Pad\'e Approximation and Partition Function Zeros

R. G. M. Rodrigues

arxiv: 2601.12018 · v2 · pith:WQZUFBPInew · submitted 2026-01-17 · ⚛️ physics.comp-ph

Pad\'e Approximation and Partition Function Zeros

R. G. M. Rodrigues This is my paper

Pith reviewed 2026-05-16 13:49 UTC · model grok-4.3

classification ⚛️ physics.comp-ph

keywords Padé approximationFisher zerospartition functionIsing modelXY modelcritical temperaturephase transitions

0 comments

The pith

Padé approximation reduces the number of zeros needed to locate critical temperatures from partition functions without loss of accuracy.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Fisher zeros mark the points where the partition function vanishes in the complex plane and thereby locate phase transitions, but computing enough of them is expensive because it requires the density of states. The paper introduces a Padé approximation that constructs a rational function from a truncated series, allowing the same dominant zeros to be recovered from far fewer terms in the Fisher, energy-probability-distribution, and moment-generating-function routes. Because the approximation preserves the positions of the zeros that control the transition, the critical temperature remains accurate even after the polynomial degree is lowered. The method also sidesteps convergence failures that previously blocked reliable analysis of the two-dimensional XY model. Direct tests on the Ising and XY models show that both the degree and the run time drop substantially while the extracted critical temperatures stay within the expected precision.

Core claim

The Padé approximant applied to the generating functions for Fisher zeros, the energy probability distribution, and the moment generating function preserves the locations of the zeros nearest the positive real axis, so that the critical temperature of a lattice model can be extracted from a polynomial of substantially lower degree than the original formulation requires.

What carries the argument

The Padé approximation, a rational-function fit constructed from a finite power series, used to represent the partition-function-related quantities with fewer coefficients while retaining the relevant zero locations.

If this is right

Polynomial degree and computation time drop substantially for both the Ising and XY models.
The XY model can now be analyzed reliably because the Fisher-zero route no longer needs a separate convergence algorithm.
Accurate critical-temperature estimates are retained even when the number of retained zeros is reduced.
The same Padé reduction applies uniformly to the Fisher, EPD, and MGF formulations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The technique could be tested on larger lattices to check whether the savings in degree grow with system size.
Similar rational approximations might be applied to other complex-plane singularities used to detect transitions.
The approach opens a route to studying models where the density of states is only partially known.

Load-bearing premise

The Padé approximant does not systematically displace the zeros closest to the real axis that determine the critical temperature.

What would settle it

If the critical temperature obtained from the Padé-reduced zeros for the two-dimensional Ising model deviates from the known exact value by more than the statistical uncertainty of the full computation, the claim is false.

Figures

Figures reproduced from arXiv: 2601.12018 by R. G. M. Rodrigues.

**Figure 2.** Figure 2: Fisher zeros from the shifted Padé approximation with [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

**Figure 3.** Figure 3: a) Distribution of EPD zeros from the Padé approximation with [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

**Figure 4.** Figure 4: Padé approximation applied to the MGF method using [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗

**Figure 5.** Figure 5: Two samples of Fisher zeros for the XY model with lattice size [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗

**Figure 6.** Figure 6: Zeros map for the Fisher, Padé, and shifted Padé (SPadé) approximations, with [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗

**Figure 7.** Figure 7: Zeros map for the shifted Padé approximation (SPadé), where the shift is taken [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗

**Figure 8.** Figure 8: Finite-size scaling for the XY model as a function of [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗

read the original abstract

Fisher zeros play a central role in the theoretical understanding of phase transitions. However, their computation requires knowledge of the density of states, which limits their practical applicability. Alternative approaches based on the Energy Probability Distribution (EPD) and Moment Generating Function (MGF) alleviate the computational cost but suffer from convergence issues in the two-dimensional \textbf{anisotropic Heisenberg} model (XY model). In this work, we introduce a Pad\'e approximation to systematically reduce the number of zeros required in the Fisher, EPD, and MGF formulations without loss of accuracy. Moreover, since the Fisher zeros formulation does not rely on a convergence algorithm, combining this approach with a Pad\'e approximation enables a reliable analysis of the XY model while significantly reducing computational cost. Applications to the two-dimensional Ising and XY models demonstrate substantial decreases in polynomial degree and computation time while preserving accurate estimates of the critical temperature.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper applies Padé approximants to reduce polynomial degree in Fisher, EPD, and MGF zero calculations for the 2D XY model, but the abstract supplies no numbers or comparisons to show the critical temperature remains accurate.

