标题： 无质量费米子在圆柱体上的模哈密顿量和模流 显示英文标题

标题： Modular Hamiltonian and modular flow of massless fermions on a cylinder

摘要： 我们明确确定了在1+1维圆柱体上的钻石区域中无质量自由费米子的模流和模哈密顿量。 我们考虑了周期性和反对称边界条件，在反对称情况下考虑基态，在周期性情况下考虑最一般的准自由零能基态族，这些态依赖于四个参数，通常是混合态。 虽然对于反对称基态和一个周期性基态（最大混合的零温态）的模数据已知，但我们在周期性情况下对于一般基态的结果是完全新的。 我们发现，通常情况下模流和模哈密顿量都是非局部的，并且我们证明了当态变为纯态的参数极限下，模数据变为局部的。 此外，即使在局部情况下，模流通常也会混合两种手性。 这种行为之前尚未被观察到。 显示英文摘要

摘要： We determine explicitly the modular flow and the modular Hamiltonian for massless free fermions in diamonds on a cylinder in 1+1 dimensions. We consider both periodic and antiperiodic boundary conditions, the ground state in the antiperiodic case and the most general family of quasi-free zero-energy ground states in the periodic case, which depend on four parameters and are generally mixed. While for the antiperiodic ground state and one periodic ground state (the maximally mixed zero-temperature state) the modular data is known, our results for the generic ground state in the periodic case are completely new. We find that generically both the modular flow and the modular Hamiltonian are non-local, and we show that in the parametric limit where the state becomes pure the modular data becomes local. Moreover, even in the local case the modular flow generically mixes the two chiralities. This kind of behavior has not been observed previously.