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arXiv 1604.07571hep-thcond-mat.stat-mechgr-qc
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纠缠熵中的边界效应

Boundary effects in entanglement entropy

Clement Berthiere, Sergey N. Solodukhin

摘要:

在$d$维闵可夫斯基时空中,我们给出了纠缠面与边界相交的情形下的Renyi熵和纠缠熵的一些显式计算。当边界是一个单一的平面,我们计算的熵的贡献,由于这种交叉,首先在诺依曼和狄利克雷边界条件的情况下,然后在一个通用的罗宾型边界条件的情况下。本文在任意维$d$中分析了Neumann相和Dirichlet相边界耦合的流动,证明了流动是单调的,并指出了$d=3$情况的特殊性。我们认为,沿纠缠表面的平移对称性被打破,由于边界的存在,这表明纠缠是不均匀的。为了定量地描述这一点,我们引入了纠缠熵的密度,并明确地计算它。这个量清楚地表明纠缠在边界附近最大。然后,我们考虑的情况下,边界是由两个平行的平面在一个有限的分离和计算的纠缠熵以及它的密度在这种情况下。由于边界对纠缠熵的完全贡献不依赖于平面之间的距离,并且在单平面边界的情况下是熵的两倍。此外,我们发现如何面积法,在熵的一部分成比例的整个纠缠表面的面积,取决于两个边界之间的分离的大小。后者示出出现在UV有限的熵的一部分。

英文摘要:

We present a number of explicit calculations of Renyi and entanglement entropies in situations where the entangling surface intersects the boundary in $d$-dimensional Minkowski spacetime. When the boundary is a single plane we compute the contribution to the entropy due to this intersection, first in the case of the Neumann and Dirichlet boundary conditions, and then in the case of a generic Robin type boundary condition. The flow in the boundary coupling between the Neumann and Dirichlet phases is analyzed in arbitrary dimension $d$ and is shown to be monotonic, the peculiarity of $d=3$ case is noted. We argue that the translational symmetry along the entangling surface is broken due the presence of the boundary which reveals that the entanglement is not homogeneous. In order to characterize this quantitatively, we introduce a density of entanglement entropy and compute it explicitly. This quantity clearly indicates that the entanglement is maximal near the boundary. We then consider the situation where the boundary is composed of two parallel planes at a finite separation and compute the entanglement entropy as well as its density in this case. The complete contribution to entanglement entropy due to the boundaries is shown not to depend on the distance between the planes and is simply twice the entropy in the case of single plane boundary. Additionally, we find how the area law, the part in the entropy proportional to the area of entire entangling surface, depends on the size of the separation between the two boundaries. The latter is shown to appear in the UV finite part of the entropy.

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