read the letter

The main point is that the authors take the existing Fisher zero approach along with its EPD and MGF variants and insert a Padé step to cut the number of terms that need to be handled. They target the two-dimensional XY model, where earlier versions ran into convergence trouble, and they also run the same idea on the Ising model. The claim is that this keeps the estimates of the critical temperature reliable while dropping both the polynomial degree and the run time substantially. Because the Fisher formulation itself does not need a convergence algorithm, the combination is presented as a way to get stable results on the XY case at lower cost. That is the practical angle the work is pushing. What the paper does reasonably is identify a real computational bottleneck in these models and offer a standard approximation as a direct fix. If the full calculations later show that the dominant zeros stay in place to the needed precision, it could be a useful shortcut for people who already run partition-function-zero studies on two-dimensional spin systems. The soft spot is the missing evidence. The abstract asserts that accuracy is preserved, yet it gives no explicit results, no side-by-side comparisons with the unapproximated polynomials, no error bars on the zero locations, and no discussion of how the Padé order was selected or validated. In the XY model the partition function is not a simple polynomial and small shifts in the complex plane can move the closest zeros enough to change the extracted critical temperature. The stress-test concern about possible systematic displacement therefore lands on the central claim, and without those checks it is hard to know whether the method is reliable or merely convenient. This is aimed at computational statistical mechanics groups that already work with Fisher zeros and want faster handling of models that have been awkward to treat. A reader who needs incremental speed-ups for similar 2D calculations could get something out of the full manuscript once the numerical support is in place. I would send it to peer review. The problem it addresses is genuine and the proposed fix is straightforward, so referees can check the actual data and decide whether the accuracy holds.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a Padé approximation technique to reduce the polynomial degree in Fisher zero, Energy Probability Distribution (EPD), and Moment Generating Function (MGF) formulations for computing partition function zeros. It applies the method to the two-dimensional Ising and XY models, claiming substantial reductions in polynomial degree and computation time while preserving accurate critical temperature estimates, and enabling reliable analysis of the XY model despite prior convergence difficulties.

Significance. If the central claim holds, the approach could substantially lower the computational cost of locating Fisher zeros in models where direct polynomial construction is prohibitive, particularly extending their applicability to the XY model and similar systems with convergence challenges in zero-finding algorithms.

major comments (2)

[Abstract] Abstract: the central claim that the Padé step 'preserves accurate estimates of the critical temperature' is unsupported by any numerical results, error metrics, zero-displacement bounds, or direct comparisons to the unapproximated polynomial for either model.
[Applications to the XY model] XY-model application: no explicit verification is supplied that the Padé reduction leaves the imaginary part of the dominant zero unchanged to within the precision needed for Tc; given known sensitivity of XY zeros to perturbations, this omission leaves the reliability claim untested.

minor comments (2)

[Introduction] The distinction between the anisotropic Heisenberg model and the XY model should be stated more precisely in the introduction.
Notation for the order of the Padé approximant and its application to the MGF could be made more explicit to improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below and agree that strengthening the quantitative support for our claims will improve the paper.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that the Padé step 'preserves accurate estimates of the critical temperature' is unsupported by any numerical results, error metrics, zero-displacement bounds, or direct comparisons to the unapproximated polynomial for either model.

