连续对称性自发破缺系统的纠缠熵
Entanglement Entropy of Systems with Spontaneously Broken Continuous Symmetry
摘要：
研究了连续对称性自发破缺系统的纠缠性质。我们发现，除了预期的面积定律行为，纠缠熵包含一个subleading的贡献，它与子系统的大小在数学上的分歧与蒙特卡罗模拟的A。Kallin et.（Phys.Rev.B84，165134（2011））。对数的系数是简单地由$N_G (d-1)/2$给出的通用数，其中$N_G$是Goldstone模式的数量，而$d$是空间维度。即使子系统边界是直的，不含角，这个术语也存在，它的起源在于戈德斯通模式的相互作用和有限体积中对称性的恢复。我们还计算了纠缠谱的“低能量”部分，并表明它具有相同的特征“状态塔”的形式作为物理低能量谱时，系统与自发破缺连续对称性被放置在一个有限的体积。
英文摘要：
We study entanglement properties of systems with spontaneously broken continuous symmetry. We find that in addition to the expected area law behavior, the entanglement entropy contains a subleading contribution which diverges logarithmically with the subsystem size in agreement with the Monte Carlo simulations of A. Kallin et. al. (Phys. Rev. B 84, 165134 (2011)). The coefficient of the logarithm is a universal number given simply by $N_G (d-1)/2$, where $N_G$ is the number of Goldstone modes and $d$ is the spatial dimension. This term is present even when the subsystem boundary is straight and contains no corners, and its origin lies in the interplay of Goldstone modes and restoration of symmetry in a finite volume. We also compute the "low-energy" part of the entanglement spectrum and show that it has the same characteristic "tower of states" form as the physical low-energy spectrum obtained when a system with spontaneously broken continuous symmetry is placed in a finite volume.
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