关联矩阵纠缠本征谱的标度行为
Scaling behavior of eigenspectrum for entanglement from correlation matrices
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中文总结 AI 辅助
本研究通过循环矩阵近似解析关联矩阵特征值，发现其体特征值按$1/L_A$标度，结合熵广延性解释了临界纠缠的$\log_2 L_A$标度，并揭示了向非临界行为的平滑交叉。
中文摘要 AI 辅助
我们研究了关联矩阵特征值的标度行为，该矩阵刻画了一个子系统与其在总纯态中的补集部分之间的纠缠。纠缠熵最显著的特征是其对一维临界系统基态子系统尺寸的对数依赖。尽管这一结果具有鲁棒的普适性并广泛关联于众多主题，但其透彻理解需要复杂的数学物理技术或共形场论。我们工作的目的是从底层特征值分布的角度阐明这一点。核心对象是关联矩阵，其形式为Toeplitz或块Toeplitz矩阵。我们发展了大矩阵维度极限下的循环矩阵近似，从而能够解析地分析单个特征值的行为。我们发现，对于自由晶格费米子和横场伊辛链，本征谱体中的特征值随子系统尺寸按$1/L_A$标度。结合熵函数的广延性，这解释了临界点处纠缠的鲁棒$\log_2 L_A$标度。从伊辛链的无纠缠极限扰动出发，我们发现向“非临界”标度的平滑交叉行为，其特征是非常缓慢的对数依赖，使其看似非临界系统预期的恒定纠缠值。
英文摘要
We study the scaling behavior of the eigenvalues of correlation matrices, which characterize the entanglement of a subsystem with its complement part of a total pure state. A most distinguishing feature of entanglement entropy is its logarithmic dependence on the subsystem size for the groundstate of one-dimensional critical systems. Despite its robust universal character and relevance to a wide range of topics, a thorough understanding of this result requires sophisticated mathematical physics techniques or conformal field theory. The aim of our work is to shed light on this from the underlying eigenvalue distribution perspective. The central object is the correlation matrix, which takes the form of Toeplitz or block-Toeplitz matrix. We develop a circulant matrix approximation in the large matrix dimension limit, thus allowing for the individual eigenvalues behavior to be analysed analytically. We find that for both free lattice fermions and transverse field Ising chain, eigenvalues in the bulk of the eigenspectrum scales as $1/L_A$ with the subsystem size. Together with the extensivity of the entropy function, it explains the robust $\log_2 L_A$ scaling of entanglement at criticality. Perturbing from the entanglement-free limit of the Ising chain, we find a smooth crossover behavior to `non-critical' scaling that is characterized by a very slow logarithmic dependence rendering it seemingly a constant entanglement value expected of non-critical systems.
发表机构
- Zhejiang Institute of Modern Physics, School of Physics, Zhejiang University（浙江大学现代物理研究所物理学院）
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