Winding number excitation detects phase transition in one-dimensional $XY$ model with variable interaction range

We numerically study the critical behavior of the one-dimensional $XY$ model of the size $N$ with variable interaction range $L$. As expected, the standard local order parameter of the magnetization is shown to well detect the mean-field-type transition which occurs at any nonzero value of $L/N$. The system is particularly interesting since the underlying one-dimensional structure allows us to study the topological excitation of the winding number across the whole system even though the system shares the mean-field transition with the globally coupled system. We propose a nonlocal order parameter based on the width of the winding number distribution which exhibits a clear signature of the transition nature of the system.

Figure 1

Magnetization $m_{\varphi }$ as a function of temperature $T$ for various system sizes $N$ and interaction ranges $L$ for (a) $L/N=0.4$ and (b) $L/N=0.1$.

Figure 2

(a) Specific heat $c_v$ and (b) Binder cumulant $U_B$ vs temperature $T$ for various system sizes $N$. The interaction range $L$ is chosen to yield the same value of the ratio $L/N=0.4$.

Figure 3

(a) $c_v$ and (b) $U_B$ vs $T$ for $L/N=0.1$. For comparison, see Fig. 2 for $L/N=0.4$.

Figure 4

Finite-size scaling collapse of the magnetization $m_{\varphi }$: $m_{\varphi }{N}^{\beta /\overline{\nu }}$ vs $(T-T_c)N^{1/\overline{\nu }}$ for (a) $L/N=0.4$ and (b) $L/N=0.1$. With $T_c=0.5,\beta /\overline{\nu }=1/4$, and $\overline{\nu }=2$ chosen, good quality of scaling collapse of data is achieved.

Figure 5

Probability distribution function $P\left(q\right)$ of the winding number $q$ (a) at $T=0.5$ for various system sizes $N$ and (b) for a given size $N=640$ at various temperatures $T$. For each $P\left(q\right)$ the width of the distribution $\sigma (N,L,T)$ is computed and then used to scale the horizontal axis as shown in (c) and (d), each obtained from (a) and (b), respectively. For (a)–(d), $L/N=0.4$ has been used.

Figure 6

(a) Standard deviation $\sigma$ as a function of temperature $T$ for various system sizes $N$. (b) The newly defined nonlocal order parameter $m_q$ in Eq. (14) vs $T$ for various values of $N$. (c) $m_q$ vs $N$ in log-log scale clearly exhibits the power-law decay form at $T=T_c\approx 0.5$, with the slope corresponding to ${\beta }_q/{\overline{\nu }}_q=0.49$. (d) Finite-size scaling collapse of $m_q$ yields ${\overline{\nu }}_q=2.00\left(5\right)$. From (c) and (d), we estimate $T_c=0.500\left(3\right),{\beta }_q/{\overline{\nu }}_q=0.49\left(4\right),$ and ${\overline{\nu }}_q=2.00\left(5\right)$. All results in (a)–(d) are for $L/N=0.4$.