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A sine-square deformation approach to quantum critical points in one-dimensional systems

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Pith's one-line read Quantum critical points in one-dimensional chains can be located by finding the parameter value where a local observable becomes site-independent in the sine-square-deformed ground state; the paper demonstrates four-decimal accuracy from sy

desk verdict A promising SSD-based heuristic for locating QCPs that works on the tested Ising chains, but the unproven converse and trivial uniform points need addressing before it is a general method. read the letter →

arxiv 2512.14149 v3 pith:5NMAEMRZ submitted 2025-12-16 cond-mat.str-el quant-ph

Yuki Miyazaki , Shiori Tanigawa , Giacomo Marmorini , Nobuo Furukawa , Daisuke Yamamoto This is my paper · ORCID
classification cond-mat.str-elquant-ph
keywords sine-squaredeformationquantumcriticalpointIsingchaininmixedfieldslong-rangeinteractionsfinite-sizescalingexponentsRydbergatomarraystranslationalinvariance
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 proposes a practical criterion for locating quantum phase transitions in one-dimensional systems: in the ground state of a Hamiltonian whose couplings are modulated by a sine-square envelope, any local observable should become translationally invariant when the system is gapless. Taking differences of the local transverse magnetization at several pairs of sites, the authors find that these differences vanish or change sign at the same field, and extrapolating their zero-crossing and minimum positions with a shared quadratic form yields quantum critical points with four-decimal accuracy from chains of up to 84 sites. The method is validated on the mixed-field antiferromagnetic Ising chain with nearest-neighbor and with 1/r^6 long-range couplings, and the resulting phase boundaries agree with earlier calculations. The same differences collapse under a standard finite-size scaling form, suggesting that critical exponents of the Ising universality class can also be extracted. A Rydberg-atom arrangement is proposed to realize the sine-square-deformed Hamiltonian experimentally.

What carries the argument

The central object is the sine-square deformation (SSD), a site-dependent modulation f_L(i)=sin^2(pi/L (i-1/2)) applied to every term of the Hamiltonian, which suppresses boundary effects while retaining open boundaries. The argument is carried by six differences Δ_1,...,Δ_6 between expectation values of the transverse magnetization at pairs of sites (L/2, L/3, L/4 relative to site 1): according to the paper's proposition, all of them vanish or change sign at the quantum critical point. The quantitative estimate comes from a least-squares fit of the crossing and minimum fields to a quadratic polynomial in 1/L with a single common intercept, which forces all series to converge to one critical

What would settle it

Compute the sine-square-deformed ground state of a model with an extended gapless phase, such as the spin-1/2 XXZ chain in its XY regime, and check whether the local magnetization differences Δ_n vanish throughout the phase. If they do, the criterion would label every point of that phase as critical, falsifying the method's identifying power; if they vanish only at isolated parameter values, the converse half of the proposition would be supported.

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Extended reading notes

Core claim

The central claim is that sine-square deformation, which multiplies each local term of an open chain by sin^2(pi/L (i-1/2)), restores translational symmetry to the ground state exactly at a quantum critical point. Under this claim, six differences Δ_1,...,Δ_6 between transverse magnetizations at special pairs of sites all tend to vanish only at criticality. Finite-size extrapolation of the crossing and minimum positions of these Δ_n, with all series constrained to a common quadratic intercept, gives h_x^c=0.40165(7) at h_z=0.5 and h_x^c=0.26080(28) at h_z=0.75 for the nearest-neighbor chain, and shifts the long-range phase boundary to lower h_x, for example h_x^c=0.488019(2) at h_z=0. The sc

Load-bearing premise

The method hinges on the unproven claim that in the thermodynamic limit the sine-square-deformed ground state of any gapless one-dimensional system yields site-independent expectation values for every local observable, and on the practical converse that the only place this happens is the quantum critical point.

Editorial extensions

If this is right

  • The quantum critical point can be estimated with four-decimal precision from chains of only 84 sites, and results remain consistent when fitting data up to 36 sites, so small simulators can map phase boundaries.
  • For the mixed-field Ising chain, the method reproduces the known phase boundary from energy-gap calculations with smaller systems, and for the long-range 1/r^6 model it finds a slightly reduced antiferromagnetic region, with h_x^c=0.488019(2) at h_z=0, about 2.4% below the nearest-neighbor value.
  • The SSD observable Δ_1 obeys a finite-size scaling collapse with exponents consistent with the (1+1)-dimensional Ising universality class, opening the possibility of extracting critical exponents from the same data.
  • Because many independent pairs of sites can be used, multiple scaling conditions emerge that together constrain the critical point tightly.
  • The sine-square-deformed J1-J2 Ising chain can be approximately implemented with Rydberg atoms in optical tweezers using recursively determined zigzag spacings, making the method accessible to current quantum simulators.

