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

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 $J&gt;0$ and $K&lt;0$ 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 $(\ln L)<sup>{-2}$, 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.

• The paper reveals that in the AT model on the UJ lattice, the critical line splits into two BKT boundaries, forming an extended critical phase.
• It employs extensive Metropolis Monte Carlo simulations to map the phase diagram and quantify sublattice-dependent critical exponents.
• The study highlights how lattice inhomogeneity and frustration stabilize quasi-long-range order and induce partial antiferromagnetic ordering.

Emergent Critical Phases in the Ashkin-Teller Model on the Union-Jack Lattice

Introduction

The investigated Ashkin-Teller (AT) model is a well-established spin system embodying two coupled Ising degrees of freedom. Traditionally, studies of the AT model on uniform 2D lattices such as square, triangular, and kagome have revealed ferromagnetic-paramagnetic transitions and partial antiferromagnetic ordering, but only along isolated critical lines for $J>0$, $K<0$. This paper demonstrates that the non-uniform Union-Jack (UJ) lattice induces a markedly different scenario: the critical line splits into two Berezinskii-Kosterlitz-Thouless (BKT) boundaries, birthing an extended critical phase. The UJ lattice, with sublattices possessing coordination numbers 4, 8, and 8, provides an ideal testbed for exploring frustration and inhomogeneity. Using extensive Metropolis Monte Carlo simulations, the authors systematically map the phase diagram and critically analyze the novel critical behaviors.

Model and Methodology

The AT Hamiltonian considered is

$$\frac{\mathcal{H}}{k_B T} = -J\sum_{\langle i,j\rangle} (\sigma_i \sigma_j + \tau_i \tau_j) - K\sum_{\langle i,j\rangle} \sigma_i \sigma_j \tau_i \tau_j,$$

with $\sigma_i, \tau_i = \pm 1$, $J$ the two-spin interaction, and $K$ the four-spin term. Spins reside on both layers of the UJ lattice, and the coupled spin variable $s_i = \sigma_i \tau_i$ is utilized to distinguish ordered phases. Sublattices are classified as $A_4$ (coordination 4), $B_8$, and $C_8$ (each coordination 8). Simulations utilize the Metropolis algorithm, updating $K<0$0, $K<0$1, and $K<0$2 sequentially. Measured observables include site-resolved magnetization, Binder ratio, susceptibility, and correlation length $K<0$3, with criticality probed via size-scaling and novel pseudo-susceptibility probes.

Phase Diagram and Characterization

Emergent Critical Phase

The primary result is the robust emergence of a critical phase for $K<0$4, $K<0$5 bounded by two BKT transitions. Unlike previously studied uniform and frustrated lattices where such critical phases are absent or require additional vortex terms/complex interactions, the UJ lattice naturally supports criticality due to its structural inhomogeneity and extensive geometric frustration. Within the critical phase, the magnetization decays algebraically with system size, $K<0$6, and the ratio $K<0$7 stays finite in the thermodynamic limit—a signature of quasi-long-range order.

Sublattice Inhomogeneity

A distinctive analytic and numerical finding is that the critical exponent $K<0$8 varies among sublattices, evidencing that lattice inhomogeneity directly modulates critical fluctuations. The $K<0$9 sublattice exhibits faster magnetization decay than the $$\frac{\mathcal{H}}{k_B T} = -J\sum_{\langle i,j\rangle} (\sigma_i \sigma_j + \tau_i \tau_j) - K\sum_{\langle i,j\rangle} \sigma_i \sigma_j \tau_i \tau_j,$$0 and $$\frac{\mathcal{H}}{k_B T} = -J\sum_{\langle i,j\rangle} (\sigma_i \sigma_j + \tau_i \tau_j) - K\sum_{\langle i,j\rangle} \sigma_i \sigma_j \tau_i \tau_j,$$1 sublattices; for $$\frac{\mathcal{H}}{k_B T} = -J\sum_{\langle i,j\rangle} (\sigma_i \sigma_j + \tau_i \tau_j) - K\sum_{\langle i,j\rangle} \sigma_i \sigma_j \tau_i \tau_j,$$2-spins, $$\frac{\mathcal{H}}{k_B T} = -J\sum_{\langle i,j\rangle} (\sigma_i \sigma_j + \tau_i \tau_j) - K\sum_{\langle i,j\rangle} \sigma_i \sigma_j \tau_i \tau_j,$$3 is consistently larger on $$\frac{\mathcal{H}}{k_B T} = -J\sum_{\langle i,j\rangle} (\sigma_i \sigma_j + \tau_i \tau_j) - K\sum_{\langle i,j\rangle} \sigma_i \sigma_j \tau_i \tau_j,$$4. For $$\frac{\mathcal{H}}{k_B T} = -J\sum_{\langle i,j\rangle} (\sigma_i \sigma_j + \tau_i \tau_j) - K\sum_{\langle i,j\rangle} \sigma_i \sigma_j \tau_i \tau_j,$$5-spins, these differences are less pronounced but observable with high-precision fits.

