The Nonclassical Regime of the Two-dimensional Long-range XY Model: a Comprehensive Monte Carlo Study

TL;DR

The paper investigates the two-dimensional long-range XY model with algebraically decaying couplings $J(r)\sim r^{-(2+\sigma)}$ using large-scale Monte Carlo simulations up to system size $L=8192$. It identifies a crossover at $\sigma_*=2$ separating long-range order with Goldstone modes (for $\sigma\le2$) from a short-range, BKT-like regime with quasi-long-range order (for $\sigma>2$), and demonstrates that critical exponents vary smoothly with $\sigma$ rather than obey Sak’s criterion. The authors provide detailed finite-size scaling analyses, show power-law divergence of the correlation length for $\sigma\le2$ and exponential growth for $\sigma>2$, and reveal logarithmic corrections at the marginal point $\sigma=2$. The results challenge previous theoretical predictions and establish a robust phase diagram with significant implications for LR interacting systems and related universality classes.

Abstract

The two-dimensional (2D) XY model plays a crucial role in statistical and condensed matter physics. With the introduction of long-range interactions, the system exhibits a richer set of physical phenomena and a crossover between non-classical and short-range universality classes.In this work, we investigate the 2D XY model with algebraically decaying interactions $\sim 1/r^{2+σ}$, and provide a comprehensive numerical analysis of its thermodynamic properties. We demonstrate that for $σ\leq 2$, the system undergoes a second-order phase transition into a ferromagnetic phase characterized by the emergence of long-range order. In the low-temperature phase, due to the presence of the Goldstone mode, the correlation function saturates to a non-zero constant in the form of a power law for $σ< 2$, with decaying exponent $2-σ$, and in the form of the inverse logarithm of distance for $σ=2$. Moreover, the critical points and exponents are also determined for various $σ$. We provide compelling evidence that the crossover between non-classical and short-range regimes occurs at $σ=2$. This work presents a detailed account of the simulation methodology, extensive numerical data, and new insights into the physics of long-range interacting systems.

Figures (22)

• Figure 1: The phase diagram of the two-dimensional long-range XY model reveals distinct behaviors depending on the interaction range $\sigma$ and the inverse temperature $\beta=1/T$. In the short-range regime ($\sigma > 2$), the system exhibits BKT transitions (brown line) into a QLRO phase. In the nonclassical regime ($1 < \sigma \leq 2$), the system undergoes a continuous phase transition (red line) into a long-range ordered phase. Lastly, in the classical regime ($\sigma \leq 1$), the transition (purple lines) is governed by the Gaussian fix point. The symbols $\beta_c^{\text{CG}}$ and $\beta_c^{\text{NN}}$ denote the critical inverse temperatures for the complete-graph and NN interaction limits, respectively.
• Figure 2: Overview of the dimensionless ratio $\xi/L$ for different $\sigma$ values. The phase transition type changes from a second-order transition for $\sigma \le2$ to a BKT-type transition for $\sigma > 2$, as shown here. The second-moment correlation length divided by $L$, $\xi/L$, is plotted as a function of inverse temperature $\beta$ for various $\sigma$ values: (a) $\sigma = 1.25$, (b) $\sigma = 1.75$, (c) $\sigma = 1.875$, (d) $\sigma = 2$, (e) $\sigma = 2.1$, and (f) $\sigma = 3$. System sizes range from $L = 64$ to $L = 4096$. For $\sigma \le 2$, the curves for different system sizes exhibits a clear intersection, indicating a second-order phase transition. For $\sigma = 3$, all $\xi/L$ curves for $L > 64$ converge after the transition point, a hallmark of a BKT transition. In the $\sigma = 2.1$ case, smaller system sizes (up to $L = 1024$) show a crossing due to the strong finite-size effect. For larger system sizes ($L = 2048$ and $L = 4096$), the $\xi/L$ values tend to converge in the low-T phase. This subtle case will be elucidated in Sec. \ref{['sigma_g2']}.
• Figure 3: Different behaviors of $(\xi/L)^2$ in the low-T phase for various $\sigma$. $(\xi/L)^2$ as a function of system size $L$ in the semi-logarithmic coordinates is shown for various $\sigma$ values at inverse temperatures $\beta = 1$ (a) and $\beta = 2$ (b). The linear behavior of the red squares demonstrates the logarithmic divergence of $(\xi/L)^2$ for the case of $\sigma = 2$. For $\sigma<2$ (the green dots in the figure), $(\xi/L)^2$ diverges faster, indicating a possible power-law divergence. This is confirmed in Sec. \ref{['Goldstone mode']}. The diverging $(\xi/L)^2$ is a signature of the LRO in the low-T phase for $\sigma\leq2$ . For $\sigma > 2$ (the blue dots), $(\xi/L)^2$ tends to converge, indicating QLRO in the system.
• Figure 4: The existence of spontaneous magnetization for $\sigma \leq 2$ in low-T phase. The squared magnetization $\langle M^2\rangle$ is plotted as a function of system size $L$ for various temperatures at $\sigma = 1.75$ (a1), $\sigma = 1.875$ (b1), and $\sigma = 2$ (c1). The $x$-axis represents the leading finite-size correction term, which varies by panel: $L^{-\eta_\ell}$ ($\eta_\ell = 0.25, 0.125$ for $\sigma = 1.75, 1.875$, respectively) in panels (a1) and (b1), $1 / \ln(L / L_0)$ in panel (c1). The fitting parameter $L_0=e^{-5.4}, e^{-6.8}, e^{-6.38}, e^{-6.2}$ for $\beta = 1, 2, 4, 8$ in panel (c1). For all three panels, the fitting lines align with the data point, and intersect with y-axis at positive values, indicating the spontaneous magnetization at low temperatures in the thermodynamic limit. To diminish the finite-size correction, the plots of $M_r^2 = \langle M^2\rangle-b\langle M_k^2\rangle$ are presented at $\sigma=1.75$ (a2), $\sigma=1.875$ (b2) and $\sigma=2$ (c2). The $x$-axis represents the leading correction term: $L^{-0.65}$ in panel (a2), $L^{-0.5}$ in panel (b2), and $L^{-0.4}$ in panel (c2). $b$ equals $22$ for panel (a2), 48 for panel (b2), $149,175,152,154$ in $\beta=1,2,4,8$ respectively for panel (c2). After subtracting a positive quantity, $M_r^2$ converges faster and continues to reach a positive value at the thermodynamic limit with smaller finite-size corrections, which is strong evidence of spontaneous magnetization.
• Figure 5: The demonstration of the presence of Goldstone mode. The log-log plots of $\chi_k$ versus system size $L$ are shown for different temperatures at $\sigma = 1.25$ (a), $1.75$ (b), and $1.875$ (c). As $L$ increases, the data points approach power-law growth with increasing system size $L$, confirming the form of the correlation function Eq. \ref{['correlation function']} and the presence of the Goldstone mode. Furthermore, the fitting lines at different temperatures are parallel for a certain $\sigma$, aligning with the prediction that the value of $\eta_\ell$ remains independent of temperature.