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arXiv 2005.01334math-phhep-thmath.MPnlin.SI
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分离变量关联函数：XXX自旋链

Correlation functions by Separation of Variables: the XXX spin chain

G. Niccoli, H. Pei, V. Terras

备注：
Some references and comments added, 52 pages
发表信息：
SciPost Phys. 10, 006 (2021)
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摘要：

我们解释了如何计算关联函数在零温度下的量子版本的分离变量（SoV）的框架内的情况下，一个简单的模型：XXX海森堡链自旋1/2扭曲（准周期性）的边界条件。我们首先详细介绍了我们的方法在反周期边界条件的情况下的所有步骤。该模型可以通过引入非均匀性参数在SoV框架中求解。局部算子对本征态的作用自然地表示为对这些非均匀性参数的多重和。我们解释了如何将这些不均匀参数的总和转化为多个轮廓积分。评估这些多重积分的积分轮廓以外的极点的残留物，我们重写这个行动作为一个涉及巴克斯特多项式的根加上贡献的极点在无穷大的总和。我们表明，在无穷远的极点的贡献消失在热力学极限，我们恢复在这个极限为零温度的关联函数的多重积分表示，以前已经通过研究的周期性的情况下，由Bethe Anglesia或通过研究的无限体积模型的q-顶点算子的方法。我们最后表明，该方法可以很容易地推广到更一般的非对角扭转的情况下：然后修改有限体积中的关联函数的不同项的相应权重，但是我们在热力学极限中恢复了与周期或反周期情况相同的多重积分表示，从而证明了关联函数的热力学极限与边界扭转的特定形式无关。

英文摘要：

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.

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