In this paper we study the phase diagram of the five-state Potts antiferromagnet on the bisected-hexagonal lattice. This question is important since Delfino and Tartaglia recently showed that a second-order transition in a five-state Potts antiferromagnet is allowed, and the bisected-hexagonal lattice had emerged as a candidate for such a transition on numerical grounds. By using high-precision Monte Carlo simulations and two complementary analysis methods, we conclude that there is a finite-temperature first-order transition point. This one separates a paramagnetic high-temperature phase, and a low-temperature phase where five phases coexist. This phase transition is very weak in the sense that its latent heat (per edge) is two orders of magnitude smaller than that of other well-known weak first-order phase transitions.
The phase diagram of the nearest-neighbor
Z(5) model on the square and triangular lattices is investigated using techniques of the real-space renormalization group. We use the recently introduced technique of Nienhuis, Berker, Riedel, and Schick, which yields the expected first-order transition on the Potts subspace of the
Z(5) model on the triangular lattice. For the general
Z(5) model we obtain a line of first-order transitions which crosses over into a line of second-order transitions. The same technique when applied to the square lattice is found to show this crossover behavior only in combination with a Migdal procedure.
Simulations of an Ising
q-state Potts model which is equivalent to the Ising model on annealed percolation clusters are used to determine the phase diagram of the model in two dimensions. Three topologically different phase diagrams are obtained: (i) for
q=2, there are two critical Ising lines meeting at
T=0 at the four-state Potts critical point; (ii) for
2<q< 4, the Ising critical line meets the
q-state critical line and a line of first-order transitions at a bicritical point; (iii) for
q>4, the Ising critical line intersects a line of first-order transitions at a critical end point.
We study a generalization of the well-known classical two-dimensional square lattice compass model of XY spins (sometimes referred to as the ${90}^{\ensuremath{\circ}}$ compass model), which interpolates between the XY model and the compass model. Our model possesses the combined
C4 lattice and spin rotation symmetry of the compass model but is free of its fine-tuned subsystem symmetries. Using both field theoretic arguments and Monte Carlo simulations, we find that our model possesses a line of critical points with continuously varying exponents of the Ashkin-Teller type terminating at the four-state Potts point. Further, our Monte Carlo study uncovers that beyond the four-state Potts point, the line of phase transition is connected to the lattice-nematic Ising phase transition in the square lattice compass model through a region of first-order transitions.
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 (nearest neighbor, ferromagnetic) and
J2>0 (second neighbor, antiferromagnetic). The striped phase breaks a
Z4 symmetry and is stabilized at low temperatures for
g=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-
J2 model has a line of very weak first-order phase transitions in the whole region
1/2<g<g∗, where ${g}^{*}=0.67\ifmmode\pm\else\textpm\fi{}0.01$. Thereafter, the transitions from
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∗ in the
J1-
J2 model. The continuous transitions near
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>1/2. 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.