On scalar products in higher rank quantum separation of variables
On scalar products in higher rank quantum separation of variables
复制标题
关于高阶量子变量分离中的标量积
DOI：
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
L. Vignoli
中科院分区：
文献类型：
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作者：
J. Maillet;G. Niccoli;L. Vignoli
利用高阶量子可积点阵模型（[1]）的量子分离变量（SoV）框架，引入了超越已获得的完全转移矩阵谱描述的一些基础，并为局部算子的矩阵元素的计算开辟了道路。这首先等于获得所谓的独立状态（转移矩阵特征状态或它们的一些简单概括）的标量积的简单表达式。在高秩的情况下，左、右SoV基预期是伪正交的，也就是说，对于给定的SoV协向量，与所选的右SoV基的向量可以有一个以上的非消失重叠。为了简单起见，我们描述了在$N$位的$mathcal{Y}(gl_3)$晶格模型的基本表示中获得这些伪正交重叠的方法，其中秩为2。精确地描述了共矢量和矢量SoV基之间的非零耦合。在相应的SoV测度保持合理的简单性和可能的实际应用的同时，我们解决了构造满足标准正交性的左右SoV基的问题。在我们的方法中，SoV碱基是通过使用守恒电荷族来构建的。这使我们在SoV基的构造上有很大的自由度，并允许我们寻找一组守恒电荷的选择，这将导致正交的协矢量/矢量SoV基。我们首先在具有简单谱和零行列式的扭转矩阵的情况下定义这种选择。然后，我们将相关的守恒电荷族和正交SoV基推广到一般的简单谱和可逆扭转矩阵。在这种选择的守恒电荷和相关的正交SoV基下，独立态的标量积大大简化，其形式类似于$mathcal{Y}(gl_2)$ 1的情况。
Using the framework of the quantum separation of variables (SoV) for higher rank quantum integrable lattice models [1], we introduce some foundations to go beyond the obtained complete transfer matrix spectrum description, and open the way to the computation of matrix elements of local operators. This first amounts to obtain simple expressions for scalar products of the so-called separate states (transfer matrix eigenstates or some simple generalization of them). In the higher rank case, left and right SoV bases are expected to be pseudo-orthogonal, that is for a given SoV co-vector, there could be more than one non-vanishing overlap with the vectors of the chosen right SoV basis. For simplicity, we describe our method to get these pseudo-orthogonality overlaps in the fundamental representations of the $mathcal{Y}(gl_3)$ lattice model with $N$ sites, a case of rank 2. The non-zero couplings between the co-vector and vector SoV bases are exactly characterized. While the corresponding SoV-measure stays reasonably simple and of possible practical use, we address the problem of constructing left and right SoV bases which do satisfy standard orthogonality. In our approach, the SoV bases are constructed by using families of conserved charges. This gives us a large freedom in the SoV bases construction, and allows us to look for the choice of a family of conserved charges which leads to orthogonal co-vector/vector SoV bases. We first define such a choice in the case of twist matrices having simple spectrum and zero determinant. Then, we generalize the associated family of conserved charges and orthogonal SoV bases to generic simple spectrum and invertible twist matrices. Under this choice of conserved charges, and of the associated orthogonal SoV bases, the scalar products of separate states simplify considerably and take a form similar to the $mathcal{Y}(gl_2)$ rank one case.
DOI：
10.1088/1751-8113/44/1/015001
发表时间：
2010-09
期刊：
Journal of Physics A: Mathematical and Theoretical
影响因子：
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作者：
H. Frahm;J. H. Grelik;A. Seel;T. Wirth
通讯作者：
H. Frahm;J. H. Grelik;A. Seel;T. Wirth