Phase transitions in frustrated $XY$ model on a square lattice

We study the phase diagram of a frustrated $XY$ model with a nematic coupling $\left(\Delta \right)$ on the square lattice by means of Monte Carlo simulation. Besides the conventional magnetic-chiral phase, the phase diagram shows an obvious region in which the magnetism is algebraically ordered but the chirality remains disordered. In addition, in the large $\Delta$ region, a nematic-chiral phase without magnetic order is identified, which is similar to the phase found in the frustrated $XY$ model on triangular lattice [J. H. Park, S. Onoda, N. Nagaosa, and J. H. Han, Phys. Rev. Lett. 101, 167202 (2008)]

Figure 1

(Color online) A schematic depiction of the transition from (a) magnetic-chiral phase to (b) nematic-chiral phase.

Figure 2

(Color online) Calculated phase diagram for the model in Eq. (1). The high-temperature paramagnetic phase is denoted by PM, the phases with algebraic correlations in magnetic and nematic order by aM and aN, respectively, and the long-range correlations in the chirality order by C. The statistical errors of all the symbols are the same as their thickness in the $T$ direction.

Figure 3

(Color online) Helicity modulus Υ as a function of $T$ for various sizes $L$ (a) at $\Delta =0.5$ and (c) $\Delta =0.9$. The straight line is $\left(2/\pi \right)\left(1+3\Delta \right)T$. The crossing temperatures of this line and Υ for each $L^{-1}$ are shown in (b) for $\Delta =0.5$ and (d) $\Delta =0.9$ with the extrapolation to $L^{-1}=0$.

Figure 4

(Color online) Binder’s fourth-order cumulant $U_L$ as a function of $T$ for different lattice sizes at (a) $\Delta =0.5$ and (b) $\Delta =0.9$.

Figure 5

(Color online) A scaling plot of $M$ and its susceptibility $\chi$ for $\Delta =0.05$, $\Delta =0.5$, and $\Delta =0.9$.

Figure 6

(Color online) (a) Simulated curves of $I$ and special heat $C$ as a function of $T$ at $\Delta =0.9$. (b) A snapshot of the nematic-chiral state at $T=0.2$ for $\Delta =0.9$. The spins in the same sublattice are shown with red arrows (gray arrows) for clarity.