central claim that Gamma/Delta_h stays finite and small as u->u_c rests on a Holstein-Primakoff expansion truncated at cubic order and a one-loop self-energy containing only the A->Phi Phi process, Eq. (32). Near the critical point the on-shell NG momentum is k0 proportional to Delta_h^2 proportional to (1-u^2), the NG Bogoliubov coefficient grows as k0^{-1/4}, and the thermal factor in Eq. (33) behaves as 4T/Delta_h. The manuscript gives no estimate of the neglected quartic HP terms, of the two-loop sunset diagram with two Phi propagators, or of the quasiparticle residue Z. A finite one-loop Gamma/Delta_h is not by itself sufficient to guarantee a sharp Higgs peak if higher-order diagrams or a vanishing Z modify the spectral function. I ask for a quantitative scaling estimate, or an explicit calculation, of the leading neglected corrections in the limit u->1.
[Sec. IV.B, Eq. (39)] The step from the exact self-energy expression, Eq. (33), to the closed form, Eq. (39), is not shown. In particular, the simplification of |M_{0,k,-k}|^2 in the long-wavelength limit and the angular integration are not displayed, so the numerical prefactor of Eq. (41) cannot be checked from the text. This prefactor controls the statement that the damping is 'sufficiently smaller than unity' near the critical point. Please provide the intermediate algebra or include it in a supplemental derivation.
[Sec. IV.B, approximations after Eq. (38)] The long-wavelength approximations for the Bogoliubov coefficients do not match the exact small-k limits of Eqs. (18)-(21). For u->1, Eq. (18) at k=0 gives u_A^2 approximately [4 sqrt(1-u^2)]^{-1}, whereas the text's u_A^2 = 1/(2 Delta_bar_h) = [8 sqrt(1-u^2)]^{-1}. Similarly, Eq. (20) with gamma_k approximately 1 - A|k| gives u_Phi^2 approximately [4 sqrt(A|k|)]^{-1}, whereas the text gives [8 sqrt(A|k|)]^{-1}. These factor-of-two discrepancies in u_A^2 and u_Phi^2 change |M_{0,k,-k}|^2 and hence the prefactor of Eq. (41). Please correct the approximations or explain the convention used.

minor comments (4)

[Eq. (14)] The Fourier transform convention is not stated. The sign convention in Eq. (14) affects the momentum-conservation delta functions in Appendix A; please specify it explicitly.
[Eq. (34)] The expansion gamma_k = 1 - A|k|^{alpha-d} is used for d < alpha < d+2. The limit alpha -> d+2, mentioned after Eq. (38), should be qualified: at alpha = d+2 logarithmic corrections can appear, and the simple power-law form requires care.
[Sec. IV.B, Eq. (42)] The statement that Eq. (42) agrees with previous works would be easier to verify if the corresponding equations in Refs. [22,29] were cited explicitly, together with the mapping z_eff -> z.
[Sec. V] The concluding paragraph mentions a possible future study of spatial inhomogeneity and Higgs bound states; this is fine, but the discussion would benefit from a sentence distinguishing mean-field u_c=1 from the numerically more accurate u_c values cited in Sec. II.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: dispersions and damping follow from the stated Hamiltonian by explicit mean-field, Bogoliubov, and one-loop self-energy algebra; self-citations are methodological or cross-checking, not load-bearing.

