arXiv 1512.022322015-12hep-thcond-mat.stat-mechcond-mat.str-elquant-ph

黑洞奇异性、广义（全息）$c$ -定理和纠缠负性

Black Hole Singularity, Generalized (Holographic) $c$-Theorem and Entanglement Negativity

Shamik Banerjee, Partha Paul

备注：: 20 pages, 3 figures, latex, references added

摘要：

在本文中，我们重新审视的问题，在什么意义上空的$AdS_{5}$黑膜几何可以被认为是RG-流。我们这样做，首先构建一个全息$c$ -函数使用因果视界的黑膜几何。$c$函数的UV值为$a_{UV}$，然后在曲率奇异处单调减小到零。直观地说，如果我们认识到对偶CFT处于热态，并且热态是有效质量的，具有由温度设置的间隙，则可以理解$c$函数的行为。在场论中，对数纠缠度是衡量混合态纠缠度的一种度量。例如，在有限温度下无限长线上的二维CFTs中，一个区间的重正化纠缠负性具有UV（Low-T）值$c_{UV}$和IR（High-T）值零。所以这是我们的$c$函数的潜在候选者。在四维空间中，我们期望同样的事情在物理基础上成立。既然因果视界在黑膜视界的后面，那么全息$c$函数对内部的物理是敏感的。相应地，场论$c$ -函数也应该包含关于内部的信息。因此，我们的结果表明，负性（或任何候选的$c$函数）的高温（IR）膨胀可能是探测奇点附近物理学的一种方法。在有限温度下的负性取决于理论的全部算符内容，因此也许这只能在特定情况下才能做到。这种$c$ -函数在整体中的存在是时空是由纠缠建立起来的范例的一个极端例子。特别地，$c$函数在曲率奇异性处达到零的事实与两个事实相关：IR场论中的量子纠缠的损失和在这种情况下是曲率奇异性的形成的体几何的结束。

英文摘要：

In this paper we revisit the question that in what sense empty $AdS_{5}$ black brane geometry can be thought of as RG-flow. We do this by first constructing a holographic $c$-function using causal horizon in the black brane geometry. The UV value of the $c$-function is $a_{UV}$ and then it decreases monotonically to zero at the curvature singularity. Intuitively, the behavior of the $c$-function can be understood if we recognize that the dual CFT is in a thermal state and thermal states are effectively massive with a gap set by the temperature. In field theory, logarithmic entanglement negativity is an entanglement measure for mixed states. For example, in two dimensional CFTs on infinite line at finite temperature, the renormalized entanglement negativity of an interval has UV (Low- T) value $c_{UV}$ and IR (High-T) value zero. So this is a potential candidate for our $c$-function. In four dimensions we expect the same thing to hold on physical grounds. Now since the causal horizon goes behind the black brane horizon the holographic $c$-function is sensitive to the physics of the interior. Correspondingly the field theory $c$-function should also contain information about the interior. So our results suggest that high temperature (IR) expansion of the negativity (or any candidate $c$-function) may be a way to probe part of the physics near the singularity. Negativity at finite temperature depends on the full operator content of the theory and so perhaps this can be be done in specific cases only. The existence of this $c$-function in the bulk is an extreme example of the paradigm that space-time is built out of entanglement. In particular the fact that the $c$-function reaches zero at the curvature singularity correlates the two facts : loss of quantum entanglement in the IR field theory and the end of geometry in the bulk which in this case is the formation of curvature singularity.

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英文摘要：

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