pith. machine review for the scientific record. sign in

arxiv: 2605.10226 · v1 · submitted 2026-05-11 · ❄️ cond-mat.supr-con · cond-mat.stat-mech· cond-mat.str-el

Recognition: 2 theorem links

· Lean Theorem

Apparent double-T_c from a single BKT transition in anisotropic phase-only models

Pei-Yuan Cai , Yi Zhou
Authors on Pith no claims yet

Pith reviewed 2026-05-12 03:10 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con cond-mat.stat-mechcond-mat.str-el
keywords BKT transitionJosephson junction arrayanisotropic superconductorsdouble transition temperatureresistively shunted junctionfinite-size scalingvortex dynamics
0
0 comments X

The pith

Anisotropic phase-only models yield a single BKT transition yet produce apparent double-Tc in linear resistance curves under nonequilibrium dynamics.

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

The paper examines a minimal model of anisotropic Josephson-junction arrays both in equilibrium and driven by resistively shunted junction dynamics with fluctuating twist boundaries. Equilibrium calculations show only one Berezinskii-Kosterlitz-Thouless transition. Out of equilibrium, anisotropy in the couplings and in the dissipation distorts the shape of linear resistance-versus-temperature curves within the finite-size and finite-current crossover window. Methods that rely on curve shape, such as Halperin-Nelson fitting or fixed-resistance thresholds, therefore report two apparent transition temperatures, while scaling-based diagnostics that use the universal exponent of three and dynamic finite-size collapse remain consistent with a single transition. This establishes a clean baseline in which transport artifacts can mimic thermodynamic splitting without any actual second transition.

Core claim

In the equilibrium anisotropic phase-only Josephson-junction array there is a single Berezinskii-Kosterlitz-Thouless transition. When the same array is driven out of equilibrium under resistively shunted junction dynamics with anisotropic dissipation and fluctuating twist boundary conditions, the linear R-T curves in the finite-size finite-current regime are reshaped so that curve-shape criteria indicate an apparent double-Tc, whereas critical-scaling criteria (the exponent alpha equals three together with dynamic finite-size scaling) continue to signal only the single TBKT.

What carries the argument

The minimal anisotropic phase-only Josephson-junction array governed by resistively shunted junction dynamics with fluctuating twist boundary conditions, whose equilibrium state supports a single BKT transition while its nonequilibrium transport is sensitive to anisotropy.

If this is right

  • Curve-shape analyses of linear R-T data can produce misleading reports of multiple transitions in anisotropic systems.
  • Universal scaling criteria remain reliable indicators of a single underlying BKT transition even when linear curves appear split.
  • Any splitting that survives into the nonlinear critical regime lies outside the physics captured by this minimal anisotropic baseline.

Where Pith is reading between the lines

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

  • The baseline can be used to test whether additional ingredients such as amplitude fluctuations or disorder are required to explain experimental reports of persistent double-Tc.
  • Similar anisotropy-induced distortions may appear in other two-dimensional systems whose transport is governed by vortex unbinding.
  • Extending the model to include weak amplitude fluctuations would provide a direct check on whether the apparent double-Tc survives or is altered.

Load-bearing premise

The minimal anisotropic phase-only Josephson-junction array under resistively shunted junction dynamics captures the essential transport physics without amplitude fluctuations or additional microscopic details.

What would settle it

An experiment on an anisotropic two-dimensional superconductor that measures both linear resistance curves and the dynamic finite-size scaling or the exponent alpha equals three; consistency of the scaling diagnostics with a single transition while the curve-shape methods show splitting would support the claim.

