奇偶还是半半？Bose-Hubbard模型中的纠缠反常
Odd/Even or Half ? Entanglement Anomaly in the Bose-Hubbard model
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中文总结 AI 辅助
本文研究一维Bose-Hubbard模型中不同空间二分方式对纠缠熵的影响，发现莫特绝缘体中交替切割违反面积律而呈体积律，超流体中两种切割均饱和，并用奴隶玻色子方法高效验证了该反常现象。
中文摘要 AI 辅助
面积律将量子多体基态的双分块纠缠熵与子系统之间边界的大小联系起来，但该边界的几何形状很少被讨论。我们在固定密度的一维Bose-Hubbard模型中，通过比较周期晶格的四种空间二分方式对此进行研究：前半部分、后半部分、偶数位点、奇数位点；这些二分方式共享相同数量的位点，但边界的排列方式不同。我们通过微扰理论找到了解析极限：在莫特绝缘体中，连续切割服从面积律，而交替切割服从体积律$S\propto N_s$，在此意义上是一种反常；对于超流体，由于态的去局域化，两种切割都坍缩到二项式饱和。我们将这些极限表述为关于多体问题的陈述，使用基于平均场并包含量子涨落的广义奴隶玻色子方法，同时通过小晶格尺寸的精确对角化（ED）和$N_s\gg 1$的密度矩阵重整化群（DMRG）模拟进行验证。奴隶玻色子高斯基态允许从约化相关矩阵计算任意所需二分方式的纠缠熵，与ED和DMRG结果一致。使用奴隶玻色子，计算成本由局部截断$n_{\max}$决定，而非希尔伯特空间维度，因此我们可以达到远超ED的晶格尺寸。我们的方法能够将依赖于分区的标度律确立为多体特征，而不仅仅是有限尺寸效应，与$N_s\in[4,10]$的ED以及更大晶格尺寸的DMRG结果高度一致。
英文摘要
The area law relates the bipartite entanglement entropy of a quantum many-body ground state to the size of the boundary between the subsystems, but the geometry of this boundary is rarely discussed. We inspect this in the 1D Bose-Hubbard model at fixed density by comparing four spatial bipartitions of the periodic lattice: first half, second half, even sites, and odd sites; sharing the same number of sites but differing in how the boundary is arranged. We find analytical limits with perturbation theory: in the Mott insulator the contiguous cut obeys the area lay while the alternating cut obeys a volume law $S\propto N_s$, in this sense an anomaly, and for the superfluid both cuts colapse to the binomial saturation due to delocalization of the state. We formulate these limits as a statement about the many-body problem using a generalized slave-boson approach based on mean-field with quantum fluctuations while verifying with Exact Diagonalization (ED) for small lattice sizes and Densitiy Matrix Renormalization Group (DMRG) simulations for $N_s\gg 1$. The slave-boson Gaussian ground state allows to compute the entanglement entropy from a reduced correlation matrix for any desired bipartition consistent with ED and DMRG results. Using slave bosons the computational cost is set by the local cutoff $n_{\max}$ rather than the Hilbert space dimension, so we can reach lattice sizes far beyond ED. Our method is capable of establishing the partition-dependent scaling laws as a many-body feature, not only a finite-size effect, in great agreement with the ED for $N_s\in[4,10]$ and DMRG for larger lattice sizes.
发表机构
- Universidad Nacional Autónoma de México（墨西哥国立自治大学）
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