Asymptotic fate of the two-peak energy histograms at the marginal four-state Potts critical point and their lattice-geometry dependence

The two-dimensional ferromagnetic $q$-state Potts model sits exactly at its marginal case at $q=4$: the transition is exactly known to be continuous, with logarithmic corrections from a marginally irrelevant operator, yet finite systems display first-order-like double-peaked energy histograms. Wolff-cluster Monte Carlo simulations at the exact critical points (Peng and Jiang, PTEP 2024) sharpened a long-standing puzzle: on the square lattice the two-peak structure of the critical energy-density histogram not only persists to $L=4096$ ($>16$ million spins) but strengthens beyond $L=512$, with the histogram free-energy barrier $\Delta F(L)$ apparently growing with $L$ — the textbook first-order signature — while on the honeycomb lattice no two-peak structure appears even at $L=64$, and an earlier Wang–Landau flat-histogram Monte Carlo study claimed the honeycomb transition is first order. The problem is to determine the true asymptotic finite-size behavior of the critical energy-density distribution at the marginal $q=4$ point (persistence, saturation, or crossover to single-peaked scaling, and the crossover scale if any), and to construct the analytic scaling theory — with the non-universal, lattice-dependent amplitude of the marginally irrelevant operator — that quantitatively explains the square-versus-honeycomb contrast and resolves the conflicting first-order claim for the honeycomb lattice. The problem remains open: a subsequent 2025 tensor-network study observed ambiguous energy-gap signals that it attributed to temperature-resolution artifacts while explicitly calling for further investigation, and no analytic calculation of the critical energy distribution at marginality exists.

Background

The $q$-state ferromagnetic Potts model assigns each site of a lattice one of $q$ spin states $\sigma_i\in\{1,\dots,q\}$ and couples nearest neighbors through the Hamiltonian $H=-J\sum_{\langle ij\rangle}\delta_{\sigma_i,\sigma_j}$ with the Kronecker $\delta$; $q=2$ is the Ising model. In two dimensions the model is exactly solved on the square, triangular, and honeycomb lattices: the free energy is known, and the critical points follow from self-duality (an exact mapping of the model at one temperature onto a dual model at another temperature, which pins the critical point of self-dual cases) and star-triangle relations (an exact lattice identity that converts the honeycomb model into the triangular one) (Baxter 1973; Baxter–Temperley–Ashley 1978; Hintermann–Kunz–Wu 1978). In the standard convention the exact critical temperatures of the $q=4$ model are $T_c=1/\ln 3$ on the square lattice and $T_c=1/\ln 5$ on the honeycomb lattice. The transition is first order for $q\ge5$ and continuous for $q\le4$; $q=4$ is the marginal case separating the two regimes.

Marginality has a precise meaning here: the $q=4$ fixed point still possesses a marginally irrelevant scaling operator (a perturbation whose coupling dies out only logarithmically slowly under renormalization), so every leading thermodynamic and finite-size singularity carries multiplicative logarithmic corrections. The scaling theory of the Potts multicritical point (Cardy–Nauenberg–Scalapino 1980; Nauenberg–Scalapino 1980), Cardy's conformal analysis (1986), and quantitative transfer-matrix tests — exact finite-size calculations on infinitely long strips (Salas–Sokal 1997) — fix, for example, the thermal exponent $y_t=3/2$ and the finite-size scaling variable $t\,L^{3/2}(\ln L)^{-3/4}$. Crucially, this theory determines singular thermodynamic quantities and scaling variables — not the shape, and in particular not the possible multimodality, of the finite-size energy probability distribution at criticality.

