Phase diagram of generalized fully frustrated $XY$ model in two dimensions

It is shown that the phase diagram of the two-dimensional generalized fully frustrated $XY$ model on a square lattice contains a crossing of the chirality transition and the Kosterlitz-Thouless (KT) transition, as well as a stable phase characterized by a finite helicity modulus ${\rm Y}$ and an unbroken chirality symmetry. The crossing point itself is consistent with a critical point without any jump in ${\rm Y}$, with the size $\left(L\right)$ scaling ${\rm Y}\sim L^{-0.63}$ and the critical index $\nu \approx 0.77$. The KT transition line remains continuous beyond the crossing but eventually turns into a first-order line. The results are established using Monte Carlo simulations of the staggered magnetization, helicity modulus, and the fourth-order helicity modulus.

Figure 1

(Color online) The phase diagram of the 2D $\mathrm{GFF}XY$ model. The phases are characterized by the helicity modulus ${\rm Y}$ and the staggered magnetization $m$. The four phases correspond to all possible combinations of finite and vanishing ${\rm Y}$ and $m$. The points correspond to data obtained from the simulations. The horizontal line $p=1$ corresponds to the usual $\mathrm{FF}XY$ model. The phase lines cross and merge in one point.

Figure 2

(Color online) MC determination of the staggered magnetization. (a) The transition across the horizontal phase line $p_c\approx 1.3479$ is confirmed by a sharp drop to zero for larger sizes $L$. (b) Size scaling of Binder cumulant $B_m$ for $p=1$. The unique crossing of curves for different sizes yields the transition temperature $T_{ch}\approx 0.454$. This method is used to determine the $m$-phase line for $p<p_c$.

Figure 3

(Color online) MC determination of the KT transition line. (a) The helicity modulus ${\rm Y}$ as a function $T$ for various sizes $L$. The broken lines correspond to the jump condition; $b=1$ is the expected universal jump. $T_{\mathit{KT}}$ is estimated by extrapolating the crossing point with the $b=1$ line and the data to $L=\infty$, as shown in (b), which shows a second-order polynomial extrapolation of the crossing points, and $T_{\mathit{KT}}=0.447$ is obtained in accord with the finding $T_{\mathit{KT}}=0.446$ in Ref. 4. (c) The difference $\Delta T\equiv \mid T_{\mathit{KT}}(b,L)-T_{\mathit{KT}}(b∕2,L)\mid$ in crossing points $T_{\mathit{KT}}$ [see (a)] for $(b,b∕2)=(1,1∕2)$ and (2, 1). It is clearly seen that the jump has to be less than twice the universal jump consistent with the universal jump. (d) The same construction for $p=1.5$. We believe that a nonuniversal jump for $p>p_c$ cannot be entirely ruled out. (a)–(c) correspond to $p=1$, while (d) is for $p=1.5$.

Figure 4

(Color online) MC simulation of the fourth-order modulus ${{\rm Y}}_4$. (a) shows the typical ${{\rm Y}}_4$ characteristic of a KT transition for $p=1$: The well minimum moves from higher $T$ toward the transition temperature (vertical broken line) with increasing $L$. The size of the minimum extrapolates to a finite value. (a) and (b) show that this typical KT feature remains intact as $p$ is increased to the vicinity of $p_c$. (c) and (d) show that this KT feature is dramatically changed in the immediate vicinity of $p_c$, while (e) and (f) show that the KT feature is recovered for $p$ values above $p_c$.

Figure 5

(Color online) MC simulation of critical size scaling for the helicity modulus ${\rm Y}$ at the crossing point of the phase lines. (a) The size scaling ${\rm Y}\sim L^{-a}$ is obeyed to very good approximation. The collapsing point of the data determines the critical temperature $T_c\approx 0.1675$. (b) With the $T_c$ from (a) the fuller scaling ${\rm Y}=L^{-a}F\left[L^{1∕\nu }(T-T_c)\right]$ with $\nu =0.77$ is born out.