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arXiv:2605.09453

Emergent critical phases of the Ashkin-Teller model on the Union-Jack Lattice

Changzhi Zhao, Wanzhou Zhang, Yuan Huang, Chengxiang Ding, Youjin Deng

Published May 10, 2026• 0 citations
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Abstract

The Ashkin-Teller (AT) model is a classic spin model in statistical mechanics. For traditional homogeneous lattices like triangular and kagome lattices, even when frustration exists, the model only has one ferromagnetic-paramagnetic critical line in the $J>0$ and $K<0$ region. However, in this paper, for the Union Jack lattice, where the lattice coordination numbers are 4, 8, and 8 and which also contains a large number of small triangular units, using Metropolis Monte Carlo method, we find that, the critical line of the AT model splits into two Berezinskii-Kosterlitz-Thouless(BKT) boundaries, and a critical phase emerges in the intermediate region. This phenomenon is the combined result of frustration, lattice inhomogeneity and the two coupled spin degrees of freedom inherent to the AT model. In detail, the novel critical phase characterized by a power-law decay of magnetization with system size, where the correlation length ratio $ξ/L$ remains finite even in the thermodynamic limit. We also introduce the susceptibility $\widetildeχ = \text{d}\langle m \rangle /\text{d}J$ as a key probe, and through this probe, pseudo-critical points $J_c(L)$ are observed to scale proportionally to $(\ln L)^{-2}$, a behavior consistent with BKT criticality. Since superfluids, superconductors, and supersolids all possess quasi-long-range order and fall into the category of critical phases, our results could also inspire the exploration of such quantum phases.

