Quantum simulating continuum field theories with large-spin lattice models

Gabriele Calliari; Hannes Pichler; Marco Di Liberto; Torsten V. Zache

REVIEW 4 major objections 5 minor 114 references

Quantum simulating continuum field theories with large-spin lattice models

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims that a lattice of large spins can quantitatively reproduce continuum scalar quantum field theory after extrapolation in spin length and system size, as demonstrated numerically for the sine-Gordon model.

desk verdict A solid, benchmark-heavy demonstration that large-spin truncation plus two-step extrapolation can reach quantitative continuum sine-Gordon physics, though the extrapolation law itself is empirical and some equilibrium comparisons are partly fitted. read the letter →

arxiv 2412.15325 v2 pith:ACZW42PV submitted 2024-12-19 quant-ph cond-mat.quant-gashep-lat

Gabriele Calliari , Marco Di Liberto , Hannes Pichler , Torsten V. Zache This is my paper · ORCID

classification quant-phcond-mat.quant-gashep-lat

keywords sine-Gordonmodelquantumfieldtheorysimulationlarge-spinlatticemodelsquditregularizationcontinuumlimitextrapolationsolitonscatteringreal-timedynamicsRydbergsimulators

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 sets out to establish that a lattice of large spins—each site a multi-level qudit, not a qubit—can faithfully simulate the continuum physics of scalar quantum field theories. The link is an operator identification: the spin ladder operators act like the vertex operators $e^{\pm i\varphi}$ and the spin component $J_z$ acts like the conjugate momentum $\pi$, so a spin of length $J$ provides a finite-dimensional regulator of a bosonic field. The authors argue that after extrapolating observables linearly in $1/[J(J+1)]$ and then quadratically in $1/N$, lattice results converge to the true continuum limit, and they demonstrate this quantitatively for the sine-Gordon model with matrix-product-state simulations. The demonstrated observables cover vacuum properties, soliton wave-packet dynamics, and soliton-antisoliton scattering shifts that agree with the quantum $S$-matrix, plus a non-integrable perturbation that shows confinement and string-breaking-like dynamics. If the approach is right, it gives analog quantum simulators—particularly Rydberg arrays—a concrete route to quantitative predictions in out-of-equilibrium quantum field theory, including problems like false-vacuum decay that currently lack classical benchmarks.

What carries the argument

The load-bearing object is the operator identification of Eq. (3), which converts the $2J+1$ eigenstates of $J_z$ into a truncated momentum-space regulator for a compactified bosonic field, with $e^{\pm i\varphi}$ acting as raising/lowering operators on that space. With couplings rescaled as $\lambda'_\kappa=\lambda_\kappa[J(J+1)]^{\kappa/2}$ and $V'_{nn}=J(J+1)V_{nn}$, the lattice Hamiltonian generates, through a Taylor expansion of the nearest-neighbor cosine term, the continuum Hamiltonian $\frac12\hat{\pi}^2+\frac12(\nabla\hat{\varphi})^2 - (M_0^2/\beta^2)\cos(\beta\hat{\varphi})$; higher-order raising terms $\lambda_\kappa(\hat{J}_+)^\kappa$ produce the Fourier components of an arbitrary even periodic potential. The sequence of extrapolations—linear in $1/[J(J+1)]$ at fixed $N$, then quadratic in $1/N$, then the small-$aM_0$ check against predicted continuum scaling—is what turns finite-spin lattice data into continuum predictions, and matching of the renormalized $\beta^2$ and $M'_0$ from the measured observables is what keeps the procedure independent of the solvability of the target theory.

What would settle it

Take a single fixed target ($\beta^2$, $M'_0$) in the simulated sine-Gordon regime and compute one observable, say the mass gap, at $J=10,14,18,20$ for fixed $N$; then check whether the extrapolated $J\to\infty$ value from a linear fit in $1/[J(J+1)]$ is stable when a quadratic term in $1/[J(J+1)]^2$ is added to the fit. A systematic shift of the extrapolated value larger than the combined error bars would falsify the linear correction law in that parameter range. A complementary check is to compute the same continuum observable with an independent regularization—for instance a bosonic field with a large local Hilbert-space cutoff on a fine lattice—and require agreement with the large-spin extrapolation before the method is used to benchmark a quantum simulator.

