集体场论中纠缠熵的非线性
Finiteness of Entanglement Entropy in Collective Field Theory
摘要：
本文讨论了引力理论中的纠缠熵的有限性问题，其中的突现空间是全息对偶的目标空间。在二维非临界弦论和$c=1$矩阵模型的对偶性研究中，这个问题已经在本征值空间中用费米子多体理论研究过了。本征值空间的一个子区域的纠缠熵是有限的，即矩阵模型中的目标空间纠缠，尺度由局域费米动量提供。然而，费米动量是一种依赖于位置的弦耦合，这在集体场论公式中很清楚。这表明有限性是一种非微扰效应。我们提供了证据，这一预期的显式计算矩阵量子力学的集体场论与消失的潜力。在累积展开的纠缠熵的领导项计算使用确切的本征态和本征值的集体哈密顿量，产生一个有限的结果，在精确的协议与费米子的答案。处理理论微扰，我们表明，在微扰展开的每一项是UV发散。然而，该级数可以被图解，产生精确的有限结果。我们的结果表明，对于高维弦论，纠缠熵的有限性也是非微扰的，其尺度由牛顿常数提供。
英文摘要：
We explore the question of finiteness of the entanglement entropy in gravitational theories whose emergent space is the target space of a holographic dual. In the well studied duality of two-dimensional non-critical string theory and $c=1$ matrix model, this question has been studied earlier using fermionic many-body theory in the space of eigenvalues. The entanglement entropy of a subregion of the eigenvalue space, which is the target space entanglement in the matrix model, is finite, with the scale being provided by the local Fermi momentum. The Fermi momentum is, however, a position-dependent string coupling, as is clear in the collective field theory formulation. This suggests that the finiteness is a non-perturbative effect. We provide evidence for this expectation by an explicit calculation in the collective field theory of matrix quantum mechanics with vanishing potential. The leading term in the cumulant expansion of the entanglement entropy is calculated using exact eigenstates and eigenvalues of the collective Hamiltonian, yielding a finite result, in precise agreement with the fermion answer. Treating the theory perturbatively, we show that each term in the perturbation expansion is UV divergent. However the series can be resummed, yielding the exact finite result. Our results indicate that the finiteness of the entanglement entropy for higher dimensional string theories is non-perturbative as well, with the scale provided by the Newton constant.
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