Floating Phases in One-Dimensional Rydberg Ising Chains

Andreas M. L\"auchli; Michael Rader

REVIEW 3 major objections 5 minor 1 cited by

Floating Phases in One-Dimensional Rydberg Ising Chains

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A one-dimensional Rydberg Ising chain with van der Waals interactions hosts extended floating phases, not just direct crystal-to-disorder transitions, and these phases sit at parameters current quantum simulators can reach.

desk verdict First quantitative map of floating phases in the 1/r^6 Rydberg chain; the central existence claim is solid, but the phase boundaries rest on a c=1 classifier that the authors themselves admit may overreach. read the letter →

arxiv 1908.02068 v1 pith:EKID6BAZ submitted 2019-08-06 cond-mat.quant-gas cond-mat.stat-mechquant-ph

Michael Rader , Andreas M. Läuchli This is my paper · ORCID

classification cond-mat.quant-gascond-mat.stat-mechquant-ph

keywords RydbergatomsfloatingphaseincommensurateorderLuttingerliquidquantumdiagramvanderWaalsinteractiontensornetworkcentralcharge

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 computes the ground-state phase diagram of a one-dimensional chain of Rydberg atoms with van der Waals $1/r^6$ interactions and argues that a third kind of phase, the floating phase, occupies substantial regions of it. In a floating phase the Rydberg density forms a gapless, incommensurate quasi-crystal: correlations decay as a power law and the ordering wavevector drifts continuously with the Rabi frequency and detuning instead of locking to the lattice. The computed floating phases appear between the known $1/q$ crystalline lobes and the disordered phase, fully coat the $1/5$ lobe, and separate the $1/3$ and $1/4$ crystals, at parameters within reach of current Rydberg quantum simulators. The result matters because these phases had been predicted theoretically but not located numerically, and their presence changes how experiments should interpret the melting of the lattice-locked crystals.

What carries the argument

The central object is the floating phase itself, identified with two numerical diagnostics. The first is the scaling of entanglement entropy $S$ with correlation length $\xi$, fitted to $S=\frac{c}{6}\log\xi+\text{const.}$, which yields $c\approx 1$ in floating regions and $c\approx 1/2$ at the Ising transition out of the $1/2$ crystal. The second is the static structure factor $S(k)$ computed from density correlations, which shows power-law diverging peaks at a wavevector that drifts incommensurately as parameters are varied and interpolates between the Bragg peaks of neighbouring crystals. The ground states come from infinite matrix product state simulations, with the long-range $1/r^6$ interaction represented as a sum of ten exponentials and gapless energies extrapolated in $1/\xi^2$.

What would settle it

Measure the static structure factor along the $x=4$ cut of the phase diagram in a Rydberg chain: if sharp Bragg peaks or an energy gap persist where the paper predicts power-law diverging peaks at drifting wavevectors, the floating phase is absent or narrower. Numerically, recomputing the boundaries without the $c=1\pm 5\%$ cutoff, for example with explicit Kosterlitz-Thouless scaling fits, would confirm or shrink the reported floating regions.

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Extended reading notes

Core claim

The paper's central claim is that the Hamiltonian $H=-\frac{\Omega}{2}\sum_j\sigma^x_j-\Delta\sum_j n_j+V\sum_{j<l}\frac{n_j n_l}{(l-j)^6}$ has extended floating phases in its ground-state phase diagram, not only direct transitions from commensurate $1/q$ crystals to a disordered phase. A floating phase is a one-dimensional incommensurate 'solid' that is gapless, has central charge $c=1$, and shows power-law density correlations whose dominant wavevector varies continuously with parameters. The numerical data place these phases between the $1/3$ and $1/4$ lobes and around the $1/5$ lobe, and suggest that all $1/q$ lobes with $q\ge 5$ are fully immersed in floating phase. The $1/2$ crystal is the exception: it melts through a direct Ising transition with central charge $c=1/2$, with no floating phase coating it.

Load-bearing premise

The classification of a region as a floating phase rests on the automatic criterion $c=1\pm 5\%$ from entanglement scaling, and because the adjoining transitions are Kosterlitz-Thouless with logarithmic corrections, this criterion may overestimate how wide the floating phases are.