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Extended reading notes

Core claim

The central claim is that the generalized Heisenberg Hamiltonian (Eq. (2)), together with the identification $e^{\pm i\varphi}\leftrightarrow \hat{J}_\pm/\sqrt{J(J+1)}$ and $\hat{\pi}\leftrightarrow \hat{J}_z$ (Eq. (3)), realizes a scalar QFT in its continuum limit after a two-step extrapolation: first $J\to\infty$ with a linear fit in $1/[J(J+1)]$, then $N\to\infty$ with a quadratic fit in $1/N$, with continuum scaling controlled by the dimensionless mass $aM_0$. The paper demonstrates this numerically for the one-dimensional sine-Gordon model, matching analytical predictions for the mass gap, ground-state energy density, and vertex-operator expectation values, including the Lukyanov-Zamolodchikov conjecture for $\langle e^{in\beta\varphi}\rangle$. It further shows how to prepare static and moving solitons by local rotations that imprint the classical phase and momentum profiles while leaving vacuum correlations intact, and it finds that the position shift in soliton-antisoliton scattering agrees with the transmissive $S$-matrix prediction. A double-frequency cosine perturbation of the sine-Gordon potential produces repeated collisions consistent with a confining meson, and at stronger perturbation, pair production and plasma oscillation of the 'electric' observable $\langle\sin(\beta\varphi)\rangle$, read as string breaking. Throughout, the renormalized couplings $\beta^2$ and $M'_0$ are extracted from the numerics rather than from the classical microscopic mapping, which the paper argues makes the regularization applicable beyond integrable benchmarks.

Load-bearing premise

The load-bearing premise is the purely empirical extrapolation ansatz: after fixing the microscopic couplings, observables are fitted linearly in $1/[J(J+1)]$ and then quadratically in $1/N$, and the continuum value is read off from those fits; no proof is given that these are the correct finite-spin and finite-size correction laws, and the paper itself notes that targets requiring larger $\kappa$ would need spin lengths far beyond the simulated $J\le 20$.

Editorial extensions

If this is right

Equilibrium observables of the continuum sine-Gordon model—mass gap, vertex-operator expectation values, ground-state energy density, and the Luttinger parameter $K$—can be recovered from finite large-spin simulations through the two-step extrapolation in $1/[J(J+1)]$ and $1/N$.
Soliton and antisoliton wave-packets can be prepared by local spin rotations that imprint classical phase and momentum profiles without altering vacuum two-point correlations, and their real-time propagation and scattering are quantitatively captured up to velocities $v\lesssim 0.5$.
The scattering position shift of a soliton-antisoliton pair matches the transmissive $S$-matrix prediction (Eq. (32)) once renormalized $\beta^2$ and $M'_0$ from the equilibrium analysis are used, demonstrating quantitative access to non-trivial real-time QFT observables.
Adding a double-frequency cosine perturbation realizes a confining potential for the soliton-antisoliton pair, and the resulting dynamics shows meson-like repeated collisions at weak perturbation and string-breaking with plasma oscillations at stronger perturbation.
The same regularization extends to general scalar field theories by adding higher-order raising terms $\lambda_\kappa(\hat{J}_+)^\kappa$: any symmetric periodic potential becomes programmable through its Fourier coefficients, and the method works in $d$ spatial dimensions for platforms with large spins.

