Nematic and Chiral Order for Planar Spins on a Triangular Lattice

We propose a variant of the antiferromagnetic $XY$ model which includes a biquadratic ($J_2$) as well as the quadratic ($J_1$) interaction on the triangular lattice. The phase diagram for large $J_2/J_1$ exhibits a phase with coexisting quasi-long-range nematic, and long-ranged vector spin chirality orders in the absence of magnetic order, which qualifies our model as the first instance of a classical spin model that exhibits a vector chiral spin liquid phase. The interplay of nematic and spin chirality orders is discussed. A variety of critical properties are derived by means of Monte Carlo simulation.

Figure 1

Phase diagram of the $J_1-J_2$ model in Eq. (1) with $J_1=1-x$ and $J_2=x$. Two closely spaced transition temperatures labeled by $T_{\mathrm{KT}}$ and $T_{\chi }$ separate the paramagnetic (PM) phase from the algebraically correlated phase at a lower temperature. aM, aN, and C stand for phases with algebraic correlations in (antiferro) magnetic and (antiferro) nematic order parameters, and the long-range correlations in the chirality order. A further transition from aN to aM occurs as an Ising transition for $x>x_c$ with $x_c\approx 0.7$. All the symbols have a thickness in the temperature direction consistent with their statistical errors. Inset: The onset of chirality order at $T_{\chi }$ (red) takes place at temperatures close to, but slightly higher than the corresponding KT transition temperature $T_{\mathrm{KT}}$ (black) for all $x$. The Ising transition temperature $T_I$ is not shown here for clarity.

Figure 2

Phase stiffness ${\rho }_s(T)$ of the $J_1-J_2$ model for $L=15–60$ and $x=0.5$ and 0.9. The straight line is $(2/\pi )(\sqrt{3}/2)(J_1+4J_2)T$. The crossing temperature of this line and ${\rho }_s(T)$ for each $L$ is shown in the inset along with the extrapolation to $L^{-1}=0$.

Figure 3

A scaling plot of chirality based on Eq. (2) and its susceptibility for $x=0.3$ and $x=0.8$, for lattice sizes $L=15–60$. The exponents used are those of $x=0$ [14]. The last row shows the behavior of the Binder cumulants at $x=0.3$ and $x=0.8$, respectively.

Figure 4

The size dependence of (a) the magnetic ($\mathcal{M}$) and (b) the nematic ($\mathcal{N}$) order parameters at $x=0.9$ are shown on the log-log plot. Insets: The critical exponent ${\eta }_{\mathcal{M}}(T)$ for $\mathcal{M}\sim 1/L^{{\eta }_{\mathcal{M}}(T)}$ and ${\eta }_{\mathcal{N}}(T)$ for $\mathcal{N}\sim 1/L^{{\eta }_{\mathcal{N}}(T)}$.

Figure 5

(a) A cartoon depicting the loss of head-tail order in going from aM to aN phase. (b) A snapshot of the chiral-nematic state at $T=0.2$ for $x=0.9$ where the Ising transition occurs at $T_I=0.177$. Within the same sublattice the “body” of the arrows, not their tips, are seen to point in the same general direction.

Figure 6

Eight possible magnetic patterns within the nematically ordered phase, which also includes configurations with the global rotation of all the spins shown here. The corresponding chirality of each spin configuration is shown inside the triangle.