Finite-size scaling of Lee-Yang zeros and its application to the 3-state Potts model and heavy-quark QCD

Kazuyuki Kanaya; Masakiyo Kitazawa; Tatsuya Wada

REVIEW 2 major objections 4 minor 31 references

Finite-size scaling of Lee-Yang zeros and its application to the 3-state Potts model and heavy-quark QCD

T0 review · 2 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read This paper proposes the Lee-Yang-zero ratio method, a finite-size scaling technique that locates critical points from the intersection of ratio curves, and demonstrates it in the 3D Ising model, the 3-state Potts model, and heavy-quark QCD.

desk verdict New crossing observable for finding critical points; the Potts/Ising checks are solid, the QCD central value has an unquantified finite-size correction. read the letter →

arxiv 2501.18904 v1 pith:FWOHN7ST submitted 2025-01-31 hep-lat

Tatsuya Wada , Masakiyo Kitazawa , Kazuyuki Kanaya This is my paper · ORCID

classification hep-lat MSC 82B2082B2782B80 PACS 05.50.+q11.15.Ha12.38.Gc64.60.F

keywords Lee-Yangzerosfinite-sizescalingcriticalpoint3-statePottsmodelheavy-quarkQCDBindercumulantreweightingmethodIsinguniversalityclass

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 proposes a general finite-size scaling method for locating critical points, based on ratios of Lee-Yang zeros (LYZR). The authors show that in the 3D Ising model, the ratio of two Lee-Yang zeros has a scaling form, $R_{nm}(t,L) = r_{nm} + c_{nm} L^{y_t} t + O(t^2)$, so curves at different lattice sizes intersect at the critical temperature $t=0$, just as Binder cumulant curves do. Monte Carlo data for the Ising model confirm the crossing, and the method is then extended by a linear coupling mapping to other systems in the Ising universality class. Applied to the 3D three-state Potts model and to heavy-quark QCD with $N_f=2$, the LYZR crossings give critical-point locations consistent with Binder-cumulant analyses. The paper argues the LYZR method suppresses finite-size operator-mixing effects faster than Binder cumulants, which would make it a useful tool for locating the QCD critical point.

What carries the argument

The Lee-Yang-zero ratio (LYZR) is the ratio $R_{nm}(t,L)=h^{(n)}_{LY}(t,L)/h^{(m)}_{LY}(t,L)$ of the imaginary parts of the $n$-th and $m$-th Lee-Yang zeros, the zeros of the partition function in the complex external-field plane. Its finite-size scaling form, $R_{nm}=r_{nm}+c_{nm}L^{y_t}t+\cdots$ with $r_{nm}=X_n/X_m$, is the machinery that carries the argument: it converts the unknown critical point into a volume-independent crossing, exactly the role Binder cumulants play, while the subleading correction $D_{nm}L^{2\bar y}$ is smaller than the Binder counterpart. For systems not written in Ising variables, the partition function is related to the Ising one by a linear coupling transformation $A$ (Eq. (12)); the LYZR then has the same crossing form up to a correction factor that vanishes as $L\to\infty$. The numerical implementation uses reweighting to evaluate the partition function at complex couplings and locate the zeros.

What would settle it

Compute $R_{21}(t,L)$ for the two-dimensional Ising model on lattices of size $L=16,32,64$ using exact transfer-matrix or Monte Carlo reweighting data; if the curves do not intersect at the known $T_c$ (or the intersection drifts with $L$ beyond the predicted $L^{2\bar y}$ correction), the claimed general crossing property is falsified.

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Extended reading notes

Core claim

The central claim is that the ratios of Lee-Yang zeros, $R_{nm}=h^{(n)}_{LY}/h^{(m)}_{LY}$, are a critical-point diagnostic with the same crossing structure as Binder cumulants. Starting from the finite-size scaling form $Z(t,h,L^{-1})=\tilde Z(L^{y_t}t, L^{y_h}h)$ and the regularity of the scaled zero functions $\tilde h^{(n)}_{LY}(\tilde t)$, the paper Taylor-expands $\tilde h^{(n)}_{LY}$ at $\tilde t=0$ and obtains $R_{nm}(t,L)=r_{nm}+c_{nm}L^{y_t}t+O(t^2)$, with $r_{nm}=X_n/X_m$ volume-independent. Hence $R_{nm}(t,L)$ curves for different $L$ cross at the critical point $t=0$, and the slope grows as $L^{y_t}$. The paper verifies this in 3D Ising Monte Carlo data (crossing $T\simeq 4.5114$, consistent with the precise $T_c=4.51152322(1)$), then applies the method to the 3-state Potts model, obtaining $\tau_c=0.549375(18)$ from $R_{21}$, and to heavy-quark QCD at $N_t=4$, obtaining $\beta^*_c=5.68595(30)$ from $R_{21}$, both consistent with Binder-cumulant estimates. In general systems the method requires a linear mapping between the physical couplings and the Ising variables $t,h$; this introduces a subleading correction factor $1+D_{nm}L^{2\bar y}$ with $\bar y=y_t-y_h<0$, which the paper notes decays faster than the corresponding Binder correction.

