连续对称性自发破缺的二维纠缠熵的普适对数修正
Universal logarithmic corrections to entanglement entropies in two dimensions with spontaneously broken continuous symmetries
摘要：
本文研究了在热力学极限下基态打破连续对称性的量子自旋模型中，嵌入二维$L\times L$晶格中的一维（线）$L$子系统的Rényi纠缠熵.利用量子MonteCarlo模拟，我们首先研究了具有反铁磁最近邻耦合$J_1>0$和铁磁次邻耦合$J_2\le 0$的$J_1 - J_2$海森堡模型。从$L=4$到$L=40$，在有限尺寸系统上SU（2）对称性破缺的特征清楚地表现为对雷尼纠缠熵的普遍加性对数修正：$l_q \ln L$和$l_q\simeq 1$，与雷尼指数和$J_2$的值无关。我们证实了这一结果，使用高精度自旋波分析（恢复自旋旋转对称性）的有限晶格上的$10^5\times 10^5$网站，允许进一步探索非通用的有限尺寸的修正和研究，此外的情况下，U（1）对称性破缺。我们的结果与Metlitski和Grover的预测$l_q=n_G/2$完全一致，其中$n_G$是Goldstone模式的数量[arXiv：1112.5166]。
英文摘要：
We explore the Rényi entanglement entropies of a one-dimensional (line) subsystem of length $L$ embedded in two-dimensional $L\times L$ square lattice for quantum spin models whose ground-state breaks a continuous symmetry in the thermodynamic limit. Using quantum Monte Carlo simulations, we first study the $J_1 - J_2$ Heisenberg model with antiferromagnetic nearest-neighbor $J_1>0$ and ferromagnetic second-neighbor couplings $J_2\le 0$. The signature of SU(2) symmetry breaking on finite size systems, ranging from $L=4$ up to $L=40$ clearly appears as a universal additive logarithmic correction to the Rényi entanglement entropies: $l_q \ln L$ with $l_q\simeq 1$, independent of the Rényi index and values of $J_2$. We confirm this result using a high precision spin-wave analysis (with restored spin rotational symmetry) on finite lattices up to $10^5\times 10^5$ sites, allowing to explore further non-universal finite size corrections and study in addition the case of U(1) symmetry breaking. Our results fully agree with the prediction $l_q=n_G/2$ where $n_G$ is the number of Goldstone modes, by Metlitski and Grover [arXiv:1112.5166].
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