Mei Zhongtao, Bolech C J
Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221-0011, USA.
Phys Rev E. 2017 Mar;95(3-1):032127. doi: 10.1103/PhysRevE.95.032127. Epub 2017 Mar 16.
Using the algebraic Bethe Ansatz, we derive a matrix product representation of the exact Bethe-Ansatz states of the six-vertex Heisenberg chain (either XXX or XXZ and spin-1/2) with open boundary conditions. In this representation, the components of the Bethe eigenstates are expressed as traces of products of matrices that act on a tensor product of auxiliary spaces. As compared to the matrix product states of the same Heisenberg chain but with periodic boundary conditions, the dimension of the exact auxiliary matrices is enlarged as if the conserved number of spin-flips considered would have been doubled. This result is generic for any non-nested integrable model, as is clear from our derivation, and we further show this by providing an additional example of the same matrix product state construction for a well-known model of a gas of interacting bosons. Counterintuitively, the matrices do not depend on the spatial coordinate despite the open boundaries, and thus they suggest generic ways of exploiting (emergent) translational invariance both for finite size and in the thermodynamic limit.
利用代数贝塞耳假设，我们推导出了具有开放边界条件的六顶点海森堡链（XXX或XXZ且自旋为1/2）的精确贝塞耳-安萨茨态的矩阵积表示。在这种表示中，贝塞耳本征态的分量被表示为作用于辅助空间张量积的矩阵乘积的迹。与具有周期边界条件的同一海森堡链的矩阵积态相比，精确辅助矩阵的维度增大了，就好像所考虑的自旋翻转守恒数加倍了一样。从我们的推导中可以清楚地看出，这个结果对于任何非嵌套可积模型都是通用的，并且我们通过为一个相互作用玻色子气体的著名模型提供相同矩阵积态构造的另一个例子进一步证明了这一点。与直觉相反，尽管有开放边界，矩阵并不依赖于空间坐标，因此它们为在有限尺寸和热力学极限下利用（涌现的）平移不变性提供了通用方法。