Figures

Figures reproduced from arXiv: 2605.10226 by Pei-Yuan Cai, Yi Zhou.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic of the anisotropic phase-only model and the RSJ [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Geometric-mean helicity modulus divided by tempera [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Dimensionless [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Dynamic finite-size scaling of the [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Zoom-in of low-resistance regions of the normalized linear [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Halperin–Nelson fits of the normalized linear resistance for [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. E [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Geometric-mean sti [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
read the original abstract

Transport experiments on two-dimensional superconductors often yield direction-dependent transition temperatures, raising the question of whether such a ``double-$T_c$'' reflects a true thermodynamic splitting or a transport artifact. To establish a baseline, we study a minimal anisotropic phase-only Josephson-junction array in equilibrium and under resistively shunted junction dynamics with fluctuating twist boundary conditions. The equilibrium model exhibits a single Berezinskii--Kosterlitz--Thouless (BKT) transition. Out of equilibrium, anisotropic Josephson couplings and anisotropic dissipation reshape the linear $R$--$T$ curves in a finite-size, finite-current crossover regime, so that curve-shape criteria such as Halperin--Nelson fits and fixed-resistance thresholds yield an apparent double-$T_c$. In contrast, critical-scaling criteria -- the universal exponent $\alpha=3$ and dynamic finite-size scaling -- remain consistent with the single $T_{\mathrm{BKT}}$. A robust splitting that persists in the nonlinear critical scaling, such as that recently reported at KTaO$_3$ interfaces, therefore points to physics beyond this clean anisotropic baseline.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript studies a minimal anisotropic phase-only Josephson-junction array to address whether direction-dependent transition temperatures in 2D superconductors reflect a true thermodynamic double-Tc or a transport artifact. Equilibrium simulations establish a single Berezinskii-Kosterlitz-Thouless (BKT) transition. Under resistively shunted junction dynamics with anisotropic Josephson couplings, anisotropic dissipation, and fluctuating twist boundary conditions, the linear R-T curves are reshaped in the finite-size/finite-current crossover regime such that curve-shape criteria (Halperin-Nelson fits, fixed-resistance thresholds) produce an apparent double-Tc, while critical-scaling criteria (universal exponent α=3 and dynamic finite-size scaling) remain consistent with a single TBKT. The work concludes that any splitting persisting into nonlinear scaling (as reported at KTaO3 interfaces) requires physics beyond this clean anisotropic baseline.

Significance. If the results hold, the paper supplies a useful parameter-free baseline for interpreting anisotropic 2D superconductor transport data. It cleanly separates apparent effects arising from anisotropy in the crossover regime from true thermodynamic splitting, using only standard Josephson-array and BKT ingredients. The explicit demonstration that universal scaling criteria survive while curve-shape criteria do not is of direct experimental relevance. Credit is given for the absence of free parameters or ad-hoc axioms and for combining equilibrium and dynamical simulations.

major comments (2)
  1. [Dynamical simulations and scaling analysis] The central distinction between curve-shape and scaling criteria is load-bearing. The dynamical simulations section should quantify the finite-size crossover more explicitly: state the linear sizes L simulated, the current densities employed, and the precise metric used to demarcate the crossover regime so that readers can verify that the reported α=3 and dynamic FSS analyses are performed outside that regime.
  2. [Model definition and limitations] The phase-only truncation is the model's defining approximation. While the manuscript correctly presents the result as a baseline, the discussion should briefly address whether amplitude fluctuations (omitted here) could renormalize the effective anisotropy or modify the I-V exponent near TBKT, citing relevant literature on amplitude-inclusive anisotropic models.
minor comments (2)
  1. [Notation] Notation for the BKT temperature is inconsistent (T_BKT vs. T_{BKT}); adopt a single form throughout text and figures.
  2. [Figures] Figure captions for R-T and scaling plots should explicitly state the anisotropy ratios (J_x/J_y and dissipation anisotropy) used in each panel.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for the constructive suggestions. We address each major comment below and will revise the manuscript accordingly to improve clarity.

read point-by-point responses

  1. Referee: [Dynamical simulations and scaling analysis] The central distinction between curve-shape and scaling criteria is load-bearing. The dynamical simulations section should quantify the finite-size crossover more explicitly: state the linear sizes L simulated, the current densities employed, and the precise metric used to demarcate the crossover regime so that readers can verify that the reported α=3 and dynamic FSS analyses are performed outside that regime.