Reading between the lines

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

Editorial inference: if the linear-in-$1/[J(J+1)]$ correction law holds beyond the simulated range, the method predicts that a single experimental run at $J\approx20$, $N\approx100$ on a Rydberg platform could already put continuum non-equilibrium QFT predictions (such as false-vacuum decay rates) to a quantitative test without classical benchmarks.
Editorial inference: the compactness of the field inherited from the spin regularization is not just a truncation artifact—it may be turned into a feature for studying periodic-target theories such as the axion or compact QED, where topological sectors and monopole physics are the object of interest.
Editorial inference: a natural testable extension is to extract the spreading velocity $\sigma_v$ of a moving soliton at several $\beta^2$ values and verify the predicted $\sigma_v\propto\beta/\sqrt{\gamma}$ scaling directly, which would independently confirm the semi-classical model without relying on the mass-gap benchmark.
Editorial inference: the resource trade-off question the paper raises—local Hilbert-space dimension versus number of sites—could be settled by a fixed-total-cost comparison of this large-spin encoding against a spin-1/2 XXZ realization of the same sine-Gordon dynamics, including noise and gate fidelities; the paper's conjecture of built-in error robustness would be tested by such a comparison.

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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

Colleague,

This paper is a solid numerical demonstration of a large-spin regularization for scalar QFTs, with the sine-Gordon model as the test case. The genuinely new piece is the systematic two-step extrapolation protocol (linear in 1/[J(J+1)], then quadratic in 1/N) plus the first MPS benchmarks of soliton-antisoliton scattering with continuum-extrapolated position shifts. The operator mapping itself is standard — Haldane and Zache et al. covered it — but the quantitative recipe is new.

What the paper does well: the benchmarks are extensive and mostly convincing. Mass gap, ground-state energy density, and vertex-operator exponents agree with exact sine-Gordon predictions after the extrapolations. The scattering shifts in Fig. 6 also land near the quantum prediction using renormalized parameters from equilibrium fits. That is a real achievement. The appendices are unusually thorough about neglected interactions, convergence, and error estimation.

Soft spots. The extrapolation ansatz is empirical; the paper never derives the leading 1/[J(J+1)] correction or bounds the m-distribution that would justify it. The stress-test note is fair: if that correction law is wrong, all continuum numbers shift. But the evidence that it works is not bad — the data collapse and the consistency across observables give it weight. The bigger soft spot is that some comparisons are not fully ab initio: renormalized beta^2 is extracted from the same simulations that are then checked against the mass formula, and Z_beta rescales the vertex operators. That puts part of the equilibrium comparison close to curve matching. The scattering comparison is more independent, using equilibrium-extracted parameters, so it carries weight. No code or data are shipped, which limits reproducibility.

Who it is for: people working on analog quantum simulation of field theories, especially Rydberg or qudit platforms. It deserves a serious referee; the central claim — that large-spin truncation plus extrapolation gives quantitative continuum physics — is well supported within the small-beta^2 regime. Larger beta^2 and kappa > 1 remain open.

Recommendation: send to peer review. A critical referee should push on the extrapolation law and on releasing data/code, but the paper is a legitimate contribution.

Referee Report

4 major / 5 minor

Summary. The manuscript proposes a large-spin (qudit) lattice regularization of scalar quantum field theories, based on the operator identification e^{±iφ} ↔ J_±/√(J(J+1)) and π ↔ J_z in the Hamiltonian of Eq. (2). The central claim is that continuum physics can be recovered quantitatively by extrapolating in spin length J, system size N, and lattice spacing a (Sec. II B). The paper supports this claim with extensive MPS/DMRG/TEBD simulations of the sine-Gordon model, including CFT and Luttinger-liquid benchmarks, the mass gap, ground-state energy density, vertex expectation values, a test of the Lukyanov-Zamolodchikov conjecture, soliton preparation and propagation, soliton-antisoliton scattering shifts, and a non-integrable perturbation exhibiting string-breaking-like dynamics.

