We explain how to compute correlation functions at zero temperature within the framework of the quantum version of the Separation of Variables (SoV) in the case of a simple model: the XXX Heisenberg chain of spin 1/2 with twisted (quasi-periodic) boundary conditions. We first detail all steps of our method in the case of anti-periodic boundary conditions. The model can be solved in the SoV framework by introducing inhomogeneity parameters. The action of local operators on the eigenstates are then naturally expressed in terms of multiple sums over these inhomogeneity parameters. We explain how to transform these sums over inhomogeneity parameters into multiple contour integrals. Evaluating these multiple integrals by the residues of the poles outside the integration contours, we rewrite this action as a sum involving the roots of the Baxter polynomial plus a contribution of the poles at infinity. We show that the contribution of the poles at infinity vanishes in the thermodynamic limit, and that we recover in this limit for the zero-temperature correlation functions the multiple integral representation that had been previously obtained through the study of the periodic case by Bethe Ansatz or through the study of the infinite volume model by the q-vertex operator approach. We finally show that the method can easily be generalized to the case of a more general non-diagonal twist: the corresponding weights of the different terms for the correlation functions in finite volume are then modified, but we recover in the thermodynamic limit the same multiple integral representation than in the periodic or anti-periodic case, hence proving the independence of the thermodynamic limit of the correlation functions with respect to the particular form of the boundary twist.

中文翻译：

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通过变量分离的相关函数：XXX自旋链

在简单模型的情况下，我们将解释如何在变量分离（SoV）的量子版本框架内在零温度下计算相关函数：自旋1/2的XXX Heisenberg链（具有扭曲的（准周期）边界）条件。我们首先详细介绍反周期边界条件下方法的所有步骤。可以通过引入不均匀性参数在SoV框架中求解该模型。然后自然地用这些不均匀性参数上的多个和表示本地操作员对本征态的作用。我们解释了如何将不均匀性参数上的这些和转换为多个轮廓积分。通过积分轮廓之外的极点残差评估这些多个积分，我们将这个动作改写为包含Baxter多项式根的和加上无穷极点的和。我们表明极点处的极点在热力学极限中消失，并且我们在该极限中恢复了零温度相关函数的多重积分表示形式，该表示形式先前是通过Bethe Ansatz或通过q顶点算子方法研究无限体积模型。最后，我们证明了该方法可以很容易地推广到更一般的非对角扭曲的情况：在有限体积中修改相关函数的不同项的相应权重，但是我们在热力学极限中恢复相同的倍数积分表示，而不是周期性或非周期性的情况，