Load-bearing premise

The load-bearing assumption is that the scaled Lee-Yang zero positions $\tilde h^{(n)}_{LY}(\tilde t)$ are analytic functions through the critical point, so the Taylor expansion in Eq. (8) with a nonvanishing leading coefficient $X_n$ is valid; if a zero used in a ratio has $X_n=0$ or the expansion fails, the volume-independent crossing does not occur.

Editorial extensions

If this is right

In any model in the Ising universality class, a handful of lattice sizes and a reweighting-based zero determination should be enough to read off the critical point from the crossing of $R_{21}$ or $R_{31}$ curves.
The consistency between $R_{21}$, $R_{31}$, and Binder crossings in the Potts model supports using LYZR as a cross-check or replacement for Binder cumulants, with a smaller leading finite-size correction.
The heavy-quark QCD result pins the deconfinement critical point at $N_t=4$ to $\beta^*_c \approx 5.686$ with an error comparable to the Binder analysis, so LYZR is usable directly on existing lattice configurations.
Because the ratios approach 1 for $t>0$ and $(2n-1)/(2m-1)$ for $t<0$, the method also gives a diagnostic of which side of the critical point a simulation is on, not just the location of the crossing.

Reading between the lines

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

Beyond the paper, the same crossing logic should apply to Lee-Yang zeros in the complex chemical-potential plane, where reweighting from imaginary $\mu_B$ is sign-problem-free; if so, LYZR could bound the QCD critical point without a real-$\mu_B$ simulation.
Beyond the paper, the slope $c_{nm}L^{y_t}$ of the ratio curves could be used to extract the thermal exponent $y_t$ from the same fits that locate the crossing, giving a self-contained measurement of critical exponents.
Beyond the paper, since the method needs only the locations of the leading zeros, it may be more robust to statistical noise in the tail of the reweighted partition function than methods that use the full distribution, a testable advantage on smaller lattices.
Beyond the paper, applying LYZR to the 2D Ising model, where exact transfer-matrix zeros are available, would provide a clean benchmark of the crossing property and the predicted $L^{2\bar y}$ correction.

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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

The thing to know: this is a short proceedings paper that introduces Lee-Yang-zero ratios (LYZR) as a crossing observable in the style of Binder cumulants. The scaling relation at the heart of it, Eq. (9), follows from a Taylor expansion of the scaled zeros around the critical point, and the three tests look right: in the 3D Ising and 3-state Potts models the crossings land on the known or Binder-based critical couplings. The method is a genuine addition to the finite-size scaling toolbox.

What is new: using ratios of multiple Lee-Yang zeros to locate a critical point is a clean idea, and Eq. (16) states explicitly why the method can beat Binder cumulants—the leading correction falls as L^{2\bar y} rather than as the Binder correction. Applying it to heavy-quark QCD is a new data point for the deconfinement critical point, even though it reuses the configurations from Kiyohara et al. [29].

Where I would be careful: the QCD fits drop the second bracket in Eq. (16), the D_nm L^{2\bar y} piece, and justify it by what is seen in the Potts model at L >= 40. In QCD the largest available aspect ratio is L_T = 12, and at L_T = 9 that correction is about 2%. Unless D_nm is small, which is not shown, the crossing point can shift by an amount comparable to the quoted statistical error. The consistency with the Binder result from the same configurations is reassuring but not independent. I would like a fit that includes the term, or at least a sensitivity estimate. The text also calls the heavy-quark fit "three-parameter" while actually listing four parameters (r, c, beta*_c, y_t); that is a typo that should be fixed. No data or code are provided, so the numerical claims are not directly reproducible; for a proceedings paper this is acceptable if the companion paper [1] carries the detail.

