调节Lorentz不变场论中的纠缠结构
Structure of entanglement in regulated Lorentz invariant field theories
摘要：
调节洛伦兹不变的量子场论满足面积法的纠缠熵$S$的一个空间子区域在基态中的$d>1$空间维度，然而，全密度矩阵包含更多的$e^{S}$非零本征值。我们问如何以及一个子区域$R$在基态的这样一个理论的状态可以近似时，只保持$e^{S}$最大的本征值的约化密度矩阵的$R$。我们认为，通过使区域$R$足够大，我们始终可以确保保持大致的$e^{S}$状态会导致迹模的有界误差，即使对于无间隙基态的子区域也是如此。我们支持这些一般的参数与明确计算的错误在半空间几何中的自由标量场在任何维度。一路上，我们表明，在任何共形场论在小Renyi参数的基态球的Renyi熵控制的常规\textit{thermal entropy density}在低温下。我们还重新获得和推广了与DMRG有关的一维基态区间Schmidt系数衰减的一些旧结果。最后，我们讨论了调节器的作用，我们的论点精确的紫外物理的不敏感性，和绝热连续性在我们的结果中的作用。
英文摘要：
Regulated Lorentz invariant quantum field theories satisfy an area law for the entanglement entropy $S$ of a spatial subregion in the ground state in $d>1$ spatial dimensions; nevertheless, the full density matrix contains many more than $e^{S}$ non-zero eigenvalues. We ask how well the state of a subregion $R$ in the ground state of such a theory can be approximated when keeping only the $e^{S}$ largest eigenvalues of the reduced density matrix of $R$. We argue that by taking the region $R$ big enough, we can always ensure that keeping roughly $e^{S}$ states leads to bounded error in trace norm even for subregions in gapless ground states. We support these general arguments with an explicit computation of the error in a half-space geometry for a free scalar field in any dimension. Along the way we show that the Renyi entropy of a ball in the ground state of any conformal field theory at small Renyi parameter is controlled by the conventional \textit{thermal entropy density} at low temperatures. We also reobtain and generalize some old results relevant to DMRG on the decay of Schmidt coefficients of intervals in one dimensional ground states. Finally, we discuss the role of the regulator, the insensitivity of our arguments to the precise ultraviolet physics, and the role of adiabatic continuity in our results.
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