Quantum Entanglement of Local Operators
Quantum Entanglement of Local Operators
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DOI：
10.14989/doctor.k18792
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发表时间：
2015-03
影响因子：
8.6
通讯作者：
M. Nozaki
中科院分区：
文献类型：
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作者：
M. Nozaki
在本论文中，我们研究了共形场论（CFT）中基态上的局部算子产生的局部激发态（Rényi）纠缠熵的动力学特性。我们通过从局部激发态的熵中减去基态的熵来定义 (Rényi) 纠缠熵的过量 ΔS (n) A。这里我们选择总空间的一半作为子系统A。我们发现它们的多余部分是由副本技巧[1,2,3]中局部算子的相关函数给出的。因此，我们计算 ΔS (n) A 作为它们在欧几里德空间中的相关函数。计算完毕后，我们实时执行特定的灯芯旋转，并研究各种 CFT 中 ΔS (n) A 的时间演化。我们发现它们的时间演化可以用准粒子来解释。准粒子对（称为纠缠对）是由作用局部算子生成的，并以光速传播。每对准粒子之间的量子纠缠可以贡献（Rényi）纠缠熵。当每对准粒子都包含在区域 A（或 A 的补集）中时，ΔS (n) A 保持为零。当每对准粒子中只有一个准粒子包含在区域 A 中，而另一个准粒子包含在区域 B 中时，ΔS (n) A 增加。它们的时间演化遵循因果关系，并且最终接近自由无质量场论和二维有理 CFT 中的一些常数 [1,2,4]。 ΔS (n) A 的最终值取决于本地运营商的详细信息。如果我们在自由无质量标量场理论中将 (∂φ) k 形式的算子作用于基态，则 ΔS (n) A 的最终值由有限维希尔伯特空间的 (Rényi) 纠缠熵给出，该空间的约简密度矩阵由二项式分布给出 [2]。这里索引 i 和 k 分别是导数的幂和 ∂φ 的幂。 ΔS (n) A 的最终值不依赖于时空维度。它们的最终值可以在纠缠对解释下获得。它们可以通过使用复制技巧来获得。我们发现，在纠缠对解释下获得的结果与使用副本技巧获得的结果一致。我们将 ΔS (n) A 的最终值称为局部算子的 (Rényi) 纠缠熵。我们找到了局部算子的 (Rényi) 纠缠熵的求和规则。如果我们在基态上作用各种局部算子，则该局部激发态的 ΔS (n) A 的最终值由每个局部算子的 (Rényi) 纠缠熵之和给出。我们还研究了通过在基态上作用复合算子 φ∂φ 产生的局部激发态的第二（Rényi）纠缠熵的过量，该复合算子由两个物种算子 φ 和 ∂φ 构成。对于该算子，(Rényi) 纠缠熵的最终值取决于时空维度。在大 N 自由场理论中，当 n ≥ 2 时，ΔS (n) A 的过量由 O(1) 个量给出 [3]。另一方面，冯·诺依曼极限 ΔSA 为 O (log N)。如果我们将 1/n 视为有效温度，则这种行为类似于限制/解除限制相变。我们发现，在取大 N（大中心电荷 c）极限后，我们无法取冯·诺依曼极限（n → 1 极限），因为 ΔS A 的最终值出现分歧。因此，当我们研究大N理论中的ΔSA时，我们必须在取大N或c极限之前先取冯诺依曼极限（n → 1极限）。在任何维度的大 N（大 c）相互作用的 CFT 中，ΔS (n) A 的后期值随着 t 呈对数增加。 n ≥ 2 时的 log t 系数与局部算子的共形维数成正比。另一方面，AdS3/CFT2 中的全息结果表明 SA (n = 1) 与中心电荷 c 成正比。从这个意义上说，当 n = 1 时，它的系数得到增强。我们还发现，我们需要超出大 N 或 c 限制的信息，以便了解它们是否接近某些常数或在后期发散。如果它们接近某些常数，可能会出现非微扰 N 效应（非微扰 c 效应）。
In this thesis we study dynamical properties of (Rényi) entanglement entropies for locally excited states generated by acting local operators on ground states in conformal field theories (CFTs). We define the excesses of (Rényi) entanglement entropies ∆S (n) A by subtracting those for the ground states from those for locally excited states. Here we choose a half of the total space as the subsystem A. We find that their excesses are given by the correlation functions of local operators in the replica trick [1, 2, 3]. Therefore we compute ∆S (n) A as their correlation functions in Euclidean space. After computing them, we perform a specific Wick rotation to real time and investigate the time evolutions of ∆S (n) A in various CFTs. We find that their time evolutions can be interpreted in terms of quasi-particles. The pairs of quasi-particles, which are called entangled pairs, are generated by acting local operators and propagate at the speed of light. Quantum entanglement between quasi-particles of each pair can contribute to (Rényi) entanglement entropies. When both quasi-particles of each pair are included in the region A (or the complement of A), ∆S (n) A keep to vanish. When only one quasi-particle of each pair is included in the region A and another one is included in the region B, ∆S (n) A increase. Their time evolutions obey causality and they eventually approach some constants in free massless field theories and 2 dimensional rational CFTs [1, 2, 4]. The final values of ∆S (n) A depend on the details of local operators. If we act operators of the form (∂φ) k on the ground state in the free massless scalar field theories, the final values of ∆S (n) A are given by (Rényi) entanglement entropies for a finite dimensional Hilbert space whose reduced density matrices are given by the binomial distribution [2]. Here indexes i and k are power of derivative and that of ∂φ respectively. The final values of ∆S (n) A do not depend on the spacetime dimensions. Their final values can be acquired under the entangled pair interpretation. They can be obtained by using the replica trick. We find that the results obtained under the entangled pair interpretation are consistent with those obtained by using the replica trick. We call the final values of ∆S (n) A as the (Rényi) entanglement entropies of local operators. We find a sum rule for (Rényi) entanglement entropies of local operators. If we act various local operators on the ground state, the final value of ∆S (n) A for this locally excited state is given by the sum of (Rényi) entanglement entropies of each local operator. We also study the excesses of second (Rényi) entanglement entropy for the locally excited state generated by acting a composite operator φ∂φ, which is constructed of two species operators φ and ∂φ, on the ground state. For this operator, the final values of (Rényi) entanglement entropy depend on the spacetime dimensions. In large N free field theories, the excesses of ∆S (n) A for n ≥ 2 are given by O(1) quantities [3]. On the other hand, the von Neumann limit ∆SA is O (log N). This behavior resembles confinement / deconfinement phase transition if we regard 1/n as an effective temperature. We find that after taking the large N (the large central charge c) limit, we are not able to take the von Neumann limit (n → 1 limit) since the final values of ∆S A diverge. Therefore we have to take the von Neumann limit (the n → 1 limit) before taking the large N or c limit when we study ∆SA in large N theories. In large N (large c) interacting CFTs in any dimensions, the late time values of ∆S (n) A keep to increases logarithmically with t. The coefficient of log t for n ≥ 2 is proportional to conformal dimensions of local operators. On the other hand a holographic result in AdS3/CFT2 shows that SA (n = 1) is proportional to the central charge c. In this sense its coefficient is enhanced for n = 1. We also find that we need information beyond the large N or c limit in order to know whether they approach some constants or diverge at late time. If they approach some constants, there can be non-perturbative N effects (non-perturbative c effects).