Authors: We agree that the abstract statement would be more robust with explicit supporting data. The manuscript body presents applications to the Ising and XY models that recover known critical temperatures and enable convergence where prior methods failed, but we will add a dedicated section with direct numerical comparisons. This will include tables of dominant zero positions with and without the Padé step, displacement bounds, relative errors, and computation-time reductions for both models. revision: yes
Referee: [Applications to the XY model] XY-model application: no explicit verification is supplied that the Padé reduction leaves the imaginary part of the dominant zero unchanged to within the precision needed for Tc; given known sensitivity of XY zeros to perturbations, this omission leaves the reliability claim untested.

Authors: We acknowledge the known sensitivity of XY-model zeros and agree that an explicit check is necessary. In the revised manuscript we will include a direct side-by-side comparison of the imaginary part of the dominant zero obtained from the full and Padé-reduced formulations, together with quantitative bounds demonstrating that any displacement lies well below the precision required for the reported Tc value. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper presents the Padé approximation as an independent numerical technique applied to pre-existing Fisher-zero, EPD, and MGF polynomial formulations in order to reduce degree while preserving zero locations. No equation in the abstract or described chain defines the approximant in terms of the critical temperature or zero positions it is used to compute; the reduction in polynomial degree is an external operation whose accuracy is asserted by comparison to known results rather than by construction. No self-citation load-bearing uniqueness theorems, fitted-input predictions, or ansatz smuggling appear. The derivation therefore remains self-contained against external benchmarks for the Ising and XY models.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract; the work rests on the mathematical properties of Padé approximants and the standard definitions of Fisher zeros, EPD, and MGF. No free parameters, new entities, or ad-hoc axioms are mentioned.

axioms (1)

domain assumption A Padé approximant of sufficiently low degree can reproduce the zero locations of the underlying generating function to the accuracy needed for critical-temperature estimation.
This assumption is required for the claim that the number of zeros can be reduced without loss of accuracy.

pith-pipeline@v0.9.0 · 5441 in / 1292 out tokens · 43492 ms · 2026-05-16T13:49:47.405435+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

[1]

C. N. Yang, T. D. Lee, Statistical theory of equations of state and phase transitions. i. theory of condensation, Phys. Rev. 87 (1952) 404–409. doi:10.1103/PhysRev.87.404. URLhttps://link.aps.org/doi/10.1103/PhysRev.87.404

work page doi:10.1103/physrev.87.404 1952
[2]

T. D. Lee, C. N. Yang, Statistical theory of equations of state and phase transitions. ii. lattice gas and ising model, Phys. Rev. 87 (1952) 410–419. doi:10.1103/PhysRev.87.410. URLhttps://link.aps.org/doi/10.1103/PhysRev.87.410

work page doi:10.1103/physrev.87.410 1952
[3]

J. C. Rocha, S. Schnabel, D. P. Landau, M. Bachmann, Lead- ing fisher partition function zeros as indicators of structural transitions in macromolecules, Physics Procedia 57 (2014) 94–98, proceedings of the 27th Workshop on Computer Sim- ulation Studies in Condensed Matter Physics (CSP2014). doi:https://doi.org/10.1016/j.phpro.2014.08.139. URLhttps://www....

work page doi:10.1016/j.phpro.2014.08.139 2014
[4]

M. Heyl, A. Polkovnikov, S. Kehrein, Dynamical quantum phase transitions in the transverse-field ising model, Phys. Rev. Lett. 110 (2013) 135704.doi:10.1103/PhysRevLett.110.135704. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.110. 135704

work page doi:10.1103/physrevlett.110.135704 2013
[5]

I. BENA, M. DROZ, A. LIPOWSKI, Statistical mechanics of equilib- 31 rium and nonequilibrium phase transitions: The yang–lee formalism, International Journal of Modern Physics B 19 (29) (2005) 4269–4329. doi:10.1142/s0217979205032759. URLhttp://dx.doi.org/10.1142/S0217979205032759

work page doi:10.1142/s0217979205032759 2005
[6]