Significance. If the extrapolation is controlled, this is a significant step toward analog quantum simulation of scalar QFTs with qudit or Rydberg platforms, and the paper's numerical evidence is extensive. The agreement with exact sine-Gordon formulas for the mass gap, ground-state energy density, vertex scaling exponents, and scattering shifts is genuinely impressive, and the soliton preparation protocol plus the truncated-Wigner semi-classical model are useful contributions in their own right. The paper does not ship code or machine-checked proofs, and the extrapolation law and operator renormalization are empirical rather than derived, so the strength of the central claim is somewhat ahead of the evidence. Nevertheless, the scattering-shift comparison provides a nontrivial predictive test that goes beyond curve fitting.

major comments (4)

[Sec. II B 2 and App. D] The linear-in-1/[J(J+1)] extrapolation is the load-bearing step for every continuum number in the paper, but it is neither derived nor bounded. Equation (4) bounds the finite-J error per momentum eigenstate only by O(m(m±1)/[J(J+1)]), and the m-distribution of the low-energy many-body state is not controlled. App. F 2 shows that the neglected higher-derivative corrections depend on β², so the leading finite-J correction is coupling-dependent rather than universal. The authors themselves state in Sec. III B 2 that larger κ requires much larger J, which indicates that J=20 is not self-evidently asymptotic. Since the mass gap, vertex expectation values, renormalized β² and M0', and the scattering shifts in Figs. 2–6 all pass through this same ansatz, a wrong leading correction law would shift every quoted continuum result. Please derive the leading correction from a systematic 1/J expansion in the relevant low-energy subspace, or provide a convergence test (e.g., additional J values and a comparison of linear versus quadratic fits in 1/[J(J+1)]) with propagated systematic errors.
[Sec. III B 2, Eqs. (20)–(21), Fig. 4] Part of the equilibrium comparison is calibration rather than prediction. The renormalized β² is extracted from the same simulations via the log-log fit of Eq. (20), and M0' is then fixed through Eq. (21) using that β². In addition, the absolute value of the vertex operator in Fig. 4(c) is rescaled by a fitted factor Z_β, and Fig. 4(d) uses (Z_β)^{n²}. The genuinely predictive checks are the ground-state energy density of Eq. (15), the scaling exponent α=β²/4π, and the scattering shifts of Sec. V. The manuscript should state this limitation explicitly and provide propagated uncertainties for β² and M0' when comparing to exact formulas.
[Sec. V B, Fig. 6(b), App. H] The scattering-shift comparison uses the renormalized β² and M0' determined from the equilibrium analysis and the quasi-particle velocity extracted from the trajectories after the scattering event. The position-shift extraction itself has O(1) lattice-site systematic uncertainties, as described in App. H. The agreement with the theoretical predictions is encouraging, but the reported error bars do not fully establish the strength of the claim 'quantitative agreement.' Please report a full error budget for δx, including the propagation of uncertainties in β² and M0', and, if possible, identify one parameter set where β² and M0' are fixed a priori from the microscopic couplings without fitting to the same observables.
[Sec. VII] The paper itself notes that for simulating an unknown QFT it will be essential to perform an error-bounding analysis, and that the proposed approach is currently restricted to small β² due to higher-derivative corrections. These self-identified limitations are directly relevant to the main claim of 'quantitative predictions in the continuum limit.' Because the extrapolation ansatz is empirical, the manuscript should at least describe how an experiment or simulation could detect breakdown of the ansatz (e.g., by varying J in situ and checking stability of the extrapolated observables), rather than treating the linear-in-1/[J(J+1)] fit as established.

minor comments (5)

[Fig. 2 caption] The caption for panel (b) states that the red circle identifies the asymptotic value extrapolated in (a), but the red circle appears to denote the final N→∞ value; please clarify which marker corresponds to which extrapolation step.
[Eq. (21)] Equation (21) is described as formally equivalent to Eq. (9); the equivalence follows by substituting (β')⁴ from Eq. (9), but the text should spell this out for readers, since Eq. (21) no longer contains V'_nn explicitly.
[App. G 5, Eq. (G18)] The proportionality in Eq. (G18) is stated without a constant or a derivation; please specify the constant or explain the scaling argument fully, since the final result σ(v)≈0.35β depends on this prefactor.
[Abstract and Sec. VII] The abstract and conclusions describe the results as 'quantitative predictions,' but β² and M0' are calibrated from the simulations in the equilibrium section. Please adjust the wording to distinguish calibrated parameters from genuinely predicted observables such as the scattering shifts.
[Figs. 7 and 8] The color scale for the normalized energy density in the third row of Fig. 8 is not specified in the caption; please state how the normalization is performed and what the color range represents.