A standard diagnostic of transition order is the histogram of the energy density $E$ at the critical point. In two dimensions a first-order transition produces phase coexistence and a symmetric double-peaked histogram whose valley-to-peak free-energy difference $\Delta F(L)$ grows linearly with the linear size $L$, whereas a continuous transition yields a distribution approaching a single-peaked form with $\Delta F(L)$ saturating (Binder–Landau 1984). The $q=4$ square-lattice model is the textbook case of "pseudo-first-order" behavior: on small and moderate lattices the critical energy histogram does show two peaks, and the expectation from the exact continuous classification is a crossover to single-peaked scaling at sufficiently large $L$. This crossover was previously quantified at the four-state-Potts endpoint of the Ashkin–Teller critical line (a one-parameter family of conformal theories with central charge $c=1$, connecting two decoupled Ising models to the $q=4$ Potts point) in the frustrated square-lattice $J_1$–$J_2$ Ising model (Ising spins with ferromagnetic nearest-neighbor coupling $J_1$ and competing antiferromagnetic next-nearest-neighbor coupling $J_2$), where the dip between histogram peaks did not deepen appreciably from $L=128$ to $L=512$ and first-order-like signals disappear only at $L\gtrsim1500$ (Jin–Sen–Sandvik 2012; Jin–Sen–Guo–Sandvik 2013; Kalz–Honecker 2012).

Peng and Jiang (Prog. Theor. Exp. Phys. 2024) simulated the pure $q=4$ model with the Wolff cluster algorithm (a single-cluster Monte Carlo update that suppresses critical slowing down) at the exact critical points, on the square lattice up to $L=4096$ and on the honeycomb lattice up to $L=2048$. Both results ran against expectations: on the square lattice the two-peak structure persists to $L=4096$ and strengthens — the dip deepens significantly beyond $L=512$ and $\Delta F(L)$ appears to increase with $L$, the standard first-order signature — while on the honeycomb lattice no two-peak structure is visible even at $L=64$, i.e. the second-order character emerges immediately with no pseudo-first-order regime at all.

Neither observation contradicts the exact solution, which fixes the thermodynamic singularity (no latent heat in the infinite-size limit) but not the finite-size histogram shape, and qualitative lattice dependence is not forbidden: the amplitude of the marginally irrelevant operator at the $q=4$ fixed point is non-universal and can even vanish, as in the pure Baxter–Wu model (a triangular-lattice model with three-site Ising interactions in the same universality class), which realizes the $q=4$ Potts fixed point without any logarithmic corrections (Blöte–Guo–Nightingale 2017). What is missing is quantitative: no analytic calculation of the critical energy distribution at the marginal point — with lattice-dependent marginal amplitudes — exists; no simulation beyond $L=4096$ decides the trend of $\Delta F(L)$; and the picture is clouded by conflicting numerics, since Wang–Landau studies (a flat-histogram Monte Carlo scheme that estimates the energy density of states directly and can sample across energy barriers) using Binder cumulants (dimensionless ratios of moments of the order parameter or energy whose size dependence distinguishes first-order from continuous transitions) and histogram analysis claimed first-order transitions for $q=4$ on the honeycomb lattice (Murtazaev–Ramazanov–Mazagaeva–Magomedov 2019) and on the triangular lattice (Murtazaev–Kurbanova–Ramazanov 2019), in conflict with the Wolff-cluster honeycomb data and with the exact continuous classification.

Problem Statement

Determine the asymptotic finite-size behavior of the energy-density probability distribution $P_L(E)$ at the critical point of the two-dimensional ferromagnetic four-state Potts model, and explain its lattice-geometry dependence, as follows. (i) Establish whether the strengthening two-peak structure and the apparently growing barrier $\Delta F(L)$ (valley-to-peak difference of $\ln P_L(E)$ on an $L\times L$ torus) observed on the square lattice up to $L=4096$ persist as $L\to\infty$, saturate, or cross over to the single-peaked scaling expected of the exactly known continuous transition — and, if a crossover occurs, determine or bound its length scale; simulations at substantially larger $L$, exact or controlled tensor-network computations (coarse-graining contraction of the partition function written as a network of index-linked local tensors), or a demonstrated sampling artifact are all admissible routes. (ii) If the first-order-like signatures established in (i) reflect genuine finite-size physics, provide the analytic scaling theory of $P_L(E)$ at the marginal $q=4$ point — the role of the marginally irrelevant operator and its non-universal, lattice-dependent amplitude — that quantitatively accounts for the lattice-geometry dependence of $P_L(E)$, including the square-versus-honeycomb contrast whose nature parts (i) and (iii) settle. (iii) Address, as part of settling the honeycomb-lattice behavior, the conflicting claim of a first-order transition for $q=4$ on the honeycomb lattice (Murtazaev et al., Wang–Landau flat-histogram sampling). The single objective uniting these points is a validated account of first-order-like finite-size signatures at marginal $q=4$ criticality. Parts (i) and (iii) are mandatory: the square- and honeycomb-lattice behavior must be settled and the conflicting honeycomb-lattice first-order claim adjudicated. Part (ii) is mandatory whenever (i) establishes that the first-order-like signatures reflect genuine finite-size physics; if (i) instead settles the question by demonstrating a sampling artifact, a quantitative account of that artifact is required in place of (ii). The account is complete when the square- and honeycomb-lattice behavior is settled and quantitatively explained on whichever of these branches applies.

