解析模流：自由费米子的工具包
Resolving modular flow: a toolkit for free fermions
摘要：
模流是与时空区域相关的可观测量代数的对称性。它与纠缠密切相关，在最近信息论、QFT和引力之间的联系中发挥了关键作用。然而，很少有人知道它的行动以外的高度对称的情况下。这项工作的主要思想是引入一个新的公式，自由手征费米子在$1+1$维度的模块流，工作直接从\textit{resolvent}，在复杂的分析标准技术。我们提出了新的结果-不固定的共形对称-不相交的区域上的平面，圆柱体和环面。根据温度和边界条件，这些显示不同的行为，从纯粹的本地到非本地的运营商在类空分离的混合。我们发现模块化的两点函数，其分析结构是在精确的协议与KMS条件，管理模块化的演变。我们的现成公式可能为探索时空和纠缠之间的联系提供新的成分。
英文摘要：
Modular flow is a symmetry of the algebra of observables associated to spacetime regions. Being closely related to entanglement, it has played a key role in recent connections between information theory, QFT and gravity. However, little is known about its action beyond highly symmetric cases. The key idea of this work is to introduce a new formula for modular flows for free chiral fermions in $1+1$ dimensions, working directly from the \textit{resolvent}, a standard technique in complex analysis. We present novel results -- not fixed by conformal symmetry -- for disjoint regions on the plane, cylinder and torus. Depending on temperature and boundary conditions, these display different behaviour ranging from purely local to non-local in relation to the mixing of operators at spacelike separation. We find the modular two-point function, whose analytic structure is in precise agreement with the KMS condition that governs modular evolution. Our ready-to-use formulae may provide new ingredients to explore the connection between spacetime and entanglement.
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