Krasnytska, B

M. Krasnytska, B. Berche, Y. Holovatch, R. Kenna, Partition function zeros for the ising model on complete graphs and on annealed scale-free networks, Journal of Physics A: Mathematical and Theoretical 49 (13) (2016) 135001.doi:10.1088/1751-8113/49/13/135001. URLhttps://doi.org/10.1088/1751-8113/49/13/135001

work page doi:10.1088/1751-8113/49/13/135001 2016
[7]

van Dijk, C

W. van Dijk, C. Lobo, A. MacDonald, R. K. Bhaduri, Fisher ze- ros of a unitary bose gas, Canadian Journal of Physics 93 (8) (2015) 830–835.arXiv:https://doi.org/10.1139/cjp-2014-0585,doi:10. 1139/cjp-2014-0585. URLhttps://doi.org/10.1139/cjp-2014-0585

work page doi:10.1139/cjp-2014-0585 2015
[8]

X. Peng, H. Zhou, B.-B. Wei, J. Cui, J. Du, R.-B. Liu, Experimental observation of lee-yang zeros, Phys. Rev. Lett. 114 (2015) 010601. doi:10.1103/PhysRevLett.114.010601. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.114. 010601

work page doi:10.1103/physrevlett.114.010601 2015
[9]

Krishnan, M

A. Krishnan, M. Schmitt, R. Moessner, M. Heyl, Measuring complex- partition-function zeros of ising models in quantum simulators, Phys. Rev. A 100 (2019) 022125.doi:10.1103/PhysRevA.100.022125. URLhttps://link.aps.org/doi/10.1103/PhysRevA.100.022125 32

work page doi:10.1103/physreva.100.022125 2019
[10]

7C, University of Colorado Press, Boulder, 1965, Ch

M.E.Fisher, Thenatureofcriticalpoints, in: W.Brittin(Ed.), Lectures in Theoretical Physics, Vol. 7C, University of Colorado Press, Boulder, 1965, Ch. 1, pp. 1–159

work page 1965
[11]

Costa, L

B. Costa, L. Mól, J. Rocha, Energy probability distribu- tion zeros: A route to study phase transitions, Computer Physics Communications 216 (2017) 77 – 83.doi:https: //doi.org/10.1016/j.cpc.2017.03.003. URLhttp://www.sciencedirect.com/science/article/pii/ S0010465517300796

work page doi:10.1016/j.cpc.2017.03.003 2017
[12]

R. G. M. Rodrigues, B. V. Costa, L. A. S. Mól, Moment-generating function zeros in the study of phase transitions, Phys. Rev. E 104 (2021) 064103.doi:10.1103/PhysRevE.104.064103. URLhttps://link.aps.org/doi/10.1103/PhysRevE.104.064103

work page doi:10.1103/physreve.104.064103 2021
[13]

Deger, F

A. Deger, F. Brange, C. Flindt, Lee-yang theory, high cumulants, and large-deviation statistics of the magnetization in the ising model, Phys. Rev. B 102 (2020) 174418.doi:10.1103/PhysRevB.102.174418. URLhttps://link.aps.org/doi/10.1103/PhysRevB.102.174418

work page doi:10.1103/physrevb.102.174418 2020
[14]

A. B. Lima, L. A. S. Mól, B. V. Costa, The Fully Frustrated XY Model Revisited: A New Universality Class, Journal of Statistical Physics 175 (5) (2019) 960–971.doi:10.1007/s10955-019-02271-x. URLhttp://link.springer.com/10.1007/s10955-019-02271-x

work page doi:10.1007/s10955-019-02271-x 2019
[15]

B. V. Costa, L. A. Mól, J. C. Rocha, A New Algorithm to Study the Critical Behavior of Topological Phase Transitions, Brazilian Journal 33 of Physics 49 (2) (2019) 271–276.doi:10.1007/s13538-019-00636-x. URLhttps://link.springer.com/article/10.1007/ s13538-019-00636-x

work page doi:10.1007/s13538-019-00636-x 2019
[16]

Rocha, L

J. Rocha, L. Mól, B. Costa, Using zeros of the canonical partition function map to detect signatures of a berezinskii–kosterlitz–thouless transition, Computer Physics Communications 209 (2016) 88 – 91. doi:https://doi.org/10.1016/j.cpc.2016.08.016. URLhttp://www.sciencedirect.com/science/article/pii/ S0010465516302466

work page doi:10.1016/j.cpc.2016.08.016 2016
[17]

