Two-dimensional XY Ferromagnet Induced by Long-range Interaction

TL;DR

This study analyzes the 2D XY ferromagnet with long-range interactions decaying as $1/r^{2+\sigma}$ using large-scale Monte Carlo simulations up to $L=8192$. It identifies a crossover at $\sigma_*=2$ between short-range BKT behavior and long-range ordered phases, with the correlation length $\xi$ and susceptibility $\chi_k$ exhibiting distinct scaling: for $\sigma<2$ the system develops spontaneous magnetization and a Goldstone mode that yields algebraic correlations with exponent $\eta_\ell=2-\sigma$, while at $\sigma=2$ logarithmic corrections appear; for $\sigma>2$ the transition resembles a BKT transition. The results are supported by finite-size scaling of $\xi/L$ and $\chi_k$, and by a specific heat–like quantity that peaks sharply for $\sigma\le 2$. The findings have potential relevance for experimental platforms such as Rydberg atom arrays and advance understanding of LR-driven criticality in low dimensions.

Abstract

The crossover between short-range and long-range (LR) universal behaviors remains a central theme in the physics of long-range interacting systems. The competition between LR coupling and the Berezinskii-Kosterlitz-Thouless mechanism makes the problem more subtle and less understood in the two-dimensional (2D) XY model, a cornerstone for investigating low-dimensional phenomena and their implications in quantum computation. We study the 2D XY model with algebraically decaying interaction $\sim1/r^{2+σ}$. Utilizing an advanced update strategy, we conduct large-scale Monte Carlo simulations of the model up to a linear size of $L=8192$. Our results demonstrate continuous phase transitions into a ferromagnetic phase for $σ\leq 2$, which exhibits the simultaneous emergence of a long-ranged order and a power-law decaying correlation function due to the Goldstone mode. Furthermore, we find logarithmic scaling behaviors in the low-temperature phase at $σ= 2$. The observed scaling behaviors in the low-temperature phase for $σ\le 2$ agree with our theoretical analysis. Our findings request further theoretical understandings and can be of practical application in cutting-edge experiments like Rydberg atom arrays.

Figures (9)

• Figure S1: The growth of the correlation length $\xi$ with respect to the reduced temperature $t = (T - T_c)/T_c$ is shown for $\sigma = 1.875$ (green dots), $2$ (blue dots), $3$ (red dots), and the NN case (black dots). (a) The semi-logarithmic plot of $\xi$ versus $b/\sqrt{t}$, where $b=1$ for $\sigma=1.875$ and $\sigma=2$, and $b=1.25$ for $\sigma=3$ and $b=1.625$ for the NN case. The straight black line indicates exponential growth, suggesting that $\xi \sim \exp(b/\sqrt{t})$ for $\sigma = 3$ and the NN case. (b) The log-log plot of $\xi$ versus $b^2/t$. The straight dark-green and dark-blue lines indicate a power-law growth, with $\xi \sim t^{-\nu}$ for $\sigma = 1.875$ and $2$, respectively.
• Figure S2: $M^2$ versus $L$ for $\sigma=1.25, 1.75, 1.875, 2$ at different temperatures, i.e., $\beta=1, 2, 4, 8$. The solid lines represent the curves fitted to the data for the corresponding temperatures, and the fitting formulas are given by Eq.\ref{['LE4']} and Eq.\ref{['LE5']}, respectively, for $\sigma<2$ and $\sigma=2$. The dashed lines represent the final value of $M^2$ in the limit of infinite system size, as determined by the fitting results.
• Figure S3: $M^2$ (a) and $\xi/L$ (b) versus $L$ for $\sigma=3$. Double logarithmic coordinates are adopted in (a), and only the horizontal axis is on a logarithmic scale in (b).
• Figure S4: $\widetilde{\chi}_k$ versus $L$ at different temperatures for $\sigma$ = 1.25, 1.75, 2, and 3. $\widetilde{\chi}_k$ represents $\chi_k$ multiplied by a constant to make the first data point at different temperatures ($L=64$) overlap. Black and red lines represent the fitting curves of data at $\beta=4$ and $\beta=\beta_c$, respectively. The insets plot the relationship between $\widetilde{\chi}_kL^{2-\eta_\ell}$ and $L$ (in (a), (b), (c)) and the relationship between $\widetilde{\chi}_k\ln{(L/L_0)}$ and $L$ (in (d)), respectively. At low temperatures, the curves flatten out and finally converge to a constant, indicating the scaling behavior: $\chi_k\sim L^{2-\eta_\ell}$ for $\sigma<2$ and $\chi_k\sim\ln{(L/L_0)}$ for $\sigma=2$. However, for $\sigma=1.75, 1.875,$ and $2$, $\chi_k$ exhibits clearly different scaling behaviors at the critical point and low temperatures.
• Figure S5: $L^2/\chi_k$ versus $L$ for $\sigma=2$ at $\beta=1, 2, 4, 8$. The horizontal axis is on a logarithmic scale. The solid lines represent straight lines fitted to the data at corresponding temperatures. The good fit demonstrates that $L^2/\chi_k$ has a linear relationship with $\ln{L}$. The inset plots data points at the critical temperature, and its exponential growth behavior indicates a power-law relationship between $L^2/\chi_k$ and $L$.