The ferromagnetic q-state Potts model on the two-dimensional Betts or maple leaf lattice (the triangular lattice with 1/7 of its sites removed, coordination number 5) was studied for q=3 by Wang and Southern (2003), and for q=4 and q=5 only in the 2006 thesis of Monsivais-Alonso. That work found a second-order q=4 transition whose fitted magnetization exponent ratio beta/nu ~ 0.056 is less than half the two-dimensional four-state Potts value 1/8, and a very weak first-order q=5 transition whose latent heat, interfacial free-energy barrier, and thermodynamic-limit parameters were never extracted. The open problem is a single linked objective: determine the asymptotic critical behavior of the q=4 model -- standard marginal four-state universality with strong corrections to scaling, or genuine lattice-specific exponents -- and quantify the q=5 first-order transition (latent heat, barrier scaling with system size, transition temperature) through larger-L, higher-statistics simulation with controlled finite-size extrapolation. The classical Potts model on this lattice has not been revisited since the 2006 source.

Background

The q-state Potts model places a spin si∈{1,…,q}s_i\in\{1,\dots,q\} on every site of a lattice with ferromagnetic nearest-neighbor Hamiltonian H=−J∑⟨ij⟩δsi,sjH=-J\sum_{\langle ij\rangle}\delta_{s_i,s_j}, J>0J>0. In two dimensions its critical behavior as a function of qq is a benchmark of statistical mechanics: the exactly solved square, triangular, and honeycomb cases show a continuous transition for q≤4q\le4 and a first-order transition for q>4q>4, with critical exponents that depend on qq but coincide across these lattices (universality) [Wu, Rev. Mod. Phys. 54, 235 (1982)]. The case q=4q=4 is marginal: the transition stays continuous with β/ν=1/8\beta/\nu=1/8, γ/ν=7/4\gamma/\nu=7/4, α/ν=1\alpha/\nu=1, 1/ν=3/21/\nu=3/2, but corrections to scaling are strongest there, so effective exponents obtained from log-log fits over a finite range of system sizes drift substantially and can be badly biased.

The lattice in question is the Betts lattice, better known as the maple leaf lattice: remove 1/7 of the sites of the triangular lattice in a 7\sqrt7-periodic pattern, leaving every remaining site with exactly 5 neighbors (between kagome with 4 and triangular with 6), with a hexagonal unit cell of 6 sites and 15 bonds and, unlike kagome or honeycomb, no inversion symmetry. It is a vertex-transitive planar lattice (every site is equivalent to every other under a lattice symmetry) and is itself one of the eleven Archimedean lattices (the uniform tilings of the plane by regular polygons whose symmetry group acts transitively on the vertices, such as the square, triangular, and kagome lattices), specifically the (34,6)(3^4,6) (snub trihexagonal) tiling, and it is physically realized as the arrangement of magnetic ions in layered materials such as the Na2Mn3O7\mathrm{Na_2Mn_3O_7} family. Because the depletion is not handled by any star-triangle or duality transformation (local graph replacements that map the Potts partition function onto that of another lattice and thereby yield exact critical points), no exact Potts critical manifold is known for this lattice.

Prior results on it are sparse. Wang and Southern [Phys. Rev. B 68, 094419 (2003)] studied the three-state model: a second-order ferromagnetic transition with exponents close to the two-dimensional three-state values (e.g. β/ν=0.132±0.002\beta/\nu=0.132\pm0.002 versus the exact 2/152/15) and a first-order antiferromagnetic one. Monsivais-Alonso [arXiv:physics/0603035 (2006)] then simulated the ferromagnetic model for q=4q=4 and q=5q=5 with Wolff cluster updates (a Monte Carlo algorithm that grows and flips a connected cluster of equally-colored spins, avoiding the critical slowing down of single-spin updates) on systems of linear size L=18L=18 to 6060 (6L26L^2 sites). For q=4q=4 the Binder cumulants (dimensionless ratios of moments of the magnetization distribution, whose crossings between different system sizes locate a continuous transition) cross at Tc≈1.126T_c\approx1.126 (second order), but the fitted exponents came out anomalous: β/ν=0.0563±0.0040\beta/\nu=0.0563\pm0.0040 against the expected 1/81/8, with α/ν=0.645\alpha/\nu=0.645 against 1 and 1/ν=1.3851/\nu=1.385 against 1.5, while γ/ν=1.756\gamma/\nu=1.756 was fine. For q=5q=5 the energy cumulants and histograms showed a very weak first-order transition at Tc≈1.0575T_c\approx1.0575, with a double-peaked energy histogram whose valley deepens with LL.

Two standard tool-sets frame what is missing. For a continuous transition, unbiased exponent estimation requires fitting finite-size data with explicit corrections to scaling (confluent powers of LL and, at the marginal q=4q=4, logarithmic factors); a raw power-law fit over L≤60L\le60 is exactly where such corrections mislead. For a first-order transition in two dimensions, the thermodynamic characterization consists of the latent heat (the discontinuity in energy density), the interfacial free-energy barrier ΔF(L)\Delta F(L) extracted from the valley-to-peak ratio of the energy histogram, which under periodic boundary conditions grows linearly with LL (ΔF≈2σL\Delta F\approx2\sigma L, two interfaces per periodic cell, with σ\sigma the line tension), and the transition temperature extrapolated to the thermodynamic limit. The thesis identified the order of both transitions but performed none of these deferred analyses.