Reading between the lines

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

  • If the underlying proposition is true in full generality, the criterion would also be satisfied at every point inside an extended gapless phase, not only at isolated critical points; the paper itself leaves the gapless-to-gapless case open, but the implication is that the method cannot separate those cases without extra input.
  • Trivial uniformity can occur away from criticality—for example at zero transverse field, where spin-flip symmetry makes the transverse magnetization site-independent, or at very large fields where the ground state is nearly fully polarized—so a practical implementation needs an additional rule to exclude such points.
  • The exact SSD-periodic-boundary equivalence proven for free-fermion chains suggests that in those models the method might locate critical points exactly, without extrapolation; testing it on the transverse-field XY chain would be a clean check.
  • The Rydberg realization is approximate, with a nonzero lower bound on J2/J1 around 1/64, so experiments would need to quantify how residual next-nearest-neighbor and third-neighbor couplings shift the apparent uniformity point.
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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 / 4 minor

Summary. The paper proposes a method to locate quantum critical points (QCPs) in one-dimensional systems using sine-square deformation (SSD). The central idea is that at a gapless critical point, the SSD ground state becomes translationally invariant in the thermodynamic limit, so local-observable differences Δ_n between sites should vanish. The authors apply this to the mixed-field Ising chain with nearest-neighbor and long-range interactions, computing ground states with DMRG and extrapolating the zero-crossing/minimum positions of Δ_n (n=1,...,6) as a function of 1/L with a common quadratic intercept. They obtain hx_c = 0.40165(7) at hz=0.5 for the nearest-neighbor model, consistent with previous DMRG, and a slightly shifted boundary for the long-range model (hx_c=0.488019(2) at hz=0). They further perform a scaling collapse of Δ_1 to extract the Ising exponents ν=1, β=1/8, and propose an experimental implementation of SSD J1–J2 couplings with Rydberg atom arrays.

Significance. If the method is valid, it offers a practical way to estimate phase boundaries from small open-boundary systems, which is relevant for quantum simulators. The numerical results are promising: the hz=0 case of the Ising chain has an exact SSD–PBC equivalence, and the nearest-neighbor QCPs agree with independent DMRG data. The long-range phase-boundary shift is a testable physical prediction. The paper also provides a concrete Rydberg-atom geometry to realize the SSD Hamiltonian, including an analytic bound on the realizable J2/J1 ratio. However, the foundational proposition is explicitly unproven, and the method's converse is not established; the paper's contribution as a 'method' is therefore currently a heuristic with supporting case studies rather than a proven algorithm. The exponent-extraction part is exploratory and would need independent verification.

major comments (3)
  1. [§2.1, Eq. (2)] The paper uses the site-independence criterion in the converse direction: a parameter is identified as the QCP when the differences Δ_n of Eq. (5) vanish or cross zero. However, the converse of proposition (2) is false without further restrictions. For the Hamiltonian (6) at hx=0, hz=0, the ground state is a product Néel state with ⟨S^x_i⟩=0 for all i, so all Δ_n vanish identically; likewise, in the limit hx→∞ the ground state is uniformly polarized along x, again giving Δ_n=0. These are non-critical parameter values where the same criterion is satisfied. The text and figures (e.g., Fig. 4) restrict the search to a small window around the known QCP, but no exclusion rule is stated. The method therefore is not well-defined as a general QCP locator; the authors should either provide an operational rule to exclude trivial uniform points or reframe the claim as a consistency/refinement test
  2. [§2.2, Eq. (7) and Appendix Tables 1–4] The selection of which Δ_n sequences contribute as zero crossings and which as local minima is made post hoc. For hz=0.5, n=1–3 are crossings and n=4 is a minimum; for hz=0.75, n=1–4 are minima (Tables 1–2). The same quantity changes type depending on hz and L, and no a priori criterion is given. This selection, together with the common-intercept quadratic extrapolation of Eq. (7), may overstate the precision: the reported errors are only statistical and do not include the uncertainty in the sequence selection. A robustness test (e.g., excluding one sequence at a time, or using a flat average over all possible selections) should be reported to establish that the QCP estimate is not an artifact of the chosen subset.
  3. [§2.4, Eq. (9)] The scaling-collapse analysis fixes hxc to the value obtained from the same SSD extrapolation for the long-range model (Sec. 2.3), so the collapse is not an independent test of the method. Moreover, the data are for hz=0, which is the special case where the SSD–PBC equivalence is exact (Sec. 2.2); this does not provide evidence that the scaling ansatz works in the interacting, non-exactly-solvable regime hz>0 where the QCP estimates are actually non-trivial. The authors acknowledge that a more systematic verification is required, but as it stands this section does not support the claim that critical exponents can be extracted. I recommend either removing this section or reworking it using independent hxc values and finite hz data.
minor comments (4)
  1. [§2.1] The sentence 'such as ˆOi = ˆSz i (widely employed here)' is inconsistent with the actual choice ˆOi = ˆS^x_i used throughout. Please correct.
  2. [§2.2] The claim of 'multiple independent scaling conditions' is overstated; the Δ_n are differences of four reference-site magnetizations and satisfy linear relations (e.g., Δ1 = Δ2 + Δ4). The common-intercept fit is still useful, but 'independent' should be replaced by 'multiple' or the relations should be acknowledged.
  3. [§2.4 / Fig. 9] The BSA results show non-monotonic finite-size behavior, and the text does not state how the four-size window is chosen or how error bars are estimated. A brief description would help reproducibility.
  4. [Throughout] There are several typos: 'futhermore' (Sec. 2.1), 'Physical Review Leters' (Ref. [9]), and missing spaces in 'interactionsand' (Sec. 3, first paragraph).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: QCP estimates are independent extrapolations benchmarked against external DMRG; the main caveat is an unproven one-way assumption, not a circular step.