Berezinskii-Kosterlitz-Thouless Transitions

The boundaries of the critical phase show hallmark BKT scaling. Pseudo-susceptibilities are introduced: $$\frac{\mathcal{H}}{k_B T} = -J\sum_{\langle i,j\rangle} (\sigma_i \sigma_j + \tau_i \tau_j) - K\sum_{\langle i,j\rangle} \sigma_i \sigma_j \tau_i \tau_j,$$6, revealing that finite-size pseudo-critical points scale as $$\frac{\mathcal{H}}{k_B T} = -J\sum_{\langle i,j\rangle} (\sigma_i \sigma_j + \tau_i \tau_j) - K\sum_{\langle i,j\rangle} \sigma_i \sigma_j \tau_i \tau_j,$$7. This scaling, confirmed by fits, aligns with BKT universality. Binder ratios and correlation length also corroborate the BKT nature and the critical phase's stability.

Partial Antiferromagnetic Order

For $$\frac{\mathcal{H}}{k_B T} = -J\sum_{\langle i,j\rangle} (\sigma_i \sigma_j + \tau_i \tau_j) - K\sum_{\langle i,j\rangle} \sigma_i \sigma_j \tau_i \tau_j,$$8, $$\frac{\mathcal{H}}{k_B T} = -J\sum_{\langle i,j\rangle} (\sigma_i \sigma_j + \tau_i \tau_j) - K\sum_{\langle i,j\rangle} \sigma_i \sigma_j \tau_i \tau_j,$$9, the model exhibits a partial antiferromagnetic phase: sublattices with higher coordination develop antiferromagnetic ordering in $\sigma_i, \tau_i = \pm 1$0-spins, while the lowest-coordination sublattice remains paramagnetic. This entropic selection is characteristic of frustrated systems with inhomogeneous interaction strengths.

Comparative Analysis: Lattice Geometry and Phase Structure

The phase diagrams of the AT model on square, triangular, and UJ lattices are juxtaposed. Uniform lattices host only isolated critical points or lines; by contrast, the UJ geometry yields an extended critical phase. The unique combination of geometric frustration and inhomogeneity is essential for stabilization of criticality without the need for vortex terms or multi-layered couplings, as is required for critical phases in discrete-variable models.

Numerical Evidence and Strong Claims

• Critical phase exists on the UJ lattice for discrete-spin AT model, in contrast to uniform or even moderately frustrated lattices.
• Critical exponent $\sigma_i, \tau_i = \pm 1$1 is sublattice-dependent, evidencing inhomogeneous critical behavior.
• Transitions into and out of the critical phase are BKT-type, demonstrated by $\sigma_i, \tau_i = \pm 1$2 scaling of pseudo-critical points.
• Magnetization shows power-law scaling with system size in the critical phase; correlation length ratio is nonzero in thermodynamic limit.
• Strong numerical evidence: critical exponents $\sigma_i, \tau_i = \pm 1$3 for $\sigma_i, \tau_i = \pm 1$4-spins in critical phase range from 0.21–0.50, vs. $\sigma_i, \tau_i = \pm 1$5 for $\sigma_i, \tau_i = \pm 1$6-spins, depending on sublattice and interaction.

Implications and Future Directions

Theoretical implications include a deeper understanding of how structural inhomogeneity and frustration cooperate to stabilize critical behavior in statistical models with discrete degrees of freedom. This methodology and results generalize to experimental systems, e.g., AT universality observed in Selenium/Ni and Rydberg atom systems, suggesting that similar critical phases could be realized or engineered by appropriate lattice design. Since quasi-long-range order is the hallmark of superconductors, superfluids, and supersolids, these findings point to possibilities for critical quantum phases in strongly correlated materials and engineered quantum simulators.

Practically, the identification of efficient pseudo-susceptibility probes and scaling relations for criticality in inhomogeneous systems broadens the toolkit available for numerical investigation of complex spin models. The observed critical behavior in the AT model on the UJ lattice may inspire novel strategies for inducing criticality in other models via controlled lattice design or coupling.

Theoretical avenues for future exploration include:

• Characterization of multiphase points and nature of transitions for $\sigma_i, \tau_i = \pm 1$7.
• Systematic study of AT model on other inhomogeneous/complex lattices.
• Quantum analogues: mapping results to 1D quantum systems, given the equivalence with 2D classical statistical models.
• Experimental pursuit of critical phases in atomic, magnetic, or quantum lattice systems with engineered inhomogeneity.

Conclusion

The Ashkin-Teller model on the Union-Jack lattice supports an emergent critical phase, distinct from conventional uniform or frustrated lattices. This phase is characterized by quasi-long-range order, power-law scaling, and nontrivial sublattice-dependent critical exponents. The phase boundaries are BKT-type and confirmed via novel susceptibility probes and scaling analysis. These results demonstrate that lattice inhomogeneity and frustration, combined with coupled spin degrees of freedom, can stabilize critical phases even in discrete spin models. The practical and theoretical implications extend to the design and understanding of critical quantum phases in diverse physical systems, laying groundwork for future studies in both statistical mechanics and condensed matter physics.

Reference: "Emergent critical phases of the Ashkin-Teller model on the Union-Jack Lattice" (2605.09453)