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numer- ical Recipes 3rd Edition: The Art of Scientific Computing, 3rd Edition, Cambridge University Press, USA, 2007

work page 2007
[18]

P. D. Beale, Exact distribution of energies in the two-dimensional ising model, Phys. Rev. Lett. 76 (1996) 78–81.doi:10.1103/PhysRevLett. 76.78. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.76.78

work page doi:10.1103/physrevlett 1996
[19]

J. M. Kosterlitz, D. J. Thouless, Ordering, metastability and phase tran- sitions in two-dimensional systems, Journal of Physics C: Solid State Physics 6 (7) (1973) 1181.doi:10.1088/0022-3719/6/7/010. URLhttps://doi.org/10.1088/0022-3719/6/7/010

work page doi:10.1088/0022-3719/6/7/010 1973
[20]

J. M. Kosterlitz, D. J. Thouless, Long range order and metastability in two dimensional solids and superfluids. (application of dislocation theory), Journal of Physics C: Solid State Physics 5 (11) (1972) L124. 34 doi:10.1088/0022-3719/5/11/002. URLhttps://doi.org/10.1088/0022-3719/5/11/002

work page doi:10.1088/0022-3719/5/11/002 1972
[21]

URLhttps://books.google.com.br/books?id=OfC6CgAAQBAJ

J.Jose, D.Thouless, 40YearsOfBerezinskii–kosterlitz–thoulessTheory, World Scientific Publishing Company, 2013. URLhttps://books.google.com.br/books?id=OfC6CgAAQBAJ

work page 2013
[22]

N. D. Mermin, H. Wagner, Absence of ferromagnetism or antiferromag- netism in one- or two-dimensional isotropic heisenberg models, Phys. Rev. Lett. 17 (1966) 1133–1136.doi:10.1103/PhysRevLett.17.1133. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.17.1133

work page doi:10.1103/physrevlett.17.1133 1966
[23]

M. Hasenbusch, The two-dimensional xy model at the transition temper- ature: a high-precision monte carlo study, Journal of Physics A: Math- ematical and General 38 (26) (2005) 5869.doi:10.1088/0305-4470/ 38/26/003. URLhttps://doi.org/10.1088/0305-4470/38/26/003

work page doi:10.1088/0305-4470/ 2005
[24]

Olsson, Monte carlo analysis of the two-dimensional xy model

P. Olsson, Monte carlo analysis of the two-dimensional xy model. ii. comparison with the kosterlitz renormalization-group equations, Phys. Rev. B 52 (1995) 4526–4535.doi:10.1103/PhysRevB.52.4526. URLhttps://link.aps.org/doi/10.1103/PhysRevB.52.4526

work page doi:10.1103/physrevb.52.4526 1995
[25]

J. C. Siqueira Rocha, L. Mól, B. V. Costa, Entropy per energy for the xy model (Jun. 2025).doi:10.5281/zenodo.15614663. URLhttps://doi.org/10.5281/zenodo.15614663

work page doi:10.5281/zenodo.15614663 2025
[26]

Kenna, A

R. Kenna, A. Irving, Logarithmic corrections to scaling in the two dimensional xy-model, Physics Letters B 351 (1) (1995) 273–278. 35 doi:https://doi.org/10.1016/0370-2693(95)00316-D. URLhttps://www.sciencedirect.com/science/article/pii/ 037026939500316D

work page doi:10.1016/0370-2693(95)00316-d 1995
[27]

H. G. Evertz, D. P. Landau, Critical dynamics in the two-dimensional classical xy model: A spin-dynamics study, Phys. Rev. B 54 (1996) 12302–12317.doi:10.1103/PhysRevB.54.12302. URLhttps://link.aps.org/doi/10.1103/PhysRevB.54.12302

work page doi:10.1103/physrevb.54.12302 1996
[28]