    Authors: We agree that explicitly quantifying the finite-size crossover will strengthen the manuscript and help readers verify the separation between curve-shape and scaling criteria. In the revised version, we will add a dedicated paragraph in the dynamical simulations section that states the linear sizes L simulated, the current densities employed, and the precise metric (e.g., the temperature window in which the linear resistance deviates from the Halperin-Nelson form or where finite-size effects begin to dominate) used to demarcate the crossover regime. We will also explicitly confirm that the α=3 exponent and dynamic finite-size scaling analyses are performed outside this regime, where the scaling is robust. This addition will allow readers to independently assess the distinction. revision: yes

  2. Referee: [Model definition and limitations] The phase-only truncation is the model's defining approximation. While the manuscript correctly presents the result as a baseline, the discussion should briefly address whether amplitude fluctuations (omitted here) could renormalize the effective anisotropy or modify the I-V exponent near TBKT, citing relevant literature on amplitude-inclusive anisotropic models.

    Authors: We agree that a brief discussion of the phase-only approximation is appropriate to contextualize the baseline nature of our results. In the revised manuscript, we will add a short paragraph (in the model section or conclusions) noting that amplitude fluctuations, while omitted here, could in principle renormalize the effective anisotropy ratio in certain regimes. However, near TBKT in clean 2D systems, the I-V exponent remains close to the universal value according to existing studies. We will cite relevant literature on amplitude-inclusive anisotropic models (e.g., works extending the XY model to include amplitude dynamics and anisotropic Ginzburg-Landau simulations) to support this point. This will reinforce that any experimentally observed splitting persisting into the nonlinear scaling regime requires physics beyond the clean anisotropic phase-only baseline. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results follow from explicit simulation of a standard phase-only model

full rationale

The paper defines a minimal anisotropic Josephson-junction array from standard BKT and RSJ ingredients, then reports numerical results for equilibrium (single BKT) and driven transport (apparent double-Tc only in curve-shape criteria). No equations reduce a claimed prediction to a fitted input by construction, no load-bearing self-citation chain is invoked to establish the single-TBKT premise, and no ansatz is smuggled via prior work. The central distinction between criteria is obtained directly from the model's dynamics rather than by renaming or self-definition.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the phase-only approximation and the validity of resistively shunted junction dynamics for capturing transport in the anisotropic case. No new entities are introduced.

axioms (2)
  • domain assumption Phase-only approximation is valid (amplitude fluctuations negligible).
    Standard assumption in Josephson-junction array models for 2D superconductors near the BKT regime.
  • domain assumption Resistively shunted junction dynamics with fluctuating twist boundary conditions adequately represent out-of-equilibrium transport.
    Invoked to generate the linear R-T curves and finite-current crossover regime.

pith-pipeline@v0.9.0 · 5504 in / 1555 out tokens · 53078 ms · 2026-05-12T03:10:34.893976+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

54 extracted references · 54 canonical work pages

  1. [1]

    To address this, we expand Eq

    Single BKT transition from the long-wavelength theory The equilibrium question is whether anisotropy can split the BKT transition into two distinct transitions. To address this, we expand Eq. (1) for smoothly varying phase configurations withA i,µ =0. The spin-wave part of the dimensionless energy is Fsw T = 1 2 Z d2r h Kx(∂xθ)2 +K y(∂yθ)2i ,(4) where Kx ...

    work page

  2. [2]

    It is convenient to introduce the twist per bond δµ ≡Φ µ/L,(15) so that Υµ(L,T)= 1 L2 ∂2F ∂δ2µ δµ=0 .(16) For the Hamiltonian in Eq

    Directional helicity modulus In the numerical simulations we evaluate the directional he- licity moduli, which are defined by Υµ(L,T)≡ ∂2F ∂Φµ2 Φµ=0 ,(14) as the free-energy response to an infinitesimal extra boundary twistΦ µ along theµdirection[5, 24, 25] under given system sizeL, temperatureTand most importantly, under PBC, i.e., Θµ =0. It is convenien...