Circularity Check

0 steps flagged · score 4.0 of 10

Partial circularity: fitted β² and a fitted vertex rescaling Z_β enter before the same sG data are presented as quantitative agreement; the mass-gap, CFT-scaling, and scattering benchmarks retain independent content.

full rationale

The central continuum benchmarks are not circular: the mass-gap scaling with M′0 (Fig. 2c), the central charge c = 1 (Fig. 3a), the ground-state energy density (Fig. 4b), and the soliton-antisoliton scattering phase shifts (Fig. 6b) are compared with external analytical results, and the finite-J linear fit in 1/[J(J+1)] followed by the quadratic fit in 1/N is an openly empirical extrapolation ansatz rather than a derivation smuggled in as a prediction. The paper itself flags the main limitation: larger κ would require much larger J, and an error-bounding analysis is left for future work; these are correctness risks, not circularity. The self-citations ([33], [45], [100]) are not load-bearing: the operator identification is derived in Eq. (4), and the Rydberg implementation is presented as a possible platform, not as the argument for the continuum limit. The partial circularity that raises the score to 4 is the parameter-matching loop: β2 is measured from the same sG simulations and then used through Eq. (21) to set M′0 for the mass-gap comparison, while the absolute vertex amplitude is rescaled by a fitted Zβ before being reported as consistent with Eq. (14). These steps reduce part of the 'quantitative agreement' to curve matching, but they do not force the mass-gap scaling, the CFT data, or the scattering shifts, so the paper is not wholly circular.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The core contribution rests on known exact sine-Gordon results and an empirical extrapolation ansatz. Two fitted multiplicative or kinematic factors (Z_beta, v_spr) and several self-citations to the Rydberg SO(4) proposal [33] moderate the ledger, but no genuinely new physical entity is introduced.

free parameters (5)

renormalized beta^2 = pi/20, pi/13, pi/10 and related values
Extracted from MPS correlations via Eq. (20) or the CFT relation beta^2 = pi kappa^2 K, and later used to set the mass scale through Eq. (21).
mass scale M0' (effective) = values ~0.1 to 0.3 in sine-Gordon units
Set through Eq. (21) using the fitted beta^2 instead of the naive microscopic mapping Eq. (9); Figure 2(c) then uses this M0' to compare with the exact mass-gap formula, so the comparison is not fully parameter-free.
operator renormalization Z_beta = Z_beta = 1 + O(beta^2), up to about 9% correction
Multiplicative factor used to match the absolute value of the vertex operator prediction Eq. (14); higher powers are rescaled by Z_beta^(n^2), making the Lukyanov-Zamolodchikov test partly normalization-fitted.
spreading velocity v_spr = v_spr = 0.273 beta
Fixed by the smallest M0'=0.15 simulation and then used as sigma_v in the semi-classical Wigner model and in the collapse of Fig. 5(e).
initial Gaussian widths sigma_rho0, sigma_cos0 = 1.237/M0 and 0.794/M0
Fitted from the t=0 charge and cosine profiles; close to the classical Gaussian approximation (1.253/M0 and 0.798/M0), but the exact numerical values are data-derived.

