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.
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.
We study the effects of quenched disorder on the first-order phase transition in the two-dimensional three-color Ashkin-Teller model by means of large-scale Monte Carlo simulations. We demonstrate that the first-order phase transition is rounded by the disorder and turns into a continuous one. Using a careful finite-size-scaling analysis, we provide strong evidence for the emerging critical behavior of the disordered Ashkin-Teller model to be in the clean two-dimensional Ising universality class, accompanied by universal logarithmic corrections. This agrees with perturbative renormalization-group predictions by Cardy. As a byproduct, we also provide support for the strong-universality scenario for the critical behavior of the two-dimensional disordered Ising model. We discuss consequences of our results for the classification of disordered phase transitions as well as generalizations to other systems.
Monte Carlo simulations in the three-dimensional (3D) Ashkin-Teller model on a cubic lattice are performed in the regions of the two-parameter space diagram where Ising-type phase transitions are expected. The scaling behavior of the Binder cumulant Q and the magnetic susceptibility in the critical region are exploited. In simulations the periodic boundary conditions and the Metropolis algorithm are used. Starting from Ising critical exponents and applying the finite-size-scaling analysis of the cumulant Q with nonlinear corrections, the accurate positions of the critical couplings on the continuous phase transition lines are calculated. For these couplings the critical exponent y(h) is calculated analyzing the magnetic susceptibility in the framework of the finite-size scaling with corrections. The values of y(h) agree with the 3D Ising model value along two lines determined by the order parameter [ssigma] .
The results of the large-scale Monte Carlo simulations of the continuous phase transitions in the three-dimensional Ashkin-Teller model on a cubic lattice are presented in the two-parameter space. The invariance of the ratio of the square of the second moment of the order parameter to its fourth moment in the critical region and the Metropolis algorithm are exploited. Using the universality hypothesis and the finite-size-scaling analysis, the accurate positions of the continuous phase-transition lines are calculated and their Ising character is confirmed. Approximate locations of the tricritical points are also estimated.
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 χ=d⟨m⟩/dJ as a key probe, and through this probe, pseudo-critical points Jc(L) are observed to scale proportionally to (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 short-time universal behavior of the two-dimensional Ashkin-Teller model at the Baxter line by performing time-dependent Monte Carlo simulations. First, as preparatory results, we obtain the critical parameters by searching the optimal power-law decay of the magnetization. Thus, the dynamic critical exponents θ_{m} and θ_{p}, related to the magnetic and electric order parameters, as well as the persistence exponent θ_{g}, are estimated using heat-bath Monte Carlo simulations. In addition, we estimate the dynamic exponent z and the static critical exponents β and ν for both order parameters. We propose a refined method to estimate the static exponents that considers two different averages: one that combines an internal average using several seeds with another, which is taken over temporal variations in the power laws. Moreover, we also performed the bootstrapping method for a complementary analysis. Our results show that the ratio β/ν exhibits universal behavior along the critical line corroborating the conjecture for both magnetization and polarization.
Anyons in a topologically ordered phase can carry fractional quantum numbers with respect to the symmetry group of the considered system, one example being the fractional charge of the quasiparticles and quasiholes in the fractional quantum Hall effect. When such symmetry-fractionalized anyons condense, the resulting phase must spontaneously break the symmetry and display a local order parameter. In this paper, we study the phase diagram and anyon condensation transitions of a \mathbb{Z}_2 ℤ 2 topological order perturbed by Ising interactions in the Toric Code. The interplay between the global (``onsite’’) Ising ( \mathbb{Z}_2 ℤ 2 ) symmetry and the lattice space group symmetries results in a non-trivial symmetry fractionalization class for the anyons, and is shown to lead to two characteristically different confined, symmetry-broken phases. To understand the anyon condensation transitions, we use the recently introduced critical torus energy spectrum technique to identify a line of emergent 2+1D XY* transitions ending at a fine-tuned (Ising ^2 2 )* critical point. We provide numerical evidence for the occurrence of two symmetry breaking patterns predicted by the specific symmetry fractionalization class of the condensed anyons in the explored phase diagram. In combination with large-scale quantum Monte Carlo simulations we measure unusually large critical exponents \eta η for the scaling of the correlation function at the continuous anyon condensation transitions, and we further identify lines of (weakly) first order transitions in the phase diagram. As an important additional result, we discuss the phase diagram of a resulting 2+1D Ashkin-Teller model, where we demonstrate that torus spectroscopy is capable of identifying emergent XY/O(2) critical behaviour, thereby solving some longstanding open questions in the domain of the 3D Ashkin-Teller models. To establish the generality of our results, we propose a field theoretical description capturing the transition from a \mathbb{Z}_2 ℤ 2 topological order to either \mathbb{Z}_2 ℤ 2 symmetry broken phase, which is in excellent agreement with the numerical results.