    work page

  3. [3]

    fluctuating twist

    Equations of motion To study finite-current transport, we evolve the same phase- only model with RSJD under FTBC [14–18, 28]. In FTBC the local phases remain periodic, θi+Lˆx=θ i, θ i+Lˆy=θ i, while the uniform twist variables∆ µ are absorbed into the bond phase differences, ϕi,µ =θ i+ˆµ−θ i −∆ µ.(21) The “fluctuating twist” means that∆ µ will also be evo...

    work page

  4. [4]

    (27) and (28) for currents applied alongxandy

    Transport criteria: q=0.6as an example The nonequilibrium analysis begins from theE–jcurves obtained by integrating Eqs. (27) and (28) for currents applied alongxandy. The integration is performed by the second- order Runge-Kutta-Helfand-Greenside algorithm for stochas- tic dynamics[34, 35]. The time step∆τis set to be 0.02, and the long-time average in E...

    work page

  5. [5]

    The critical- scaling criteria are collected in Table III, while the curve- shape-based temperatures extracted from ther lin–Tcurves are collected in Table IV

    Varying the anisotropy parameter q We now keep the dissipative channel isotropic,r0,x =r 0,y = 1, and compareq=0.6 withq=0.7 on the sameL=100 lattice, usingq=0.5 as the isotropic reference. The critical- scaling criteria are collected in Table III, while the curve- shape-based temperatures extracted from ther lin–Tcurves are collected in Table IV. Theα=3 ...

    work page

  6. [6]

    double-Tc

    Turning on anisotropy in the dissipative channel We next introduce anisotropy in the dissipative channel by setting r0,x =1,r 0,y = 1 2 , and studyq=0.5, 0.6, and 0.7 on the sameL=100 lattice. Becauser 0,µ enters only the RSJD dynamics, based on the analysis in Sec. II B, this change does not modify the equilib- rium transition. However, it can further re...

    work page

  7. [7]

    Derivation of Equation(17) Starting from Eq. (14), we fix the direction of the extra twist to beν=x,yand write the free energy as F=−TlnZ,Z= Z Dθexp[−βH(θ,Φ ν)], whereH(θ,Φ ν) incorporates a uniform twist per bondδ ν ≡ Φν/Lonly on bonds parallel to the chosen directionν: H(θ,Φ ν)=− X i X µ=x,y EJ,µ cos θi+ˆµ−θ i +δ νδµν .(A1) The first derivative of the f...

    work page

  8. [8]

    (14)) is defined at zero boundary twist and is the quantity that enters the equi- librium BKT criterion, Eq

    Remark on finite boundary twists The helicity modulus in the main text (Eq. (14)) is defined at zero boundary twist and is the quantity that enters the equi- librium BKT criterion, Eq. (13). For afiniteboundary twistΘ µ =A µL,0, one can define two similar objects:

    work page

  9. [9]

    [43] from the free- energy difference, Υ(1) µ (L,T;Θ µ)≡ 2[F(Θ µ)−F(0)] Θ2µ ,(A4)

    the phase stiffness defined in Ref. [43] from the free- energy difference, Υ(1) µ (L,T;Θ µ)≡ 2[F(Θ µ)−F(0)] Θ2µ ,(A4)

    work page

  10. [10]

    (17) but with averages taken in the finite-twisted ensemble

    the second derivative of the free energy at a nonzero boundary twist, Υ(2) µ (L,T;Θ µ)≡ 1 L2 ∂2F Θµ +Lδ µ ∂δ2µ δµ=0 ,(A5) which also admits an expression analogous to Eq. (17) but with averages taken in the finite-twisted ensemble. These need not coincide with the standardΥµ defined under zero boundary twist. Indeed, Ref. [43] demonstrated that in the two...

    work page

  11. [11]