assumptions (6)

domain assumption The identification e^{±i phi_i} ~ J_pm/sqrt(J(J+1)) and pi_i ~ J_z is exact only for states with |m| << J, with corrections of order m(m±1)/(J(J+1)).
Invoked in Eq. (4) and used to justify the large-spin limit.
domain assumption The continuum derivative term is obtained by truncating cos(phi_i - phi_j) to second order; higher-order terms are assumed to only renormalize couplings and not change the universality class.
Section II C and App. F 2 identify the neglected terms and attribute the beta^2 renormalization to them.
ad hoc to paper Extrapolations are controlled by linear-in-1/J(J+1) and quadratic-in-1/N fits.
Empirical fit forms introduced in Sec. II B 2 and App. D; no derivation shows these are the correct corrections.
domain assumption Ising and next-nearest-neighbor dipole terms are negligible in the J to infinity limit or produce corrections that can be absorbed into renormalized parameters.
App. F 1 shows numerically that the Ising term vanishes as 1/J(J+1) and NNN shifts are up to 9%, but the continuum claim assumes these corrections remain controlled.
standard math Known exact results for the sine-Gordon mass spectrum, vertex expectation values, and S-matrix are used as benchmarks.
Eqs. (12)-(15) and (31)-(32) come from the integrability literature and are accepted as external targets.
ad hoc to paper The truncated Wigner model restricts to the single-soliton sector and assumes independent Gaussian distributions in position and velocity fluctuations.
Eq. (27) and App. G 4 introduce this phenomenological model; it is used to explain the observed delocalization rather than derived from first principles.

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Cite this review

Pith. "Pith review of Quantum simulating continuum field theories with large-spin lattice models." pith.science (2026). https://pith.science/paper/ACZW42PV

@misc{pith2026241215325,
  author       = {Pith},
  title        = {Pith review of: Quantum simulating continuum field theories with large-spin lattice models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ACZW42PV}},
  note         = {Machine review of arXiv:2412.15325}
}

read the original abstract

Simulating the real-time dynamics of quantum field theories (QFTs) is one of the most promising applications of quantum simulators. Regularizing a bosonic QFT for quantum simulation purposes typically involves a truncation in Hilbert space in addition to a discretization of space. Here, we discuss how to perform such a regularization of scalar QFTs by explicitly constructing suitable many-body lattice Hamiltonians using multi-level or qudit systems, and show that this enables quantitative predictions in the continuum limit by extrapolating results obtained for large-spin models. With extensive matrix-product state simulations, we numerically demonstrate the sequence of extrapolations that leads to quantitative agreement of observables for the integrable sine-Gordon (sG) QFT. We further show how to prepare static and moving soliton excitations, and analyze their scattering dynamics in the continuum limit, in agreement with a semi-classical model and with quantitative analytical predictions. Finally, we illustrate how a non-integrable perturbation of the sG model gives rise to dynamics reminiscent of string breaking and plasma oscillations in gauge theories. Our methods are directly applicable in state-of-the-art analog quantum simulators, opening the door to quantitatively investigating a wide variety of scalar field theories and tackling long-standing questions in non-equilibrium QFT like the fate of the false vacuum.

Figures

Figures reproduced from arXiv: 2412.15325 by the authors.

**Figure 1.** Overview. (a) d-dimensional square lattice of particles with nearest-neighbor interactions. We assume each particle to be described by a large spin of length J, with ⟨Jˆ+⟩ ∼ ⟨e iφˆ ⟩ and ⟨Jˆz⟩ ∼ ⟨πˆ⟩ identifying field theory and lattice operators, as described in Sec. II. Continuum physics is extracted by a sequence of extrapolations in spin length J, system size N and lattice spacing a [see [PITH_FULL_IMAGE:figure… view at source ↗

**Figure 2.** Extrapolating to the continuum limit of the simulated sine-Gordon model. (a)-(c) Extrapolation to the continuum value of the mass gap ∆E. We consider the d = 1 lattice Hamiltonian Hˆlatt Eq. (2) and determine the mass gap ∆E in the continuum limit by simulating the model at several finite J = 12, ..., 20 and N = 30, ..., 100. (a) For fixed N = 100, V ′ nn/χ = 40.53, λ′ κ/χ = 0.20 we perform a linear fit (solid black… view at source ↗

**Figure 3.** Characterization of the CFT. (a) The energy gap ∆Elatt/χ in lattice units extracted from our lattice regularization Hˆlatt clearly shows a gapless phase at small χ/V ′ nn. In the same region, the central charge c (inset) is compatible with 1. The dotted black line delimitates the simulability region. (b) The Luttinger parameter K p of the CFT as a function of the microscopic parameters χ/V ′ nn (gray squares) fo… view at source ↗