The central argument holds up. The regularity assumption behind Eq. (8) is plausible and is the standard kind of assumption in FSS crossing methods. The Ising and Potts results are genuine tests, not just demonstrations. I would send this to a competent lattice referee. With the QCD fit caveat and the typo addressed, it should appear in the proceedings. A reader working on the QCD critical point or on Lee-Yang methods will want to cite it.

Referee Report

2 major / 4 minor

Summary. This manuscript proposes locating critical points via ratios of Lee-Yang zeros (LYZR). Starting from the finite-size scaling form of the partition function, Eq. (2), and a Taylor expansion of the scaled zeros, the paper derives Eq. (9), R_nm(t,L) = r_nm + c_nm L^{y_t} t + O(t^2), so that curves for different L cross at the critical temperature. The method is verified in the 3D Ising model by reweighting, then extended to general Ising-universality systems through the linear mapping Eq. (12), leading to Eq. (16) with a correction factor [1 + D_nm L^{2\bar y}]. The paper applies the method to the 3-state Potts model and to N_f=2 heavy-quark QCD on N_t=4 lattices, reporting tau_c = 0.549375(18)/0.549373(17) and beta*_c = 5.68595(30)/5.68603(31), which are compared with Binder-cumulant results.

Significance. If the LYZR construction is sound, it adds a useful observable to the critical-point-location toolbox: ratios of Lee-Yang zeros have Binder-like crossings, the leading finite-size correction in Eq. (16) decays faster than the corresponding Binder correction, and the method combines naturally with reweighting in theories where complex couplings can be simulated. The paper's strengths are that the scaling argument is concise and testable, and the numerical tests are anchored to independent results: the Ising crossing is compared with the high-precision T_c of Ref. [17], and the Potts and QCD crossings are compared with Binder cumulant analyses. The main weaknesses are that the headline QCD fit values rest on an unquantified neglected correction, and the analyticity assumption behind the Taylor expansion is not justified; both are addressable within the scope of the manuscript.

major comments (2)

[Sec. 3.3, Eq. (16)] The quoted values beta*_c = 5.68595(30) and 5.68603(31) are obtained by fitting R_n1(beta*,L_T) = r + c(beta* - beta*_c)L_T^{y_t} and explicitly neglecting the second factor [1 + D_nm L_T^{2\bar y} + ...] in Eq. (16). This is not justified at the volumes used: with L_T = 9, 10, 12 and \bar y = y_t - y_h \simeq -0.894, L_T^{2\bar y} is approximately 0.020, 0.016, and 0.012, so the factor varies by about 8 x 10^{-3} across the fitted volumes. Since dR/dbeta* is approximately C L_T^{y_t} with L_T^{y_t} \simeq 30, a coefficient D_nm = O(1) changes the crossing by roughly 2-3 x 10^{-4}, i.e., the size of the quoted statistical errors; the manuscript contains no estimate of D_nm and no QCD check analogous to the 'no clear deviation for L >= 40' statement used in Sec. 3.2. The agreement with the Binder value from Ref. [29] is reassuring but uses the same reweighted configurations and the same action, so it does not by itself bound this finite-size systematic. Please include the correction term in the fit or provide an independent estimate of its size before quoting these CP values.
[Sec. 2.1, Eq. (8)] The derivation of the crossing property assumes that the scaled zero function \tilde h^{(n)}_{LY}(\tilde t) is analytic at \tilde t = 0 and admits the Taylor expansion of Eq. (8). The text justifies this by the regularity of the finite-L zeros h^{(n)}_{LY}(t,L) in t, but regularity of finite-L zeros does not automatically survive the scaling limit that defines \tilde h^{(n)} in Eq. (3); zeros can collide or the limiting function can develop non-analytic behavior. If \tilde h^{(n)} contains a term such as |\tilde t|^p with 0 < p < 1, Eq. (9) and the claimed volume-independent intersection would fail. The Ising data in Fig. 3 provide empirical support, but the paper should state the analyticity assumption explicitly and add a quantitative check, for example fitting the measured slope dR_n1/dT at the crossing to the predicted L^{y_t} behavior.

minor comments (4)