Problem Statement

For the nearest-neighbor ferromagnetic Potts model on the Betts (maple leaf, 1/7-depleted triangular) lattice, complete the finite-size-scaling characterization of the marginal-to-first-order regime that the 2006 source left unfinished, in its two linked parts:

(1) The q=4q=4 exponent anomaly. Establish the asymptotic critical behavior of the ferromagnetic four-state model on this lattice: determine whether the transition belongs to the standard two-dimensional four-state Potts universality class, so that the fitted β/ν≈0.056\beta/\nu\approx0.056 is an artifact of finite system sizes and uncorrected scaling fits, or whether the depleted, inversion-asymmetric geometry genuinely changes the magnetization exponent (or breaks universality), as the source conjectured. This requires simulations at substantially larger linear sizes and higher statistics than L=60L=60, with the fits of β/ν\beta/\nu, α/ν\alpha/\nu, γ/ν\gamma/\nu, 1/ν1/\nu, and TcT_c including explicit correction-to-scaling terms (the logarithmic corrections appropriate to the marginal q=4q=4 case) and testing that standard four-state scaling form against the data -- for example through fit residuals -- rather than assuming it, and an extrapolation with quantified uncertainties.

(2) The q=5q=5 first-order quantification. Quantify the very weak first-order ferromagnetic transition: extract the thermodynamic-limit latent heat (energy-density discontinuity), the interfacial free-energy barrier as a function of LL and hence the line tension, and the thermodynamic-limit transition temperature, using energy-histogram analyses and controlled finite-size extrapolation of the pseudocritical quantities (the system-size-dependent locations of the specific-heat or susceptibility maxima).

The common objective is to close the q=3,4,5q=3,4,5 sequence for this lattice, i.e. to determine how the marginal boundary of two-dimensional Potts criticality behaves on a five-coordinated depleted triangular lattice, deciding between standard qq-dependent universality with strong corrections and lattice-specific critical behavior.

References

  1. M. del P. Monsivais-Alonso, 'Simulations in statistical physics and biology: some applications', arXiv:physics/0603035 (2006), DOI: 10.48550/arXiv.physics/0603035, https://arxiv.org/abs/physics/0603035
  2. Z. F. Wang and B. W. Southern, 'Three-state Potts model on the maple leaf lattice', Physical Review B 68, 094419 (2003), DOI: 10.1103/PhysRevB.68.094419, https://doi.org/10.1103/PhysRevB.68.094419
  3. Z.-F. Wang and L. Chen, 'Potts model on maple leaf lattice with pure three-site interaction', Communications in Theoretical Physics 43, 175 (2005), DOI: 10.1088/0253-6102/43/1/033, https://doi.org/10.1088/0253-6102/43/1/033
  4. D. Schmalfuß, P. Tomczak, J. Schulenburg, and J. Richter, 'The spin-1/2 Heisenberg antiferromagnet on a 1/7-depleted triangular lattice: ground-state properties', Physical Review B 65, 224405 (2002), DOI: 10.1103/PhysRevB.65.224405, https://doi.org/10.1103/PhysRevB.65.224405
  5. F. Y. Wu, 'The Potts model', Reviews of Modern Physics 54, 235 (1982), DOI: 10.1103/RevModPhys.54.235, https://doi.org/10.1103/RevModPhys.54.235
  6. A. Codello, 'Exact Curie temperature for the Ising model on Archimedean and Laves lattices', Journal of Physics A: Mathematical and Theoretical 43, 385002 (2010), DOI: 10.1088/1751-8113/43/38/385002, arXiv:1008.4720, https://arxiv.org/abs/1008.4720
  7. C. R. Scullard and J. L. Jacobsen, 'Potts-model critical manifolds revisited', Journal of Physics A: Mathematical and Theoretical 49, 125003 (2016), DOI: 10.1088/1751-8113/49/12/125003, arXiv:1511.04374, https://doi.org/10.1088/1751-8113/49/12/125003
  8. C. Ding, Y.-c. Wang, and Y. Li, 'Potts and percolation models on bowtie lattices', Physical Review E 86, 021125 (2012), DOI: 10.1103/PhysRevE.86.021125, arXiv:1203.2244, https://arxiv.org/abs/1203.2244
  9. D. J. J. Farnell, R. Darradi, R. Schmidt, and J. Richter, 'Spin-half Heisenberg antiferromagnet on two Archimedian lattices: from the bounce lattice to the maple-leaf lattice and beyond', Physical Review B 84, 104406 (2011), DOI: 10.1103/PhysRevB.84.104406, arXiv:1104.2572, https://arxiv.org/abs/1104.2572
  10. J. Sonnenschein, A. Maity, C. Liu, R. Thomale, F. Ferrari, and Y. Iqbal, 'Candidate quantum spin liquids on the maple-leaf lattice', Physical Review B 110, 014414 (2024), DOI: 10.1103/PhysRevB.110.014414, arXiv:2404.05617, https://arxiv.org/abs/2404.05617