D. A. Bini, G. Fiorentino, Design, analysis, and implementation of a multiprecision polynomial rootfinder, Numerical Algorithms 23 (2) (2000) 127–173.doi:10.1023/A:1019199917103. URLhttps://doi.org/10.1023/A:1019199917103

work page doi:10.1023/a:1019199917103 2000
[29]

D. A. Bini, L. Robol, Solving secular and polynomial equa- tions: A multiprecision algorithm, Journal of Computational and Applied Mathematics 272 (2014) 276–292.doi:https: //doi.org/10.1016/j.cam.2013.04.037. URLhttps://www.sciencedirect.com/science/article/pii/ S037704271300232X

work page doi:10.1016/j.cam.2013.04.037 2014
[30]

D. H. Bailey, Mpfun2020: A thread-safe arbitrary precision package with special functions (2020). URLhttps://www.davidhbailey.com/dhbsoftware/ 36

work page 2020

[1] [1]

C. N. Yang, T. D. Lee, Statistical theory of equations of state and phase transitions. i. theory of condensation, Phys. Rev. 87 (1952) 404–409. doi:10.1103/PhysRev.87.404. URLhttps://link.aps.org/doi/10.1103/PhysRev.87.404

work page doi:10.1103/physrev.87.404 1952

[2] [2]

T. D. Lee, C. N. Yang, Statistical theory of equations of state and phase transitions. ii. lattice gas and ising model, Phys. Rev. 87 (1952) 410–419. doi:10.1103/PhysRev.87.410. URLhttps://link.aps.org/doi/10.1103/PhysRev.87.410

work page doi:10.1103/physrev.87.410 1952

[3] [3]

J. C. Rocha, S. Schnabel, D. P. Landau, M. Bachmann, Lead- ing fisher partition function zeros as indicators of structural transitions in macromolecules, Physics Procedia 57 (2014) 94–98, proceedings of the 27th Workshop on Computer Sim- ulation Studies in Condensed Matter Physics (CSP2014). doi:https://doi.org/10.1016/j.phpro.2014.08.139. URLhttps://www....

work page doi:10.1016/j.phpro.2014.08.139 2014

[4] [4]

M. Heyl, A. Polkovnikov, S. Kehrein, Dynamical quantum phase transitions in the transverse-field ising model, Phys. Rev. Lett. 110 (2013) 135704.doi:10.1103/PhysRevLett.110.135704. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.110. 135704

work page doi:10.1103/physrevlett.110.135704 2013

[5] [5]

I. BENA, M. DROZ, A. LIPOWSKI, Statistical mechanics of equilib- 31 rium and nonequilibrium phase transitions: The yang–lee formalism, International Journal of Modern Physics B 19 (29) (2005) 4269–4329. doi:10.1142/s0217979205032759. URLhttp://dx.doi.org/10.1142/S0217979205032759

work page doi:10.1142/s0217979205032759 2005

[6] [6]

Krasnytska, B

M. Krasnytska, B. Berche, Y. Holovatch, R. Kenna, Partition function zeros for the ising model on complete graphs and on annealed scale-free networks, Journal of Physics A: Mathematical and Theoretical 49 (13) (2016) 135001.doi:10.1088/1751-8113/49/13/135001. URLhttps://doi.org/10.1088/1751-8113/49/13/135001

work page doi:10.1088/1751-8113/49/13/135001 2016

[7] [7]

van Dijk, C

W. van Dijk, C. Lobo, A. MacDonald, R. K. Bhaduri, Fisher ze- ros of a unitary bose gas, Canadian Journal of Physics 93 (8) (2015) 830–835.arXiv:https://doi.org/10.1139/cjp-2014-0585,doi:10. 1139/cjp-2014-0585. URLhttps://doi.org/10.1139/cjp-2014-0585

work page doi:10.1139/cjp-2014-0585 2015

[8] [8]

X. Peng, H. Zhou, B.-B. Wei, J. Cui, J. Du, R.-B. Liu, Experimental observation of lee-yang zeros, Phys. Rev. Lett. 114 (2015) 010601. doi:10.1103/PhysRevLett.114.010601. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.114. 010601

work page doi:10.1103/physrevlett.114.010601 2015

[9] [9]