Editorial extensions

If this is right

For the $1/q$ crystals with $q\ge 3$, crystal melting is generically a two-stage process: commensurate crystal, then floating phase, then disordered phase, rather than a single direct transition.
The $1/5$ lobe is fully coated by a floating phase, so a sweep from the disordered phase toward this crystal crosses an extended gapless incommensurate region before ordering sets in.
Kibble-Zurek scaling measured on existing Rydberg simulators must be read with these intermediate gapless regions in mind, since the transition out of the floating phase is not the same universality class as the direct transition.
The $1/2$ lobe remains a special case: its melting is direct and Ising-like, so the floating phase does not appear there.

Reading between the lines

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

Not drawn in the paper: the same floating phases should appear in finite chains large enough to accommodate the incommensurate period, so a chain of roughly fifty sites could detect them as a slowly drifting ordering peak when sweeping across the $1/5$ lobe.
A testable extension would be to measure the wavevector of the structure-factor peak as a function of detuning and compare its approach to the commensurate value against the square-root singularity expected at the crystal edge.
The $c=1\pm 5\%$ cutoff leaves room for a tighter classification: a follow-up using Kosterlitz-Thouless finite-entanglement scaling could sharpen the boundaries and test whether the floating phase truly touches the $1/4$ and $1/3$ lobes.

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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

Bottom line: this is the first quantitative phase diagram of the van der Waals Rydberg chain that includes the floating phase, and the central identification is credible. The paper deserves a real referee. But the plotted phase boundaries are less secure than the figure suggests, and I would want that addressed before quoting numbers.

What's new: Weimer-Buchler and Sela et al. predicted floating phases in this class of models, and hard-boson/dipolar studies saw slivers, but nobody had mapped them in the actual 1/r^6 Rydberg chain. The authors do it with iMPS, using central charge from entanglement scaling plus structure factor peak drift. The interior of the floating phase is supported by c~1 points and drifting incommensurate peaks. That part is solid.

Where it gets soft: the automatic classifier is c = 1 ± 5%, and they admit this may overestimate the floating phase extent. The stress-test note is right that this is not a footnote: the disorder-to-floating transition is KT, so with finite bond dimension a gapped near-critical state can masquerade as c=1 over a sizable window. That means the gray regions in Fig. 1, especially near the KT boundary and the fully coated 1/5-lobe, should be treated as provisional. The 1/5-lobe coating itself is partly extrapolated, not directly simulated. Also, boundaries are guides to the eye with no error bars. None of this kills the main result—there is a genuine floating phase—but it does mean the quantitative extent is not yet nailed down.

Citation pattern looks honest. They cite the relevant prior predictions and related models, and they flag where their results disagree (no floating phase near the 1/2 lobe, contrary to Sela et al.). No code or raw data shipped, which is a shame for a numerical paper, but the method is standard TeNPy and reproducible in principle.

Who should read it: anyone planning Rydberg-chain experiments, and people working on commensurate-incommensurate transitions in 1D. I would bring it to the reading group.

Verdict: send it to peer review. Ask the authors to add a more careful boundary determination—for example, scaling of the structure factor exponent or a KT collapse—or at least show how the c=1 region moves with bond dimension.

Referee Report

3 major / 5 minor

Summary. The paper reports a quantitative ground-state phase diagram for the one-dimensional van der Waals Rydberg Ising chain described by Eq. (1), using infinite matrix product state (iMPS) simulations with iDMRG and the TeNPy library. The authors identify the usual gapped crystalline 1/q lobes and the disordered phase, and additionally locate extended gapless, incommensurate "floating" phases that interpolate between crystalline lobes and the disordered regime. The floating phase is diagnosed by extracting a central charge c from the entanglement-entropy scaling relation S = (c/6) log ξ + const. (Eq. (2)) and by tracking the parameter-dependent drift of structure-factor peaks (Eq. (3)). They conclude that floating phases are prominent in experimentally accessible parameter ranges and that the 1/5-lobe is fully coated by a floating phase. The paper is a numerical study without analytic derivations, and it includes explicit caveats about the automatic classification criterion.