Figures from the paper (11 more)

**Figure 4.** Figure 4: Renormalized sG QFT. (a) Coupling β 2 as a function of the microscopic parameters κ 2p χ/V ′ nn. The colored dots resp. the yellow diamonds represent the β 2 values extracted in the sG model for κ = 1 resp. κ = 2 with a fit according to Eq. (20) [see inset]. The extra…

**Figure 5.** Figure 5: Soliton preparation and dynamics. (a)-(b) Preparation of the semi-classical soliton state: (a) we first apply site-dependent z-rotations with an angle βφ(i) s (green shaded area) to rotate in the equatorial plane the spins (initially pointing in +x-direction). (b) We …

**Figure 6.** Figure 6: Soliton-antisoliton scattering. (a) Heatmap of the excess lattice energy density ¯εi as a function of lattice site i and time t, with quasi-particle trajectories indicated by the black solid lines. To estimate the scattering position shift, we fit the trajectories with…

**Figure 7.** Figure 7: Beyond integrability. (a) Interaction potential V ( ˆφ)β 2 /M2 0 of Eq. (35) for p = 0 (gray line) and p = 0.5 (green line). (b) Energy density εi (in sine-Gordon units) of the soliton-antisoliton static pair (dashed black) for the unperturbed p = 0 (solid gray) and t…

**Figure 8.** Figure 8: Dynamics of a soliton-antisoliton pair in the perturbed sine-Gordon model. We investigate the dynamics of an initially static soliton and antisoliton for the parameters J = 22, β2 = π/20, M′ 0 = 0.2, ∆s¯s/a = 40. The perturbation strength p of the perturbed sine-Gordon…

**Figure 9.** Figure 9: Conformal field theory and impact of nextnearest-neighbor interaction. (a) Energy gap in lattice units ∆Elatt and (inset) central charge c as a function of the microscopic parameters ratio p χ/V ′ nn for nearest-neighbors only (NN, blue dots), and next-nearest-neighbo…

**Figure 10.** Figure 10 [PITH_FULL_IMAGE:figures/full_fig_p023_10.png]

**Figure 11.** Figure 11: Extrapolation of the mass M′ 0. The black dots represent the values M′ 0 determined according to Eq. (21) from the effective β 2 parameter. The colored crosses are instead the values extrapolated upon inverting Eq. (13). Different shaded of green corresponds to diff…

**Figure 12.** Figure 12: One- and two-point correlation functions of the prepared soliton state. Comparison of selected expectation values for different initial states: the solid black line represents the ground state of the quantum sG model, the blue circles the semi-classical static solito…

**Figure 14.** Figure 14: (a) we observe that the topological charge at time t = 0 follows a quadratic dependence in γM′ 0 in the large-spin (J → ∞) and classical (β 2 → 0) limit. Moreover, the topological charge is conserved over time up to t ≈ 50, the collision time of the free excitations …

**Figure 15.** Figure 15: Approximation with Gaussian functions. (a) Topological charge density ρ(x) of the classical soliton at t = 0 Eq. (G3) (solid blue line) and comparison with the Gaussian approximation Eq. (G5) (dashed orange line). (b) 1 − cos(βφˆ) of the classical soliton at t = 0 Eq.…

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In App. F 4 we discuss the Lukyanov- Zamolodchikov conjecture
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(2)], obtained from the Rydberg Hamiltonian ˆHRyd [Eq

Neglected terms in the dipole-dipole interaction The sG model can be simulated with the 1D lattice Hamiltonian ˆHlatt [Eq. (2)], obtained from the Rydberg Hamiltonian ˆHRyd [Eq. (B1)] by neglecting higher-order dipole-dipole interaction term Vij (j > i+ 1) and the Ising contri...
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We further illustrate the dependency of the neglected terms on the microscopic parameter κ

