发表机构
Ludwig-Maximilians-Universität München; Technische Universität Wien（慕尼黑大学; 维也纳工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对张量环优化困难，提出双位点Ring-DMRG算法，通过低秩环境和交替优化实现高效截断，在临界系统中保持立方缩放且所需键维数更小，并推广了置信传播。
AI 中文摘要
张量环（TR）分解为周期系统提供了自然的表示，但由于封闭几何结构阻碍了全局规范形式的建立，导致环境代价高昂且病态，使得优化变得困难。现有的周期密度矩阵重正化群（DMRG）方法通过将长程环境压缩为低秩表示来缓解这一困难，将局部操作的复杂度降至$\mathcal{O}(p\chi^3)$，其中$p$为保留的环境秩。在此，我们将该方法扩展为高效的双位点Ring-DMRG算法，并通过适当的规范变换和广义Davidson求解器提高其数值鲁棒性。双位点公式中的一个核心挑战是截断步骤，该步骤必须考虑周围环境。当环境足够可分离时，适当的坐标变换可将截断简化为普通的奇异值分解（SVD）。当环境不可分离时，我们采用保留完整环境的交替优化方法。张量环相对于张量训练的计算优势取决于所需环境秩$p$随键维数和系统尺寸的缩放。对于临界系统，发散的相关长度使得长程观测量在接近热力学极限时仍对周期几何敏感。在此情形下，达到观测量固定精度所需的环境秩不随系统尺寸增长。因此，Ring-DMRG保留了与标准DMRG相同的键维数三次方缩放，但所需键维数显著更小。低秩环境构造还为置信传播（BP）提供了系统化的推广，在秩为1的极限下可恢复标准BP。
英文摘要
Tensor-ring (TR) decompositions provide a natural representation of periodic systems but are difficult to optimize because the closed geometry prevents a global canonical form and leads to costly, ill conditioned environments. Existing periodic DMRG methods alleviate this difficulty by compressing long environments to a low-rank representation, reducing local operations to $\mathcal{O}(pχ^3)$, where $p$ is the retained environment rank. Here, we extend this approach to an efficient two-site Ring-DMRG algorithm and improve its numerical robustness through appropriate gauge transformations and a generalized Davidson solver. A central challenge in the two-site formulation is the truncation step, which must account for the surrounding environment. When this environment is sufficiently separable, a suitable change of frame reduces the truncation to an ordinary SVD. When it is not separable, we instead use an alternating optimization that retains the full environment. The computational advantage of tensor rings over tensor trains depends on the scaling of the required environment rank $p$ with bond dimension and system size. For critical systems, the divergent correlation length makes long-range observables remain sensitive to the periodic geometry as the thermodynamic limit is approached. In this regime, the environment rank required to reach a fixed accuracy in the observable does not grow with the system size. Ring-DMRG therefore retains the same cubic scaling with bond dimension as standard DMRG, but at a substantially smaller bond dimension. The low-rank environment construction also provides a systematic generalization of belief propagation (BP), with standard BP recovered in the rank-one limit.