Significance. If the reported phase diagram is quantitatively correct, this is an important result for the Rydberg-array quantum simulator community: it predicts a gapless, incommensurate phase in a parameter regime that existing experiments can reach, and it clarifies the nature of the melting transitions of the crystalline lobes. The work uses standard and independently motivated numerical diagnostics (central charge scaling and structure-factor peak drift) rather than fitting to a preconceived phase diagram, and it explicitly acknowledges the main methodological limitation. The existence of at least some floating phase is well supported by interior points with c ≈ 1 and by the smoothly drifting structure-factor peaks. The main weakness is that the quantitative extent of the floating phase, which is the paper's central quantitative claim, rests on a c = 1 ± 5% threshold that the authors themselves state may overestimate the floating-phase extent, and the phase boundaries in Fig. 1 are drawn as guides to the eye without error estimates.

major comments (3)

[Methods (automatic classification) and Fig. 1] The automatic classification of floating phase via the c = 1 ± 5% criterion in Eq. (2) is load-bearing for the quantitative extent of the gray regions in Fig. 1. The authors note that the extent "might be slightly overestimated" because the disorder-to-floating transition is Kosterlitz-Thouless, with logarithmic corrections. This caveat is stronger than "slight" in practice: for a KT transition the correlation length diverges exponentially, and with finite bond dimension χ the iMPS correlation length is bounded, so a gapped phase sufficiently close to the KT point can mimic c ≈ 1 over a sizable parameter window. I request a quantitative assessment: for representative parameter points near the gray boundaries, show convergence of c as χ increases, or replace/augment the c criterion with a more discriminating diagnostic (e.g., the Luttinger parameter K or the power-law exponent of the structure-factor peak) and provide error bars on the phase boundaries.
[Fig. 1 and phase-diagram determination] The phase transition lines in Fig. 1 are explicitly described as "guides to the eye," and the transition points are said to come from "a series of simulations," but the procedure for locating each transition point and the associated uncertainty are not given. Since the paper's stated goal is a "quantitative ground state phase diagram," the lack of error bars or a defined crossing criterion for the dots undermines the quantitative claim. Please specify how each transition point is identified (e.g., maximum of some susceptibility, crossing of energy estimates, or drift of density imbalance) and report uncertainties, at least for representative cuts such as x = 4 and y = 3.6.
[Discussion of the 1/5-lobe and q ≥ 5 lobes] The statement that the 1/5-lobe is "fully coated" and that "all 1/q-lobes for q≥5 are fully immersed into the floating phase" goes beyond the simulation data shown: Fig. 1 shows only a finite region, and the text justifies the q≥5 extrapolation by arguments in Refs. [15, 16]. This is a plausible conjecture but should be explicitly labeled as an extrapolation rather than as a numerical result. In its current form, the abstract and conclusion present this as an established part of the phase diagram, which overstates the evidence.

minor comments (5)

[Footnote [39]] There is a typographical error: "R = 10000 for the remaining. phases" should read "R = 10000 for the remaining phases."
[Method (unit-cell notation)] The symbol "Luc" is used for the iMPS unit-cell size, but it is not typeset as a subscript; please use "L_uc" for readability and define it before first use.
[Eq. (3)] The structure factor is defined up to proportionality, which is fine, but the normalization convention (including R and possible site factors) should be stated explicitly, since the comparison of peak sharpness between different R values in Figs. 3 and 4 is qualitative.
[Structure-factor interpretation] The structure-factor panels are interpreted qualitatively as showing power-law diverging peaks, but no power-law exponent is extracted from the finite-R data (R up to 10000 in Fig. 3 and Fig. 4). A quantitative extraction of the peak exponent as a function of x or y would strengthen the identification, even if only for one representative cut.
[Fig. 2] The two examples in Fig. 2 are useful sanity checks (c ≈ 1 in the floating phase and c ≈ 0.5 at the Ising transition), but the figure would benefit from indicating the range of χ values used in each fit and the fit residuals, so the reader can judge the quality of the linear scaling.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: floating-phase diagnosis uses independent central-charge and structure-factor data; the stated KT caveat affects quantitative precision, not the derivation.

