Spontaneous Symmetry Breaking in Two-dimensional Long-range Heisenberg Model

Abstract

The introduction of decaying long-range (LR) interactions $1/r^{d+σ}$ has drawn persistent interest in understanding how system properties evolve with $σ$. The Sak's criterion and the extended Mermin-Wagner theorem have gained broad acceptance in predicting the critical and low-temperature (low-T) behaviors of such systems. We perform large-scale Monte Carlo simulations for the LR-Heisenberg model in two dimensions (2D) up to linear size $L=8192$, and show that, as long as for $σ\leq 2$, the system exhibits spontaneous symmetry breaking, via a single continuous phase transition, and develops a generic long-range order. We then introduce an LR simple random walk (LR-SRW) with the total walk length fixed at O($L^d$), satisfying the extensivity of statistical systems, and observe that the LR-SRW can faithfully characterize the low-T scaling behaviors of the LR-Heisenberg model in both 2D and 3D, as induced by Goldstone-mode fluctuations. Finally, based on insights from LR-SRW, we propose a general criterion for the phase transition and the low-T properties of LR statistical systems with continuous symmetry in any spatial dimension.

Figures (3)

• Figure 1: The Phase diagram of the 2D LR-Heisenberg model, which includes: mean-field regime for $\sigma\leq1$ with critical behaviors governed by the Gaussian fixed point, nonclassical regime for $1<\sigma\leq2$ with $\sigma$-dependent critical exponents, and SR regime for $\sigma>2$ with neither LRO nor phase transition for $T>0$. The correlation function $g(r)$ in the LRO phase behaves as $\sim g_0 + ar^{2-d-\eta_\ell}$ for $\sigma<2$, with $\eta_\ell=2-\sigma$, and decays logarithmically as $\sim g_0 + a/\ln(r/r_0)$ for $\sigma=2$. For $\sigma>2$, the correlation length diverges as $\xi\sim\exp(b\beta)$, with $b$ being a non-universal constant.
• Figure 2: Existence of the finite-T phase transition and emergence of LRO for the LR-Heisenberg model with $\sigma=2$. (a) Semi-log plot of pseudo-critical points $\beta_L$ versus $L$, determined by $\xi/L=1$. For the NN case ($\sigma \to \infty)$, $\beta_L$ diverges logarithmically as $\beta_L \simeq (1/2\pi) \ln L$. For $\sigma=2$, however, $\beta_L$ decreases and rapidly converges as $\beta_L = \beta_c+ bL^{-\omega}$ with $\beta_c = 1.27(2)$ and $\omega\approx0.73$ (blue curve). (b) Different scaling behaviors of the correlation length $\xi$ for $\sigma=2$ and for $\sigma>2$. Data points for different system sizes are marked by different symbols. For $\sigma>2$, $\xi$ remains finite for any $T>0$, and, as $T\to 0$, it diverges as $\sim \exp(a \beta+b)$, where constants $(a,b)$ are $(2\pi,0)$ for the NN case and $(2.07,-1.09)$ for $\sigma=3$ in the plot. For $\sigma=2$, $\xi$ diverges as $\sim (\beta_c-\beta)^{-\nu}$ with $\nu\approx8$, where $\beta_c\approx 1.27$ can be roughly located in the inset. The inset plots $\xi/L$ versus $\beta$ for various $L$ ranging from $64$ to $2048$ in a doubling sequence, where the approximately common intersection can be observed. (c) The LRO at $\beta=2,4,8$ for $\sigma=2$. The main plot demonstrates that the squared magnetization $\langle M^2\rangle$ converges to positive values: $0.25$, $0.61$, $0.81$, respectively, following a logarithmic decay: $\sim 1/\ln(L/L_0)$. The inset further displays the residual squared magnetization $M_r^2=\langle M^2\rangle-b\langle M_k^2\rangle$, where the constant $b>0$ is taken so that $M^2 > M_r^2$ for any finite $L$. With $b \approx 153, 182$, and $182$ respectively, $M_r^2$ converge to non-vanishing values with corrections scaling as $L^{-\omega} \approx L^{-0.41}$, further showing the emergence of LRO for $L \to \infty$.
• Figure 3: The faithful characterization of the Goldstone-mode physics in the 2D and 3D LR-Heisenberg models at low-T by the fixed-$\mathcal{L}$ LR-SRW. The rescaled quantity $D_k=L^2/\chi_k$ is plotted versus $L$ (in log scale), for (a) the 2D LR-Heisenberg model, (b) the 2D fixed-$\mathcal{L}$ LR-SRW, (c) the 3D fixed-$\mathcal{L}$ LR-SRW, and (inset of (c)) the 3D LR-Heisenberg model; the inset of panel (a) is for $\sigma=2$ at different low temperatures. In all cases, $D_k$ follows a power law as $D_k \sim L^{2-\sigma}$ for $\sigma<2$, and scales logarithmically as $D_k \sim \ln L$ for $\sigma=2$. For $\sigma>2$, both the 3D LR-Heisenberg and fixed-$\mathcal{L}$ LR-SRW models have $\chi_k\sim L^2$.