full rationale

The central QCP estimates are obtained by locating zero crossings/minima of SSD local-magnetization differences Δ_n (Eq. (5)) and extrapolating them with the polynomial Eq. (7) to a common intercept p^(0). This is a genuine finite-size extrapolation of directly computed quantities, not a fit to a target critical field. The nearest-neighbor results (e.g., hx_c=0.40165(7) at hz=0.5) are checked against an independent DMRG phase boundary (Ref. [52]), and the hz=0 benchmark rests on the exact SSD–PBC equivalence proven in the external literature (Ref. [39]). No load-bearing premise is justified by a self-citation: the exact-solvability references are by other authors and are machine-checkable analytic results. In Sec. 2.4, hx_c from the same analysis is used as an input to the Δ_1 data collapse, which makes the exponent extraction self-consistent; however, the exponents c1,c2 are then fitted by Bayesian scaling analysis and compared with the known Ising values, so no quantity is defined in terms of the result it is supposed to predict. The paper's phrase 'we determine the QCP as the location where a local observable becomes site-independent' implicitly assumes the converse of Eq. (2), and the paper does not discuss trivial uniform states (e.g., hx=0 or hx→∞). This is an unproven-assumption / uniqueness concern, not a circular derivation, and the Conclusions candidly state that SSD–PBC equivalence is proven only for exactly solvable models.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central QCP estimates rest on the unproven uniformity proposition, a chosen quadratic extrapolation, and an assumption of DMRG convergence; the exponent-extraction claim adds a tentative scaling ansatz. No new entities are invented. The paper is honest that the proposition is not proven.

free parameters (3)
  • Common intercept p0 = 0.40165(7) (NN,hz=0.5); 0.26080(28) (NN,hz=0.75); 0.38934(6) (LR,hz=0.5); 0.24833(14) (LR,hz=0.75)
    Shared thermodynamic-limit intercept for the quadratic fits of all (hx_cross/min)_n; this intercept is the reported QCP.
  • Per-sequence coefficients p1_n, p2_n = Listed in Appendix Tables 1-4
    Free parameters of Eq. (7); they absorb the finite-size behavior of each sequence while sharing the common p0.
  • Scaling exponents c1, c2 = c1=1, c2=1/8 in Eq. (9); BSA sliding-window estimates trend toward these values
    Used to collapse Δ1 data; either fixed to Ising values or fitted by BSA. The exponent-extraction claim depends on them.
assumptions (6)
  • domain assumption Gapless 1D SSD ground state becomes translationally invariant for local observables in the thermodynamic limit
    Stated as a proposition in the Abstract and Sec. 2.1; not proven, only supported by exactly solved cases.
  • domain assumption SSD ground state of a CFT is an exact eigenstate of the PBC Hamiltonian
    Invoked to motivate Eq. (2), citing Ref. [39].
  • ad hoc to paper The zero-crossing/minimum positions of Δ_n extrapolate to the QCP quadratically in 1/L with a common intercept
    Eq. (7) is chosen for fitting; no scaling derivation is provided.
  • ad hoc to paper The scaling ansatz Eq. (9) applies to Δ1
    The paper calls it tentative: 'not a priori clear' and 'no theoretical guarantee'.
  • domain assumption DMRG with bond dimension 600 yields converged ground states for all L and parameters
    No truncation-error or convergence analysis is reported.
  • domain assumption In the Rydberg proposal, third- and higher-neighbor couplings can be neglected
    Sec. 3 shows they are ~2 orders smaller for J2/J1≲1, but not exactly zero.