[Sec. 3.1, Eqs. (14)-(15)] Please clarify whether the O(L^{2\bar y}) terms in Eqs. (14) and (15) are meant as absolute corrections or as relative corrections. As written, the O(L^{2\bar y}) term in Eq. (15) appears to dominate the displayed leading term L^{-y_h} for large L, which cannot be intended; rewriting these equations with an explicit bracket factor would remove the ambiguity.
[Sec. 3.3] The fit in Sec. 3.3 is described as a 'three-parameter FSS fit', but four parameters r, c, beta*_c, and y_t are listed; please correct this wording and state whether y_t is fixed to the Ising value or left free.
[Sec. 2.2] The claim that the crossing point in the Ising model is at T \simeq 4.5114 is made by visual inspection of Fig. 3; please provide a numerical estimate of the crossing point with its statistical error so that the consistency with T_c = 4.51152322(1) can be judged quantitatively.
[Sec. 2.2, Fig. 2] The data collapse in the right panel of Fig. 2 is shown qualitatively; please specify the values of y_t and y_h used in the rescaling and, if possible, quantify the quality of the collapse.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the LYZR crossing is derived from finite-size scaling, and the Potts/QCD critical couplings are fit outputs, not fitted inputs; self-citations are not load-bearing.

full rationale

The paper's derivation chain is self-contained. Eq. (2) is the standard FSS form, Eq. (8) is an explicit analyticity assumption, and Eq. (16) follows by algebra; the crossing property R_nm(tau_c,L)=r_nm is a consequence of the scaling form, not an input. In the Ising test, the crossing T~4.5114 is checked against the independent high-precision Tc=4.51152322(1) from Ref. [17]. For the Potts model and heavy-quark QCD, tau_c and beta*_c are free parameters of least-squares fits to the LYZR data with the derived ansatz; they are not obtained by tuning to the Binder values quoted for comparison. The Binder comparisons in Secs. 3.2 and 3.3 use the same configurations (Refs. [1] and [29]), so they are consistency checks rather than independent external benchmarks, but they are not used to define the LYZR prediction and thus do not make the derivation circular. Refs. [1], [24], [29], and [30] are self-citations, but no load-bearing uniqueness theorem or ansatz is imported from them; Eq. (16) is derived in the text. The neglect of the D_nm L^{2ybar} correction in the finite-L fits is an unquantified finite-size systematic, especially for L_T=9 in QCD, but that is a correctness risk, not circularity, because the extracted critical couplings are not by construction equal to any input parameter. Overall, no circular step is present; the score reflects only minor, non-load-bearing self-citations.

Assumptions & free parameters 5 free parameters · 8 assumptions · 0 invented entities

The paper's method rests on standard finite-size scaling and Lee-Yang theory plus a regularity assumption on the scaled zero functions; it does not introduce new particles, forces, or conserved quantities. The main fitted quantities are the crossing parameters and the slope and exponent of the FSS ansatz in the Potts and QCD applications.

free parameters (5)

r_nm (universal zero ratio at CP) = not reported
Fitted as the crossing value in the LYZR ansatz R=r+c(tau-tau_c)L^{y_t}; appears in Eq. (9) and Eq. (16).
c (slope coefficient) = not reported
Fitted slope of the linear-in-delta_tau term in the same ansatz in Sections 3.2 and 3.3.
y_t (thermal exponent) = fitted; Ising value 1.588 used for scaling collapse
Treated as a free parameter in the Potts and QCD fits, rather than fixed to the known Ising value.
tau_c for Potts model = 0.549375(18) from R21, 0.549373(17) from R31
Critical coupling extracted from the crossing fit; the target output of the method.
beta*_c for heavy-quark QCD = 5.68595(30) from R21, 5.68603(31) from R31
Critical coupling in heavy-quark QCD extracted from the crossing fit on N_t=4 lattices.

