三维$O(3)$临界下的Shannon-Rényi熵和参与谱
Shannon-Rényi entropies and participation spectra across 3d $O(3)$ criticality
摘要：
利用量子蒙特卡罗模拟，研究了（2+1）维$O(3)$临界点上自旋- $\frac{1}{2}$相互作用系统的多体基态Shannon-Rényi熵标度的普适性质.考虑到完整系统和线形子系统，$SU(2)$对称性破缺的Néel有序侧的过渡的特征在于存在对数项的Shannon-Rényi熵的标度，这是不存在的无序间隙相。这种标度的差异允许使用香农-雷尼熵来捕获嵌入$L\times L$环面中的长度为$L$的线形子系统的量子临界点，因为较小的子系统熵在数值上比整个系统的精度高得多。最有趣的是，在量子相变处，在香农-雷尼熵线$S_\infty^\text{line}$的临界标度中出现了一个附加的子引导常数$b_\infty^{*\rm line}=0.41(1)$。这个数字似乎是普遍的三维$O(3)$临界，在三维反铁磁自旋$\frac{1}{2}$海森堡模型的有限温度转变证实。此外，相位和相变可以在参与谱的几个特征中检测到，参与谱由线路子系统的约化密度矩阵的对角元素组成。特别地，Néel有序转变可以通过铁磁畴壁的限制机制在$\{S^z\}$基础上简单地理解。
英文摘要：
Universal features in the scalings of Shannon-Rényi entropies of many-body groundstates are studied for interacting spin-$\frac{1}{2}$ systems across (2+1) dimensional $O(3)$ critical points, using quantum Monte Carlo simulations on dimerized and plaquettized Heisenberg models on the square lattice. Considering both full systems and line shaped subsystems, $SU(2)$ symmetry breaking on the Néel ordered side of the transition is characterized by the presence of a logarithmic term in the scaling of Shannon-Rényi entropies, which is absent in the disordered gapped phase. Such a difference in the scalings allows to capture the quantum critical point using Shannon-Rényi entropies for line shaped subsystems of length $L$ embedded in $L\times L$ tori, as the smaller subsystem entropies are numerically accessible to much higher precision than for the full system. Most interestingly, at the quantum phase transition an additive subleading constant $b_\infty^{*\rm line}=0.41(1)$ emerges in the critical scaling of the line Shannon-Rényi entropy $S_\infty^\text{line}$. This number appears to be universal for 3d $O(3)$ criticality, as confirmed for the finite-temperature transition in the 3d antiferromagnetic spin-$\frac{1}{2}$ Heisenberg model. Additionally, the phases and phase transition can be detected in several features of the participation spectrum, consisting of the diagonal elements of the reduced density matrix of the line subsystem. In particular the Néel ordering transition can be simply understood in the $\{S^z\}$ basis by a confinement mechanism of ferromagnetic domain walls.
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