Comment on "Nature of Long-Range Order in Stripe-Forming Systems with Long-Range Repulsive Interactions".
In a recent Letter, Mendoza-Coto et al. [1] presented a study focusing on the nature of phase transitions in twodimensional (2D) stripe-forming systems with competing short-range attractive and long-range (1=rα) repulsive interactions. In particular, they conclude that for dipolar interactions (α 1⁄4 3), the isotropic-nematic phase transition is in the Kosterlitz-Thouless (KT) universality class. The authors support their findings by mapping an effective Hamiltonian into models which behavior resembles the 2D XY model at low temperatures [2], and a finite-size scaling analysis from Langevin simulations. However, the validity of their conclusion is hindered by the lack of numerical evidence due to the relative small lattice sizes [3], and more generally because a nonuniversal behavior is expected for such 2D systems [4]. Aswe illustrate here, tiny lattice effects are enough to alter the transition scenario completely. Below we present a comparative analysis between the microcanonical caloric curves βðEÞ obtained for the 2D dipolar Ising model [5], which displays an isotropicnematic transition, and the 2D XY model [6], which shows a KT transition. The Hamiltonians H2D dip and H 2D XY are given in terms of the parameters δ and J as in [5,6], respectively. We evaluate βðEÞ via the statistical temperature-weighted histogram analysis method (ST-WHAM) [7] from data produced by Monte Carlo simulations with cluster updates for the XY model and replica exchange method for the dipolar Ising model. Figure 1 clearly shows that caloric curves for the two models exhibit distinct behaviors. While there are two S-shaped curves in βðEÞ for the dipolar Ising model, a monotonic decreasing behavior is observed for the XY model. The two S-shaped curves in Fig. 1(a) corresponds to two transitions that separates the isotropic phase (which is stable for E=N≳−1.086 and temperatures T>1=βIN≃0.8), the nematic phase (snapshot in the middle), and the striped phase (which is stable for E=N≲−1.187 and T<1=βNS ≃ 0.767, and displays ground-state configurations with 18 stripes for N 1⁄4 72). The S-shaped curves in βðEÞ are due to the presence of first-order phase transitions [8]. The results for the XY model in Fig. 1(b) shows the region near the KT transition at T 1⁄4 1=βKT ≃ 0.893. Accordingly, there is no signal of S-shaped curves in βðEÞ because the KT transition is an infinite order transition [6]. In conclusion, our analysis indicates that the isotropicnematic transition in the dipolar Ising model is a first-order phase transition instead of a KT transition. By considering this example and the expected nonuniversality, we argue that both mapping and numerical results in [1] are insufficient to determine the nature of isotropic-nematic transition in 2D systems with competing short-range and dipolar interactions. We thank Y. Levin for the clarifying discussions, F. C. Poderoso for helping us with the simulations of the 2D X model, and the support from CNPq (Brazil, Grant No. 165153/2014-8) and LCCA-USP.