References

1. R. J. Baxter, "Potts model at the critical temperature", J. Phys. C: Solid State Phys. 6, L445 (1973), DOI 10.1088/0022-3719/6/23/005, https://doi.org/10.1088/0022-3719/6/23/005
2. R. J. Baxter, H. N. V. Temperley, S. E. Ashley, "Triangular Potts model at its transition temperature, and related models", Proc. R. Soc. Lond. A 358, 535 (1978), DOI 10.1098/rspa.1978.0026, https://doi.org/10.1098/rspa.1978.0026
3. A. Hintermann, H. Kunz, F. Y. Wu, "Exact results for the Potts model in two dimensions", J. Stat. Phys. 19, 623 (1978), DOI 10.1007/BF01011773, https://doi.org/10.1007/BF01011773
4. F. Y. Wu, "The Potts model", Rev. Mod. Phys. 54, 235 (1982), DOI 10.1103/RevModPhys.54.235, https://doi.org/10.1103/RevModPhys.54.235
5. J. L. Cardy, "Logarithmic corrections to finite-size scaling in strips", J. Phys. A: Math. Gen. 19, L1093 (1986), DOI 10.1088/0305-4470/19/17/008, https://doi.org/10.1088/0305-4470/19/17/008
6. J. L. Cardy, M. Nauenberg, D. J. Scalapino, "Scaling theory of the Potts-model multicritical point", Phys. Rev. B 22, 2560 (1980), DOI 10.1103/PhysRevB.22.2560, https://doi.org/10.1103/PhysRevB.22.2560
7. M. Nauenberg, D. J. Scalapino, "Singularities and scaling functions at the Potts-model multicritical point", Phys. Rev. Lett. 44, 837 (1980), DOI 10.1103/PhysRevLett.44.837, https://doi.org/10.1103/PhysRevLett.44.837
8. J. Salas, A. D. Sokal, "Logarithmic corrections and finite-size scaling in the two-dimensional 4-state Potts model", J. Stat. Phys. 88, 567 (1997), DOI 10.1023/b:joss.0000015164.98296.85, https://doi.org/10.1023/b:joss.0000015164.98296.85
9. K. Binder, D. P. Landau, "Finite-size scaling at first-order phase transitions", Phys. Rev. B 30, 1477 (1984), DOI 10.1103/PhysRevB.30.1477, https://doi.org/10.1103/PhysRevB.30.1477
10. U. Wolff, "Collective Monte Carlo updating for spin systems", Phys. Rev. Lett. 62, 361 (1989), DOI 10.1103/PhysRevLett.62.361, https://doi.org/10.1103/PhysRevLett.62.361
11. S. Jin, A. Sen, A. W. Sandvik, "Ashkin-Teller criticality and pseudo-first-order behavior in a frustrated Ising model on the square lattice", Phys. Rev. Lett. 108, 045702 (2012), DOI 10.1103/PhysRevLett.108.045702, arXiv:1110.5874, https://doi.org/10.1103/PhysRevLett.108.045702
12. S. Jin, A. Sen, W. Guo, A. W. Sandvik, "Phase transitions in the frustrated Ising model on the square lattice", Phys. Rev. B 87, 144406 (2013), DOI 10.1103/PhysRevB.87.144406, arXiv:1212.5339, https://doi.org/10.1103/PhysRevB.87.144406
13. A. Kalz, A. Honecker, "Location of the Potts-critical end point in the frustrated Ising model on the square lattice", Phys. Rev. B 86, 134410 (2012), DOI 10.1103/PhysRevB.86.134410, https://doi.org/10.1103/PhysRevB.86.134410