full rationale

The paper does not reduce any claimed result to its inputs. The floating phase is identified through two independent diagnostics: the central-charge scaling S=(c/6) log xi + const. (Eq. 2) and the structure factor S(k) (Eq. 3), while crystalline phases are distinguished by density imbalance. The c=1±5% threshold in the Methods section is a classification rule, not a parameter fitted to the reported phase diagram; changing the threshold would shift boundary locations but would not make the existence of a c≈1 region an identity. The authors' own caveat that 'the extent of the floating phase might be slightly overestimated using this criterion' concerns Kosterlitz-Thouless finite-entanglement accuracy, not circularity. No load-bearing self-citation or imported uniqueness theorem is used; citations to prior work [15,16,19] supply external predictions that this work tests. Thus there is no circular step.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new entities or parameters beyond the model constants. The main ledger items are the hand-chosen c = 1 ± 5% classification threshold and the ten-exponential approximation of the long-range interaction, both of which affect the quantitative phase boundaries.

free parameters (2)

Central charge classification threshold = c = 1 ± 5%
Hand-chosen criterion to declare a system floating; the authors note it may overestimate the floating-phase extent due to KT logarithmic corrections.
Number of exponentials in interaction decomposition = 10
The 1/r^6 interaction is approximated by a sum of ten exponentials; the accuracy of this fit is not quantified for this model.

assumptions (5)

domain assumption The van der Waals 1/r^6 interaction approximated by a sum of ten exponentials faithfully represents the infinite-range Hamiltonian.
Method section: accurate decompositions into a sum of exponentials are a known way to proceed; no error estimate is given for this specific model.
standard math The finite-entanglement scaling S = (c/6) log(ξ) + const applies at the simulated points and yields the central charge.
Used to extract c from iMPS; based on Pollmann et al. [38].
domain assumption Central charge c = 1 identifies the floating phase (Luttinger liquid), and c = 1/2 identifies the Ising transition.
Standard CFT classification, used throughout the analysis.
standard math The spin-chain mapping and the symmetry (Δ,n)->(2ζ(6)-Δ,1-n) justify restricting the study to Δ ≤ ζ(6).
Footnote [22] establishes this mapping and symmetry.
domain assumption The chosen unit cell L_uc of the iMPS is large enough that the variational ground state captures the correct phase.
Method section: incompatible unit cells can stabilize wrong phases; the authors compare energies to select the true ground state.

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Cite this review

Pith. "Pith review of Floating Phases in One-Dimensional Rydberg Ising Chains." pith.science (2026). https://pith.science/paper/EKID6BAZ

@misc{pith2026190802068,
  author       = {Pith},
  title        = {Pith review of: Floating Phases in One-Dimensional Rydberg Ising Chains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EKID6BAZ}},
  note         = {Machine review of arXiv:1908.02068}
}

read the original abstract

We report on the quantitative ground state phase diagram of a van der Waals interacting chain of Rydberg atoms. These systems are known to host crystalline phases locked to the underlying lattice as well as a gapped, disordered phase. We locate and characterize a third type of phase, the so called floating phase, which can be seen as a one-dimensional 'crystalline' phase which is not locked to the lattice. These phases have been theoretically predicted to exist in the phase diagram, but were not reported so far. Our results have been obtained using state-of-the-art numerical tensor network techniques and pave the way for the experimental exploration of floating phases with existing Rydberg quantum simulators.

Figures

Figures reproduced from arXiv: 1908.02068 by the authors.

**Figure 2.** FIG. 2. Scaling of the entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** FIG. 3. Vertical strips show the structure factor [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. Vertical strips show the structure factor [PITH_FULL_IMAGE:figures/full_fig_p003_4.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Schauß, M

P. Schauß, M. Cheneau, M. Endres, T. Fukuhara, S. Hild, A. Omran, T. Pohl, C. Gross, S. Kuhr, and I. Bloch, Obser- vation of spatially ordered structures in a two-dimensional Ry- dberg gas, Nature 491, 87 (2012)

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