Dirac理论中$U(1)$扭曲场的混合态形状因子
Mixed-state form factors of $U(1)$ twist fields in the Dirac theory
摘要：
利用有质量狄拉克理论的“刘维尔空间”（算符空间）定义了$U(1)$扭曲场的混合态形状因子。在渐近粒子基下考虑了具有对角密度矩阵的混合态。这包括热吉布斯态以及狄拉克理论的所有广义吉布斯系综。当混合态为热Gibbs态时，利用Riemann-Hilbert问题和低温展开，得到了$U(1)$扭曲场的有限温度形状因子.在此基础上，我们给出了一般对角混合态下$U(1)$扭曲场的形状因子的表达式。我们验证了这些形状因子满足一个非线性泛函微分方程组，这是来自混合状态形状因子的迹定义。最后，在密度矩阵特征值的弱解析条件下，写出了这些扭转场的两点关联函数的大距离形状因子展开式。利用狄拉克模型和伊辛模型之间的关系，这提供了对角混合态中伊辛模型中R $\acute{\text{e}}$ nyi熵（对于整数R $\acute{\text{e}}$ nyi参数）的大距离展开。
英文摘要：
Using the "Liouville space" (the space of operators) of the massive Dirac theory, we define mixed-state form factors of $U(1)$ twist fields. We consider mixed states with density matrices diagonal in the asymptotic particle basis. This includes the thermal Gibbs state as well as all generalized Gibbs ensembles of the Dirac theory. When the mixed state is specialized to a thermal Gibbs state, using a Riemann-Hilbert problem and low-temperature expansion, we obtain finite-temperature form factors of $U(1)$ twist fields. We then propose the expression for form factors of $U(1)$ twist fields in general diagonal mixed states. We verify that these form factors satisfy a system of nonlinear functional differential equations, which is derived from the trace definition of mixed-state form factors. At last, under weak analytic conditions on the eigenvalues of the density matrix, we write down the large distance form factor expansions of two-point correlation functions of these twist fields. Using the relation between the Dirac and Ising models, this provides the large-distance expansion of the R$\acute{\text{e}}$nyi entropy (for integer R$\acute{\text{e}}$nyi parameter) in the Ising model in diagonal mixed states.
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