    V . L. Berezinskii, Sov. Phys. JETP32, 493 (1971). [2] J. M. Kosterlitz and D. J. Thouless, Journal of Physics C: Solid 15 State Physics6, 1181 (1973)

    work page 1971

  12. [12]

    J. M. Kosterlitz, Journal of Physics C: Solid State Physics7, 1046 (1974)

    work page 1974

  13. [13]

    D. R. Nelson and J. M. Kosterlitz, Phys. Rev. Lett.39, 1201 (1977)

    work page 1977

  14. [14]

    M. E. Fisher, M. N. Barber, and D. Jasnow, Phys. Rev. A8, 1111 (1973)

    work page 1973

  15. [15]

    Minnhagen, Rev

    P. Minnhagen, Rev. Mod. Phys.59, 1001 (1987)

    work page 1987

  16. [16]

    Ambegaokar, B

    V . Ambegaokar, B. I. Halperin, D. R. Nelson, and E. D. Siggia, Phys. Rev. Lett.40, 783 (1978)

    work page 1978

  17. [17]

    Ambegaokar, B

    V . Ambegaokar, B. I. Halperin, D. R. Nelson, and E. D. Siggia, Phys. Rev. B21, 1806 (1980)

    work page 1980

  18. [18]

    B. I. Halperin and D. R. Nelson, J. Low Temp. Phys.36, 599 (1979)

    work page 1979

  19. [19]

    X. Hua, Z. Zeng, F. Meng, H. Yao, Z. Huang, X. Long, Z. Li, Y . Wang, Z. Wang, T. Wu, Z. Weng, Y . Wang, Z. Liu, Z. Xiang, and X. Chen, Nature Physics20, 957 (2024)

    work page 2024

  20. [20]

    Huang, Z

    Z. Huang, Z. Wang, X. Hua, H. Wang, Z. Li, S. Liu, Z. Wang, F. Quan, Z. Wang, J. Tao, J. J. He, Z. Xiang, and X. Chen, Directional-dependent Berezinskii-Kosterlitz-Thouless transi- tion at EuO/KTaO 3(111) interfaces (2026), arXiv:2604.00608 [cond-mat.supr-con]

    work page arXiv 2026

  21. [21]

    Z.-X. Li, S. A. Kivelson, and D.-H. Lee, Theory of an infinitely anisotropic phase of a two-dimensional superconductor (2024), arXiv:2407.10269 [cond-mat.supr-con]

    work page arXiv 2024

  22. [22]

    Z. Xu, K. Jiang, and J. Hu, Anisotropic vortex mo- tion and two-dimensional superconducting transition (2025), arXiv:2506.05830 [cond-mat.supr-con]

    work page arXiv 2025

  23. [23]

    K. K. Mon and S. Teitel, Phys. Rev. Lett.62, 673 (1989)

    work page 1989

  24. [24]

    J. S. Chung, K. H. Lee, and D. Stroud, Phys. Rev. B40, 6570 (1989)

    work page 1989

  25. [25]

    B. J. Kim, P. Minnhagen, and P. Olsson, Phys. Rev. B59, 11506 (1999)

    work page 1999

  26. [26]

    M. Y . Choi, G. S. Jeon, and M. Yoon, Phys. Rev. B62, 5357 (2000)

    work page 2000

  27. [27]

    Medvedyeva, B

    K. Medvedyeva, B. J. Kim, and P. Minnhagen, Phys. Rev. B62, 14531 (2000)

    work page 2000

  28. [28]

    Olsson, Phys

    P. Olsson, Phys. Rev. B52, 4511 (1995)

    work page 1995

  29. [29]

    Olsson, Phys

    P. Olsson, Phys. Rev. B52, 4526 (1995)

    work page 1995

  30. [30]

    Weber and P

    H. Weber and P. Minnhagen, Phys. Rev. B37, 5986 (1988)

    work page 1988

  31. [31]

    D. M. H ¨ubscher and S. Wessel, Phys. Rev. E87, 062112 (2013)

    work page 2013

  32. [32]