full rationale

The paper's central results, Eqs. (35)-(36) and (41), are derived in the text from the model Hamiltonian (1). The mean-field phase boundary is obtained from minimizing E_MF in Eq. (3), and the linear-term condition A1=0 fixes u = c1^2 - s1^2; the gap Delta_h = 4 J z_eff sqrt(1-u^2) (Eq. (26)) is a Bogoliubov result, not an imposed input. The long-range dispersions follow from substituting the standard lattice-sum expansion gamma_k ~ 1 - A |k|^(alpha-d) into the diagonalized quadratic Hamiltonian; A is supported by the external Ewald-summation reference [44] and by Refs. [41,42] in addition to the self-reference [43]. The damping rate is obtained by evaluating the explicit one-loop self-energy of Eqs. (32)-(33) in the long-wavelength approximation, giving Eq. (39) and its alpha=3, d=2 specialization Eq. (41). There is no fitted parameter renamed as a prediction, and no result is defined in terms of the quantity it is supposed to predict. The agreement of the alpha->5, d=3 limit with earlier short-range work is a post-hoc check, not an input. Self-citations (e.g., Refs. [29,30,43]) supply earlier short-range formalism and dispersion estimates, but the relevant calculations are repeated in the present text rather than used as black-box premises. The main caveats - truncation of the Holstein-Primakoff expansion at cubic order and a one-loop self-energy near a 2D critical point - concern approximation accuracy and robustness, not circular reduction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to data; u=Δ/(4J z_eff) and T are Hamiltonian/thermal control parameters, while z_eff and A are computed lattice sums. No new particles or forces are introduced; the Higgs and NG modes are standard collective excitations.

assumptions (5)

domain assumption Rydberg three-state system is well described by the spin-1 XY Hamiltonian (1) with only the XY exchange term.
Sec. II: the authors note that state-dependent dipole interactions produce additional terms but 'this deviation is relatively small, we neglect the additional terms.' This is a physical modeling assumption, not a theorem.
domain assumption The variational product state (2) plus Holstein-Primakoff expansion truncated at cubic order is a valid starting point near the quantum critical point.
Sec. III.A-C: the expansion is assumed to be valid; the paper states it is 'valid only at d≥2' but does not estimate the size of neglected higher-order terms near u_c, where fluctuations grow.
standard math The lattice band structure satisfies γ_k ≈ 1 - A|k|^{α-d} for d<α<d+2.
Eq. (34), citing Refs [41-43]; for α=3,d=2, A=2π/z_eff is fixed by Ewald summation. This non-analytic low-momentum behavior is the origin of the modified dispersions.
domain assumption At one loop, only A→ΦΦ decay contributes to the k=0 Higgs self-energy; the other cubic terms ̃S^(3) do not contribute.
Sec. III.E: 'the former contributes to the damping ... but the latter does not' due to energy-momentum conservation. This assumes no competing one-loop channel of comparable strength.
domain assumption Long-range order survives at finite temperature for d=2, α=3, so an expansion around the symmetry-broken state is meaningful.
Sec. III.C invokes Refs [27,28] (Mermin-Wagner evasion for α<2d). If this failed, the entire finite-T damping calculation would lack an ordered reference state.

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Pith. "Pith review of Higgs and Nambu-Goldstone modes in a spin-1 $XY$ model with long-range interactions." pith.science (2026). https://pith.science/paper/BEALATNW

@misc{pith2026251224557,
  author       = {Pith},
  title        = {Pith review of: Higgs and Nambu-Goldstone modes in a spin-1 $XY$ model with long-range interactions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BEALATNW}},
  note         = {Machine review of arXiv:2512.24557}
}

abstract

We theoretically study the collective excitations in a spin-1 $XY$ model with a quadratic Zeeman term and a long-range interaction that decays algebraically with the distance. Using the quantum-field theory based on the finite-temperature Green's function formalism, we analyze properties of the Nambu-Goldstone (NG) and Higgs modes in order to analytically evaluate the damping rate of the Higgs mode in the $XY$ ferromagnetic ordered phase near the quantum phase transition to the disordered phase. When the power of the algebraic decay is 3 as in the case of dipole-dipole interactions in Rydberg-atom systems, we show that at two dimensions the excitation energy of the Higgs mode exhibits a linear dispersion whereas the dispersion of the NG mode becomes proportional to the square root of the momentum. We find that the damping of the Higgs mode is significantly suppressed by the long-range interaction. We also propose how to excite and probe the Higgs mode in Rydberg-atom experiments.

Figures

Figures reproduced from arXiv: 2512.24557 by the authors.

**Figure 2.** FIG. 2. Dispersion relations of the Higgs (blue solid line) and [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3. Higgs gap ∆ [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. Beliaev damping rate of the Higgs mode as a function [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

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