量子晶格系统经典模拟中角转移矩阵和角张量的探索
Exploring corner transfer matrices and corner tensors for the classical simulation of quantum lattice systems
摘要：
在本文中，我们探讨了实际使用的角转移矩阵及其高维推广，角张量，开发张量网络算法的经典模拟的量子晶格系统的无限大。这种探索主要在一维和二维空间（1d和2d）中进行。我们描述了一些基于角矩阵和张量的数值算法来近似这些系统的不同基态性质。所提出的方法还利用矩阵乘积算子和投影纠缠对算子，并自然地保持系统的空间对称性，如平移不变性。为了评估我们的算法的有效性，我们提供了初步的基准计算自旋1/2量子伊辛模型在横向领域在1D和2D。我们的方法是其他成熟的张量网络方法的合理替代方案，例如1d中的iDMRG和iTEBD，以及2d中的iPEPS和TERG。所提出的算法的计算复杂性也被认为是，在2D中，重要的差异被发现取决于所选择的模拟方案。我们还讨论了进一步的可能性，如三维量子晶格系统，周期性边界条件，和实时演化。这个讨论导致我们重新解释标准的iTEBD和iPEPS算法的角转移矩阵和角张量。我们的论文还提供了一个角度的角转移矩阵和它的高维推广的光的新张量网络方法的许多属性。
英文摘要：
In this paper we explore the practical use of the corner transfer matrix and its higher-dimensional generalization, the corner tensor, to develop tensor network algorithms for the classical simulation of quantum lattice systems of infinite size. This exploration is done mainly in one and two spatial dimensions (1d and 2d). We describe a number of numerical algorithms based on corner matri- ces and tensors to approximate different ground state properties of these systems. The proposed methods make also use of matrix product operators and projected entangled pair operators, and naturally preserve spatial symmetries of the system such as translation invariance. In order to assess the validity of our algorithms, we provide preliminary benchmarking calculations for the spin-1/2 quantum Ising model in a transverse field in both 1d and 2d. Our methods are a plausible alternative to other well-established tensor network approaches such as iDMRG and iTEBD in 1d, and iPEPS and TERG in 2d. The computational complexity of the proposed algorithms is also considered and, in 2d, important differences are found depending on the chosen simulation scheme. We also discuss further possibilities, such as 3d quantum lattice systems, periodic boundary conditions, and real time evolution. This discussion leads us to reinterpret the standard iTEBD and iPEPS algorithms in terms of corner transfer matrices and corner tensors. Our paper also offers a perspective on many properties of the corner transfer matrix and its higher-dimensional generalizations in the light of novel tensor network methods.
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