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Pith. "Pith review of A sine-square deformation approach to quantum critical points in one-dimensional systems." pith.science (2026). https://pith.science/paper/5NMAEMRZ

@misc{pith2026251214149,
  author       = {Pith},
  title        = {Pith review of: A sine-square deformation approach to quantum critical points in one-dimensional systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5NMAEMRZ}},
  note         = {Machine review of arXiv:2512.14149}
}
abstract

We propose a method to determine the quantum phase boundaries of one-dimensional systems using sine-square deformation (SSD). Based on the proposition, supported by several exactly solved cases though not proven in full generality, that "if a one-dimensional system is gapless, then the expectation value of any local observable in the ground state of the Hamiltonian with SSD exhibits translational symmetry in the thermodynamic limit," we determine the quantum critical point as the location where a local observable becomes site-independent, identified through finite-size scaling analysis. As case studies, we consider two models: the antiferromagnetic Ising chain in mixed transverse and longitudinal magnetic fields with nearest-neighbor and long-range interactions. We calculate the ground state of these Hamiltonians with SSD using the density-matrix renormalization-group algorithm and evaluate the local transverse magnetization. For the nearest-neighbor model, we show that the quantum critical point can be accurately estimated by our procedure with systems of up to 84 sites, or even smaller, in good agreement with results from the literature. For the long-range model, we find that the phase boundary between the antiferromagnetic and paramagnetic phases is slightly shifted relative to the nearest-neighbor case, leading to a reduced region of antiferromagnetic order. Moreover, we propose an experimental procedure to implement the antiferromagnetic $J_1$-$J_2$ Ising couplings with SSD using Rydberg atom arrays in optical tweezers, which can be achieved within a very good approximation. Because multiple independent scaling conditions naturally emerge, our approach enables precise determination of quantum critical points and possibly even the extraction of additional critical phenomena, such as critical exponents, from relatively small system sizes.

Figures

Figures reproduced from arXiv: 2512.14149 by the authors.

Figure 1
Figure 1. Continuous form of the modulation function [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Schematic illustrations of the site pairs in the definitions of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. (a, b) ∆n (n = 1–6) as functions of h x for the ground state of the Hamilto￾nian (6) with h z = 0 and L = 12. (c) ∆1 for L = 4. The gray dashed line indicates the QCP. sign changes or local minima depending on L. These differences are attributed to the parity combinations of the reference sites, such as L/2, L/3, and L/4, that enter the definitions of ∆n . For each ∆n , we denote by (h x cross)n the value of h x at … view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: ∆n (n = 1–6) for the ground state of the nearest-neighbor model with SSD [Eq. (6)] at h z = 0.5, shown as functions of h x . Panels (a)–(p) are arranged in a 4 × 4 grid, where each row corresponds to a different ∆n [∆1,2,3 in the first row, ∆4 in the second, ∆5 in the …
Figure 5
Figure 5. Figure 5: (h x cross)n (circles) and (h x min)n (triangles) for the ground state of the nearest￾neighbor model with SSD [Eq. (6)], shown as functions of 1/L at (a) h z = 0.5 and (b) h z = 0.75. The color code is as follows: red (n = 1), blue (n = 2), green (n = 3), orange (n = 4…
Figure 6
Figure 6. Figure 6: Ground-state phase diagram of the mixed-field Ising chain in the [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: Same as Figs. 5(a) and 5(b), but for the long-range interaction model with SSD [Eq. (8)] at (a) h z = 0.5 and (b) h z = 0.75 (see Appendix A for the fitting parameters). the QCP. Moreover, ∆n is defined within the SSD-deformed Hamiltonian rather than the orig￾inal unif…
Figure 8
Figure 8. Figure 8: Finite-size scaling collapse of ∆1 L c2 versus (h x − h x c )L c1 for (a) nearest￾neighbor and (b) long-range interactions at zero longitudinal field (h z = 0). The scaling exponents used are (c1 ,c2 ) = (1, 1/8). parameter, so there is no a priori reason for it to fol…
Figure 9
Figure 9. Figure 9: Optimized scaling exponents in Eq. (9) obtained from the Bayesian scaling analysis [55,56], plotted as functions of 1/L for (a) the nearest-neighbor and (b) the long-range interaction models at zero longitudinal field. The red and blue dashed lines indicate the referen…
Figure 10
Figure 10. Figure 10: Schematic illustration of the three-atom arrangement corresponding to [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 11
Figure 11. Figure 11: Configurations of Rydberg atoms and the corresponding spatial profiles [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]

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