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).
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).
We investigate the symmetric Ashkin-Teller (AT) model on the triangular lattice in the antiferromagnetic two-spin coupling region (J0, 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).
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.
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 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.
Monte Carlo simulations have been used to study a model consisting of three Ising models with nearest neighbor exchange J on a face-centered-cubic lattice, which are coupled together by a constraint. The constraint requires that the vector sum of the four spins located at the corners of any elementary tetrahedron of the lattice not add to zero. Finite-size scaling analysis of the Monte Carlo results for L×L×L lattices with L=16, 32, and 64 has been used to find that Tc /J=10.84±0.01. The finite-size scaling functions for the magnetization and magnetic susceptibility near Tc are consistent with standard m=3 Heisenberg critical exponents, as predicted by renormalization group theory. Strengthening the four-spin constraint increases the value of the effective negative Ashkin–Teller four-spin coupling, and drives Tc to infinity.
The critical coarsening dynamics of the spin S =1/2, 3/2 antiferromagnetic Ising model on a triangular lattice is studied by the dynamic Monte Carlo simulation using a heat bath algorithm. The triangular antiferromagnetic Ising (TAI) model possesses an intrinsic geometrical frustration and a large degeneracy of ground state which may affect the equilibrium and non-equilibrium critical behaviors. The S =1/2 TAI has no phase transition at a finite temperature, but it was suggested that the S =3/2 TAI has the Kosterlitz–Thouless (KT)-type phase transition at a finite temperature. We employ a finite size scaling approach for the correlation function from the short-time dynamics of the S =1/2, 3/2 TAI to calculate the values of the static critical exponent η and the dynamic exponent z. For the S =1/2 TAI, our dynamic scaling analysis provides η =0.498±0.006 and z =2.278±0.020 at T =0, agreeing with the previous results. For the S =3/2 TAI, after identifying a KT-transition temperature TKT =0.51±0.01, we find the temperature-dependent η ranging from 0.301±0.008 at T =0.51 to 0.224±0.016 at T =0 along the KT-line whereas the value of z =2.20±0.06 is constant for T≤TKT. It is shown that the spin S =3/2 TAI model and the two-dimensional XY model, sharing the KT-type phase transition, exhibit similar static critical and coarsening dynamics behavior. For both the S =1/2, 3/2 TAI, the value of z (η) is compatible with (larger than) that of the Ising model at Tc and the XY model for T≤TKT in two-dimension. Our results imply that although the quasi-long-range order disappears with ηXY =0 in the two-dimensional XY model at T =0, the S =3/2 TAI still maintains it with η TAI =0.224 due to the effect of a frustration and a high degeneracy of ground state.