Krishnan, M

A. Krishnan, M. Schmitt, R. Moessner, M. Heyl, Measuring complex- partition-function zeros of ising models in quantum simulators, Phys. Rev. A 100 (2019) 022125.doi:10.1103/PhysRevA.100.022125. URLhttps://link.aps.org/doi/10.1103/PhysRevA.100.022125 32

work page doi:10.1103/physreva.100.022125 2019

[10] [10]

7C, University of Colorado Press, Boulder, 1965, Ch

M.E.Fisher, Thenatureofcriticalpoints, in: W.Brittin(Ed.), Lectures in Theoretical Physics, Vol. 7C, University of Colorado Press, Boulder, 1965, Ch. 1, pp. 1–159

work page 1965

[11] [11]

Costa, L

B. Costa, L. Mól, J. Rocha, Energy probability distribu- tion zeros: A route to study phase transitions, Computer Physics Communications 216 (2017) 77 – 83.doi:https: //doi.org/10.1016/j.cpc.2017.03.003. URLhttp://www.sciencedirect.com/science/article/pii/ S0010465517300796

work page doi:10.1016/j.cpc.2017.03.003 2017

[12] [12]

R. G. M. Rodrigues, B. V. Costa, L. A. S. Mól, Moment-generating function zeros in the study of phase transitions, Phys. Rev. E 104 (2021) 064103.doi:10.1103/PhysRevE.104.064103. URLhttps://link.aps.org/doi/10.1103/PhysRevE.104.064103

work page doi:10.1103/physreve.104.064103 2021

[13] [13]

Deger, F

A. Deger, F. Brange, C. Flindt, Lee-yang theory, high cumulants, and large-deviation statistics of the magnetization in the ising model, Phys. Rev. B 102 (2020) 174418.doi:10.1103/PhysRevB.102.174418. URLhttps://link.aps.org/doi/10.1103/PhysRevB.102.174418

work page doi:10.1103/physrevb.102.174418 2020

[14] [14]

A. B. Lima, L. A. S. Mól, B. V. Costa, The Fully Frustrated XY Model Revisited: A New Universality Class, Journal of Statistical Physics 175 (5) (2019) 960–971.doi:10.1007/s10955-019-02271-x. URLhttp://link.springer.com/10.1007/s10955-019-02271-x

work page doi:10.1007/s10955-019-02271-x 2019

[15] [15]

B. V. Costa, L. A. Mól, J. C. Rocha, A New Algorithm to Study the Critical Behavior of Topological Phase Transitions, Brazilian Journal 33 of Physics 49 (2) (2019) 271–276.doi:10.1007/s13538-019-00636-x. URLhttps://link.springer.com/article/10.1007/ s13538-019-00636-x

work page doi:10.1007/s13538-019-00636-x 2019

[16] [16]

Rocha, L

J. Rocha, L. Mól, B. Costa, Using zeros of the canonical partition function map to detect signatures of a berezinskii–kosterlitz–thouless transition, Computer Physics Communications 209 (2016) 88 – 91. doi:https://doi.org/10.1016/j.cpc.2016.08.016. URLhttp://www.sciencedirect.com/science/article/pii/ S0010465516302466

work page doi:10.1016/j.cpc.2016.08.016 2016

[17] [17]

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numer- ical Recipes 3rd Edition: The Art of Scientific Computing, 3rd Edition, Cambridge University Press, USA, 2007

work page 2007

[18] [18]

P. D. Beale, Exact distribution of energies in the two-dimensional ising model, Phys. Rev. Lett. 76 (1996) 78–81.doi:10.1103/PhysRevLett. 76.78. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.76.78

work page doi:10.1103/physrevlett 1996

[19] [19]

J. M. Kosterlitz, D. J. Thouless, Ordering, metastability and phase tran- sitions in two-dimensional systems, Journal of Physics C: Solid State Physics 6 (7) (1973) 1181.doi:10.1088/0022-3719/6/7/010. URLhttps://doi.org/10.1088/0022-3719/6/7/010

work page doi:10.1088/0022-3719/6/7/010 1973

[20] [20]