    M. R. Beasley, J. E. Mooij, and T. P. Orlando, Phys. Rev. Lett. 42, 1165 (1979)

    work page 1979

  33. [33]

    Ohta and D

    T. Ohta and D. Jasnow, Phys. Rev. B20, 139 (1979)

    work page 1979

  34. [34]

    Teitel and C

    S. Teitel and C. Jayaprakash, Phys. Rev. B27, 598 (1983)

    work page 1983

  35. [35]

    J. V . Jos´e, L. P. Kadanoff, S. Kirkpatrick, and D. R. Nelson, Phys. Rev. B16, 1217 (1977)

    work page 1977

  36. [36]

    Schultka and E

    N. Schultka and E. Manousakis, Phys. Rev. B49, 12071 (1994)

    work page 1994

  37. [37]

    C. J. Lobb, D. W. Abraham, and M. Tinkham, Phys. Rev. B27, 150 (1983)

    work page 1983

  38. [38]

    W. C. Stewart, Applied Physics Letters12, 277 (1968)

    work page 1968

  39. [39]

    D. E. McCumber, Journal of Applied Physics39, 3113 (1968)

    work page 1968

  40. [40]

    A. A. Golubov, M. Y . Kupriyanov, and E. Il’ichev, Rev. Mod. Phys.76, 411 (2004)

    work page 2004

  41. [41]

    Ambegaokar and B

    V . Ambegaokar and B. I. Halperin, Phys. Rev. Lett.22, 1364 (1969)

    work page 1969

  42. [42]

    Wolff, Phys

    U. Wolff, Phys. Rev. Lett.62, 361 (1989)

    work page 1989

  43. [43]

    Helfand, Bell System Technical Journal58, 2289 (1979)

    E. Helfand, Bell System Technical Journal58, 2289 (1979)

    work page 1979

  44. [44]

    H. S. Greenside and E. Helfand, Bell System Technical Journal 60, 1927 (1981)

    work page 1927

  45. [45]

    D. S. Fisher, M. P. A. Fisher, and D. A. Huse, Phys. Rev. B43, 130 (1991)

    work page 1991

  46. [46]

    Medvedyeva, B

    K. Medvedyeva, B. J. Kim, and P. Minnhagen, Physica C: Su- perconductivity355, 6 (2001)

    work page 2001

  47. [47]

    P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys.49, 435 (1977)

    work page 1977

  48. [48]

    The qualitative comparison between the two current directions is based on the anisotropic shape of the underlying transport curves, rather than the special value of this tolerance

    Using a narrower tolerance gives fewer accepted data points near the low-resistance side of the transition, whereas a broader tolerance gives a larger operational window. The qualitative comparison between the two current directions is based on the anisotropic shape of the underlying transport curves, rather than the special value of this tolerance

    work page

  49. [49]

    This procedure gives up to 25 window variants for each current direction

    If the central window contains the temperature interval [Tleft,T right], we generate shifted windows by moving the left and right endpoints independently by 0,±1,±2 temperature- grid points, while retaining only successful fits with positive resistance data and with all points inside the linear-response regime. This procedure gives up to 25 window variant...

    work page

  50. [50]

    M. V . Simkin and J. M. Kosterlitz, Phys. Rev. B55, 11646 (1997)

    work page 1997

  51. [51]

    Benfatto, C

    L. Benfatto, C. Castellani, and T. Giamarchi, Phys. Rev. B80, 214506 (2009)

    work page 2009

  52. [52]

    Khairnar and T

    G. Khairnar and T. V ojta, Phys. Rev. E111, 024114 (2025)

    work page 2025

  53. [53]

    Metropolis and S

    N. Metropolis and S. Ulam, Journal of the American Statistical Association44, 335 (1949), pMID: 18139350

    work page 1949

  54. [54]

    Metropolis, A

    N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, The Journal of Chemical Physics21, 1087 (1953)

    work page 1953