DOI
10.48550/arxiv.2605.09453

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The Ashkin-Teller (AT) model is a classic spin model in statistical mechanics. For traditional homogeneous lattices like triangular and kagome lattices, even when frustration exists, the model only has one ferromagnetic-paramagnetic critical line in the J>0J>0J>0 and K<0K<0K<0 region. However, in this paper, for the Union Jack lattice, where the lattice coordination numbers are 4, 8, and 8 and which also contains a large number of small triangular units, using Metropolis Monte Carlo method, we find that, the critical line of the AT model splits into two Berezinskii-Kosterlitz-Thouless(BKT) boundaries, and a critical phase emerges in the intermediate region. This phenomenon is the combined result of frustration, lattice inhomogeneity and the two coupled spin degrees of freedom inherent to the AT model. In detail, the novel critical phase characterized by a power-law decay of magnetization with system size, where the correlation length ratio ξ/Lξ/Lξ/L remains finite even in the thermodynamic limit. We also introduce the susceptibility χ~=d⟨m⟩/dJ\widetildeχ = \text{d}\langle m \rangle /\text{d}Jχ​=d⟨m⟩/dJ as a key probe, and through this probe, pseudo-critical points Jc(L)J_c(L)Jc​(L) are observed to scale proportionally to (ln⁡L)−2(\ln L)^{-2}(lnL)−2, a behavior consistent with BKT criticality. Since superfluids, superconductors, and supersolids all possess quasi-long-range order and fall into the category of critical phases, our results could also inspire the exploration of such quantum phases.
We investigate the symmetric Ashkin-Teller (AT) model on the triangular lattice in the antiferromagnetic two-spin coupling region (J0J 0J0, with K the four-spin interaction. Further, we find that, in the J→−∞J \rightarrow -\inftyJ→−∞ limit, the critical line terminates at the decoupled point K=0K=0K=0. From the numerical results and the exact mapping, we conjecture that this `tricritical' point (J→−∞,K=0J \rightarrow -\infty, K=0J→−∞,K=0) is Berezinsky-Kosterlitz-Thouless-like and the logarithmic correction is absent. The dynamic critical exponent of the worm algorithm is estimated as z=0.28(1)z=0.28(1)z=0.28(1) near (J→−∞,K=0)(J \rightarrow -\infty, K=0)(J→−∞,K=0).
We investigate the symmetric Ashkin-Teller (AT) model on the triangular lattice in the antiferromagnetic two-spin coupling region (J<0). In the J→-∞ limit, we map the AT model onto a fully packed loop-dimer model on the honeycomb lattice. On the basis of this exact transformation and the low-temperature expansion, we formulate a variant of worm-type algorithms for the AT model, which significantly suppress the critical slowing down. We analyze the Monte Carlo data by finite-size scaling, and locate a line of critical points of the Ising universality class in the region J<0 and K>0, with K the four-spin interaction. Further, we find that, in the J→-∞ limit, the critical line terminates at the decoupled point K=0. From the numerical results and the exact mapping, we conjecture that this "tricritical" point (J→-∞,K=0) is Berezinsky-Kosterlitz-Thouless-like and the logarithmic correction is absent. The dynamic critical exponent of the worm algorithm is estimated as z=0.28(1) near (J→-∞,K=0).
Abstract Phase transitions in condensed matter are a source of exotic emergent properties. We study the fully frustrated bilayer Heisenberg antiferromagnet to demonstrate that an applied magnetic field creates a previously unknown emergent criticality. The quantum phase diagram contains four states with distinctly different symmetries, all but one pair separated by first-order transitions. We show by quantum Monte Carlo simulations that the thermal phase diagram is dominated by a wall of discontinuities extending between the dimer-triplet phases and the singlet-containing phases. This wall is terminated at finite temperatures by a critical line, which becomes multicritical where the Berezinskii-Kosterlitz-Thouless (BKT) transition of the dimer-triplet antiferromagnet and the thermal Ising transition of the singlet-triplet crystal phase also terminate. The combination of merging symmetries leads to a 4-state Potts universality not contained in the microscopic Hamiltonian, which we interpret within the Ashkin-Teller model. Our results represent a systematic step in understanding emergent phenomena in quantum magnetic materials, including the “Shastry-Sutherland compound” SrCu2(BO3)2.
We consider the thermal phase transition from a paramagnetic to a stripe-antiferromagnetic phase in the frustrated two-dimensional square-lattice Ising model with competing interactions J1<0{J}_{1}<0J1​<0 (nearest neighbor, ferromagnetic) and J2>0{J}_{2}>0J2​>0 (second neighbor, antiferromagnetic). The striped phase breaks a Z4{Z}_{4}Z4​ symmetry and is stabilized at low temperatures for g=J2/∣J1∣>1/2g={J}_{2}/|{J}_{1}|>1/2g=J2​/∣J1​∣>1/2. Despite the simplicity of the model, it has proved difficult to precisely determine the order and the universality class of the phase transitions. This was done convincingly only recently by Jin et al. [Phys. Rev. Lett. 108, 045702 (2012)]. Here, we further elucidate the nature of these transitions and their anomalies by employing a combination of cluster mean-field theory, Monte Carlo simulations, and transfer-matrix calculations. The J1{J}_{1}J1​-J2{J}_{2}J2​ model has a line of very weak first-order phase transitions in the whole region 1/2<g<g∗1/2<g<{g}^{*}1/2<g<g∗, where ${g}^{*}=0.67\ifmmode\pm\else\textpm\fi{}0.01$. Thereafter, the transitions from g=g∗g={g}^{*}g=g∗ to $g\ensuremath{\rightarrow}\ensuremath{\infty}$ are continuous and can be fully mapped, using universality arguments, to the critical line of the well-known Ashkin-Teller model from its four-state Potts point to the decoupled Ising limit. We also comment on the pseudo-first-order behavior at the Potts point and its neighborhood in the Ashkin-Teller model on finite lattices, which in turn leads to the appearance of similar effects in the vicinity of the multicritical point g∗{g}^{*}g∗ in the J1{J}_{1}J1​-J2{J}_{2}J2​ model. The continuous transitions near g∗{g}^{*}g∗ can therefore be mistaken for first-order transitions, and this realization was the key to understanding the paramagnetic-striped transition for the full range of g>1/2g>1/2g>1/2. Most of our conclusions are based on Monte Carlo calculations, while the cluster mean-field and transfer-matrix results provide useful methodological benchmarks for weakly first-order behaviors and Ashkin-Teller criticality.
The Ashkin-Teller model plays a central role in studying the phase transitions of statistical mechanics models and condensed matter systems. Nevertheless, the phase transitions of the three-dimensional Ashkin-Teller model with competing interactions have long been an enigma. Here, we employ a methodology conceptually based on emergent symmetry and perform embedding cluster Monte Carlo simulations for unprecedentedly large systems. We find that the distribution of a two-dimensional magnetization vector displays emergent O(2) symmetry, which appears along the critical line separating the paramagnetic phase from the Baxter and mixed phases. Sign-reversal finite-size effects are visible in the deviations from O(2) symmetry, but the thermal and magnetic renormalization exponents are universally close to the exponents of the three-dimensional XY model. The newly unveiled "extraordinary-log" critical phase and "special" transition of the XY model are realized in a plane defect of the critical Ashkin-Teller model. Thus, for the three-dimensional Ashkin-Teller model, our study provides complementary evidence of an XY critical line, which arises from the emergence of O(2) symmetry.
Monte Carlo studies of the Z(4)Z(4)Z(4) (or Ashkin-Teller) model on the triangular lattice were carried out. In particular, we investigated whether the massless phase, which was shown to exist on the boundary of the physical parameter space, may extend into a region of finite width. The existence of such a phase would imply that Nc{N}_{c}Nc​, the critical value above which Z(N)Z(N)Z(N) models have intermediate phases, is nonuniversal, i.e., lattice and interaction dependent. Although, on the basis of our Monte Carlo results we cannot completely rule out this possibility, most aspects of our findings indicate consistency with existing theories of the Z(4)Z(4)Z(4) model which exclude the possibility of such an intermediate massless phase.