β2-renormalization and κ-dependence Here we explicitly discuss the neglected terms in the Taylor expansion of the cosine term, which we expect to be responsible for the renormalization of the β2 parame- ter. We further illustrate the dependency of the neglected terms on the mi...
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(21) from the effective β2 parameter

The black dots represent the values M ′ 0 determined according to Eq. (21) from the effective β2 parameter. The colored crosses are in- stead the values extrapolated upon inverting Eq. (13). Differ- ent shaded of green corresponds to different ratios 2 p λ′κ/V ′nn, i.e., to di...
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We further observed that the value of the bare mass M ′ 0 also differs from the microscopic identification Eq

M ′ 0 renormalization In the main text we discussed the extrapolation of the effective β2 parameter of the simulated sG model. We further observed that the value of the bare mass M ′ 0 also differs from the microscopic identification Eq. (9), and that it can be fixed in depend...
[107]

Lukyanov-Zamolodchikov conjecture In the main text we introduce the expectation value ⟨einβ ˆφ⟩ and discuss the agreement of numerical results with the unproven conjecture formulated by Lukyanov and Zamolodchikov in Ref. [48]. According to their work, the expectation value ⟨ei...
[108]

classical

State preparation and correlation functions In the main text we discuss how, by applying two sets of gates { ˆU (i) p , ˆU (i) m } on the ground state of the sG Hamil- tonian, we prepare the system in a state whose expecta- tion values of the phase ⟨ ˆφ⟩ = φs and of the moment...
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In the following, we study the soliton dynamics for N = 101, J= {16, 18, 20}, β2 = {π/20, π/13, π/10}, and M ′ 0 = {0.1, 0.15, 0.2, 0.25, 0.3}

Time evolution of the static soliton Here we provide additional details regarding the time evolution of the static soliton. In the following, we study the soliton dynamics for N = 101, J= {16, 18, 20}, β2 = {π/20, π/13, π/10}, and M ′ 0 = {0.1, 0.15, 0.2, 0.25, 0.3}. According...
[110]

Similarly to the static soliton, in Fig

Time evolution of the moving soliton We now instead investigate the quench dynamics of the moving soliton ( v ̸= 0) for N = 101 , J = {16, 18, 20}, β2 = {π/20, π/13, π/10}, and M ′ 0 = {0.15, 0.2, 0.25}. Similarly to the static soliton, in Fig. 14(a) we observe that the topolo...
[111]

In this section 27 we provide additional details

Semi-classical phenomenological model As discussed in the main text, we can engineer a semi- classical phenomenological model, which captures the nu- merically observed spreading dynamics. In this section 27 we provide additional details. a. Theoretical model We start by consi...
[112]

The resulting linear dependence of σ(v) on β explains the numerically observed spreading of observables in time

V elocity fluctuations from ground-state quantum fluctuations Here we illustrate the relation between the variance of ground-state spin operators and the standard deviation of the velocity σ(v). The resulting linear dependence of σ(v) on β explains the numerically observed spr...
[113]

Due to the mutual cancellation of the mass term, we obtain X i,j∈F ⟨ ˆJ (i) z ˆJ (j) z ⟩c ∝ NF M ′ 0 β2 ∝ M ′ 0/M′ 0 β2 = 1 β2 . (G24) In particular, for κ = 1 we get (neglecting a subleading M ′ 0-dependence ≈ −1.4M0) X i,j∈F ⟨ ˆJ (i) z ˆJ (j) z ⟩c ≈ 5 β2 , (G25) and thus for...
[114]

(G27) Note that the mass term M ′ 0 in the numerator does not get an additional γ factor, because it is directly set by the quantum ground state for which no velocity is defined

sites, and leading to X i,j∈F ⟨ ˆJ (i) z ˆJ (j) z ⟩c ∝ NF M ′ 0 β2 ∝ 1 β2γ . (G27) Note that the mass term M ′ 0 in the numerator does not get an additional γ factor, because it is directly set by the quantum ground state for which no velocity is defined. Therefore, the spread...

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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