标题： 有理自旋链的MPS重叠公式的推导 显示英文标题

标题： Derivations for the MPS overlap formulas of rational spin chains

摘要： 我们推导了一个普遍公式，用于计算在具有$\mathfrak{gl}_{N}$对称自旋链中的可积矩阵乘积态（MPS）与贝特本征态之间的重叠。 该公式将归一化重叠表示为一个 与MPS无关的Gaudin行列式比值和一个由MPS相关的$K$-矩阵定义的交换算子的特征值构造出的标量因子的乘积。 我们的证明完全独立于表示，并且仅仅依赖于代数贝特Ansatz技术和$KT$-关系。 我们也提出了重叠公式的推广，适用于$\mathfrak{so}_{N}$和$\mathfrak{sp}_{N}$自旋链，这些推广基于代数嵌入和低秩同构的支持。 这些结果显著扩展了可以使用精确重叠公式的可积初始状态类，这对量子猝发和缺陷CFT有影响。 显示英文摘要

摘要： We derive a universal formula for the overlaps between integrable matrix product states (MPS) and Bethe eigenstates in $\mathfrak{gl}_{N}$ symmetric spin chains. This formula expresses the normalized overlap as a product of a MPS-independent Gaudin-determinant ratio and a MPS-dependent scalar factor constructed from eigenvalues of commuting operators, defined via the $K$-matrix associated with the MPS. Our proof is fully representation-independent and relies solely on algebraic Bethe Ansatz techniques and the $KT$-relation. We also propose a generalization of the overlap formula to $\mathfrak{so}_{N}$ and $\mathfrak{sp}_{N}$ spin chains, supported by algebra embeddings and low-rank isomorphisms. These results significantly broaden the class of integrable initial states for which exact overlap formulas are available, with implications for quantum quenches and defect CFTs.