14. Jhao-Hong Peng, Fu-Jiun Jiang, "Phase transitions of the four-state Potts model on the square and honeycomb lattices", Prog. Theor. Exp. Phys. 2024, 013A04 (2024), DOI 10.1093/ptep/ptad158, https://doi.org/10.1093/ptep/ptad158, arXiv:2212.12272, https://arxiv.org/abs/2212.12272
15. Yuan-Heng Tseng, Shang-Wei Li, Fu-Jiun Jiang, "A Bond-Weighted Tensor Renormalization Group Study of the q-State Ferromagnetic Potts Models on the Square Lattice", Prog. Theor. Exp. Phys. 2025, 073A01 (2025), DOI 10.1093/ptep/ptaf086, arXiv:2503.12361, https://doi.org/10.1093/ptep/ptaf086
16. Yihua Sun, Yuchen Fan, "Relevant–marginal field mixing resolves the four-state Potts endpoint in the square-lattice J1–J2 Ising model", arXiv:2605.25946 (2026), https://arxiv.org/abs/2605.25946
17. A. K. Murtazaev, M. K. Ramazanov, M. K. Mazagaeva, M. A. Magomedov, "Phase Transitions and Thermodynamic Properties of the Potts Model with Spin States Number q = 4 on a Hexagonal Lattice", J. Exp. Theor. Phys. 129, 421 (2019), DOI 10.1134/S1063776119080053, https://doi.org/10.1134/S1063776119080053
18. A. K. Murtazaev, D. R. Kurbanova, M. K. Ramazanov, "Phase Transitions and the Thermodynamic Properties of the Potts Model with the Number of Spin States q = 4 on a Triangular Lattice", Phys. Solid State 61, 2172 (2019), DOI 10.1134/S1063783419110234, https://doi.org/10.1134/S1063783419110234
19. H. W. J. Blöte, WenAn Guo, M. P. Nightingale, "Scaling in the vicinity of the four-state Potts fixed point", J. Phys. A: Math. Theor. 50, 324001 (2017), DOI 10.1088/1751-8121/aa7b53, arXiv:1710.06088, https://doi.org/10.1088/1751-8121/aa7b53
20. N. Schreiber, J. Adler, "Monte Carlo study of the pure and dilute Baxter–Wu model", J. Phys. A: Math. Gen. 38, 7253 (2005), DOI 10.1088/0305-4470/38/33/004, https://doi.org/10.1088/0305-4470/38/33/004
21. Wei Liu, Fangfang Wang, Pengwei Sun, Jincheng Wang, "Pseudo-Phase Transitions of Ising and Baxter-Wu Models in Two-Dimensional Finite-Size Lattices", J. Stat. Mech. (2022) 093206, DOI 10.1088/1742-5468/ac8e5a, arXiv:2203.06563, https://doi.org/10.1088/1742-5468/ac8e5a
22. Seung Ki Baek, Harri Mäkelä, Petter Minnhagen, Beom Jun Kim, "Critical temperatures of the three- and four-state Potts models on the kagome lattice", Phys. Rev. E 83, 061104 (2011), DOI 10.1103/PhysRevE.83.061104, arXiv:1106.1296, https://doi.org/10.1103/PhysRevE.83.061104
23. Satoshi Morita, Naoki Kawashima, "Tensor Renormalization Group Calculations of Partition-Function Ratios", J. Phys. Soc. Jpn. 95, 044001 (2026), DOI 10.7566/JPSJ.95.044001, arXiv:2512.03395, https://doi.org/10.7566/JPSJ.95.044001