assumptions (8)

domain assumption Finite-size scaling hypothesis: Z(t,h,L^{-1}) = Ztilde(L^{y_t}t,L^{y_h}h) near the critical point (Eq. 2).
Standard scaling hypothesis for the 3D Ising universality class; used in Section 2.1 to derive the scaling of individual zeros.
standard math Lee-Yang zeros of the Z2-symmetric Ising model lie on the imaginary h-axis for real t.
Cites Lee-Yang theorem (Ref [18]) in Section 2.1; needed to label zeros by imaginary part.
domain assumption For t<0, LYZ are equally spaced as h_n = a(t)(2n-1)/L^3 (Eq. 4).
Taken from Ref [19]; gives the t<0 limit of the ratio.
domain assumption For t>0, LYZ converge to a finite Lee-Yang edge singularity (Eq. 5).
Known behavior from Ref [20]; gives the t>0 limit of the ratio.
standard math Scaled zero functions htilde_n(ttilde) are analytic at ttilde=0 with Taylor expansion Eq. (8).
Regularity assumption needed for the volume-independent crossing; not proved in the paper.
domain assumption Pairwise linear mapping A between (tau,xi) and Ising scaling fields (Eqs. 11-12).
Universality and mixing ansatz for Potts and QCD; cites Refs [25-27].
domain assumption Hopping parameter expansion through NLO is accurate near the heavy-quark QCD critical point.
Relies on Ref [31]; needed to trust the reweighted action in Section 3.3.
domain assumption Reweighting overlap is sufficiently controlled in the simulations.
Asserted in Section 2.2 for Ising; no quantitative overlap diagnostics shown.

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Cite this review

Pith. "Pith review of Finite-size scaling of Lee-Yang zeros and its application to the 3-state Potts model and heavy-quark QCD." pith.science (2026). https://pith.science/paper/FWOHN7ST

@misc{pith2026250118904,
  author       = {Pith},
  title        = {Pith review of: Finite-size scaling of Lee-Yang zeros and its application to the 3-state Potts model and heavy-quark QCD},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FWOHN7ST}},
  note         = {Machine review of arXiv:2501.18904}
}

read the original abstract

We propose a new general method to study critical points (CP) using the finite-size scaling of Lee-Yang zeros (LYZ). We first study the LYZ in the three-dimensional Ising model on finite lattices. We show that the ratios of multiple LYZ (Lee-Yang-zero ratios: LYZR) have useful scaling properties similar to the Binder cumulants, providing us with a novel method to study CP. In numerical simulations of the Ising model, we confirm that this method works well. We then apply the method to analyze the CP in the three-dimensional three-state Potts model and finite-temperature QCD in heavy-quark region, which are believed to belong to the same universality class as the Ising model. In these models, the partition function at complex parameters can be evaluated by the reweighting method, which allows us to determine the LYZ by varying coupling parameters continuously around the CP. We demonstrate that the LYZR method is powerful in determining the location of the CP in these models.

Figures

Figures reproduced from arXiv: 2501.18904 by the authors.

**Figure 1.** (a) Schematic plot of Lee-Yang zero ratio 𝑅𝑛𝑚(𝑡, 𝐿) as function of 𝑡 [1]. Blue, purple, and brown curves represent 𝑅𝑛𝑚(𝑡, 𝐿) at small, medium, and large 𝐿’s, respectively. In the limit 𝐿 → ∞, 𝑅𝑛𝑚(𝑡, 𝐿) approach the red step function. (b) Schematic phase diagram of the three-dimensional three-state Potts model (10). The black-solid line shows the first-order phase transition that terminates at a CP denoted by the red… view at source ↗

**Figure 2.** Left: Imaginary part of the LYZ in the three-dimensional Ising model as a function of 𝑇 for 𝐿 = 24, 48, and 64. Right: The same data after the rescaling according to Eq. (3) [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** LYZR 𝑅21 (𝑡, 𝐿) (left) and 𝑅31 (𝑡, 𝐿) (right) in the three-dimensional Ising model as functions of 𝑇 for 𝐿 = 24, 48, and 64. 3. LYZR method for CP in general systems 3.1 LYZR in general systems Next, we extend the previous argument to deal with CP in general systems that belong to the same universality class as the Ising model. For clarity of presentation, we first consider the three-dimensional three-state Potts mo… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: LYZR R21 (𝜏, 𝐿) (left) and R31 (𝜏, 𝐿) (center), together with the Binder cumulant B4 (𝜏, 𝐿) (right) in the three-dimensional three-state Potts model [1]. The diamonds are the fit results of the intersection point. In [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: LYZR R21 (𝜏, 𝐿) (left) and R31 (𝜏, 𝐿) (right) in heavy-quark QCD at 𝑁𝑡 = 4 for various aspects ratios 𝐿𝑇 = 𝑁𝑠/𝑁𝑡 . The circles are the results of FSS fits. the next-to-leading order contributions of the HPE exactly using the reweighting method [29]. The effects of yet …

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