Figure 1

(a) A lowest order diagram for the XY model. This diagram has no contribution to ${\widetilde{{\rm Y}}}_p$. (b) A lowest order diagram which has a nonzero contribution to ${\widetilde{{\rm Y}}}_p$ for the $p$-state clock model.Reuse & Permissions

Figure 2

Twist dependence of the helicity modulus ${{\rm Y}}_p$ for the $p$-state clock model in disordered phase. ${{\rm Y}}_p$ is zero if the twist angle $\Delta$ is an integral multiple of $2\pi /p$. The finiteness of ${\widetilde{{\rm Y}}}_p$ with respect to an infinitesimal twist reflects the curvature at $\Delta =0$ of the free energy difference, $F_p\left(\Delta \right)-F_p\left(0\right)$, in Eq. (6).Reuse & Permissions

Figure 3

(a)–(d) Temperature dependence of ${{\rm Y}}_p$ and ${\widetilde{{\rm Y}}}_p$ for $p=5$ and $p=6$. In both cases, ${\widetilde{{\rm Y}}}_p$ with respect to an infinitesimal twist does not reflect each phase appropriately. In comparison, for ${{\rm Y}}_p$ with respect to the finite twist $2\pi /p$, we can find three regions: blue, purple, and beige regions correspond to ordered, BKT, and disordered phase, respectively. In addition, as $L$ is increased, ${{\rm Y}}_p$ approaches the universal value on the higher and lower transition points, with the universal relations (7) and (8) shown, respectively, by yellow-green and aqua lines. Colored lines are for a guide to the eyes. (e) and (f) Finite-size scaling of ${{\rm Y}}_5$ near $T_{\mathrm{BKT}}^{\mathrm{high}}$ and $T_{\mathrm{BKT}}^{\mathrm{low}}$. All the data points collapse into universal curves in both cases. The estimated scaling parameters are $(T_{\mathrm{BKT}},L_0)=(0.944,0.8),(0.908,1.9)$ for (e) and (f), respectively.Reuse & Permissions

Figure 4

(a) and (b) Finite-size scaling of ${{\rm Y}}_6$ near $T_{\mathrm{BKT}}^{\mathrm{high}}$ and $T_{\mathrm{BKT}}^{\mathrm{low}}$. All the data points collapse into universal curves in both cases. The estimated scaling parameters are $(T_{\mathrm{BKT}},L_0)=(0.904,0.5),(0.700,0.9)$ for (a) and (b), respectively.Reuse & Permissions