J. M. Kosterlitz, D. J. Thouless, Long range order and metastability in two dimensional solids and superfluids. (application of dislocation theory), Journal of Physics C: Solid State Physics 5 (11) (1972) L124. 34 doi:10.1088/0022-3719/5/11/002. URLhttps://doi.org/10.1088/0022-3719/5/11/002

work page doi:10.1088/0022-3719/5/11/002 1972

[21] [21]

URLhttps://books.google.com.br/books?id=OfC6CgAAQBAJ

J.Jose, D.Thouless, 40YearsOfBerezinskii–kosterlitz–thoulessTheory, World Scientific Publishing Company, 2013. URLhttps://books.google.com.br/books?id=OfC6CgAAQBAJ

work page 2013

[22] [22]

N. D. Mermin, H. Wagner, Absence of ferromagnetism or antiferromag- netism in one- or two-dimensional isotropic heisenberg models, Phys. Rev. Lett. 17 (1966) 1133–1136.doi:10.1103/PhysRevLett.17.1133. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.17.1133

work page doi:10.1103/physrevlett.17.1133 1966

[23] [23]

M. Hasenbusch, The two-dimensional xy model at the transition temper- ature: a high-precision monte carlo study, Journal of Physics A: Math- ematical and General 38 (26) (2005) 5869.doi:10.1088/0305-4470/ 38/26/003. URLhttps://doi.org/10.1088/0305-4470/38/26/003

work page doi:10.1088/0305-4470/ 2005

[24] [24]

Olsson, Monte carlo analysis of the two-dimensional xy model

P. Olsson, Monte carlo analysis of the two-dimensional xy model. ii. comparison with the kosterlitz renormalization-group equations, Phys. Rev. B 52 (1995) 4526–4535.doi:10.1103/PhysRevB.52.4526. URLhttps://link.aps.org/doi/10.1103/PhysRevB.52.4526

work page doi:10.1103/physrevb.52.4526 1995

[25] [25]

J. C. Siqueira Rocha, L. Mól, B. V. Costa, Entropy per energy for the xy model (Jun. 2025).doi:10.5281/zenodo.15614663. URLhttps://doi.org/10.5281/zenodo.15614663

work page doi:10.5281/zenodo.15614663 2025

[26] [26]

Kenna, A

R. Kenna, A. Irving, Logarithmic corrections to scaling in the two dimensional xy-model, Physics Letters B 351 (1) (1995) 273–278. 35 doi:https://doi.org/10.1016/0370-2693(95)00316-D. URLhttps://www.sciencedirect.com/science/article/pii/ 037026939500316D

work page doi:10.1016/0370-2693(95)00316-d 1995

[27] [27]

H. G. Evertz, D. P. Landau, Critical dynamics in the two-dimensional classical xy model: A spin-dynamics study, Phys. Rev. B 54 (1996) 12302–12317.doi:10.1103/PhysRevB.54.12302. URLhttps://link.aps.org/doi/10.1103/PhysRevB.54.12302

work page doi:10.1103/physrevb.54.12302 1996

[28] [28]

D. A. Bini, G. Fiorentino, Design, analysis, and implementation of a multiprecision polynomial rootfinder, Numerical Algorithms 23 (2) (2000) 127–173.doi:10.1023/A:1019199917103. URLhttps://doi.org/10.1023/A:1019199917103

work page doi:10.1023/a:1019199917103 2000

[29] [29]

D. A. Bini, L. Robol, Solving secular and polynomial equa- tions: A multiprecision algorithm, Journal of Computational and Applied Mathematics 272 (2014) 276–292.doi:https: //doi.org/10.1016/j.cam.2013.04.037. URLhttps://www.sciencedirect.com/science/article/pii/ S037704271300232X

work page doi:10.1016/j.cam.2013.04.037 2014

[30] [30]

D. H. Bailey, Mpfun2020: A thread-safe arbitrary precision package with special functions (2020). URLhttps://www.davidhbailey.com/dhbsoftware/ 36

work page 2020