AI 中文总结
针对复杂晶格DMRG枚举排序问题，提出以切割宽度和平均相互作用范围为目标的SAT/CP优化方法，显著优于经典启发式，并给出严格界限。
AI 中文摘要
密度矩阵重整化群（DMRG）是一种一维张量网络技术，但它并不局限于一维系统：它可以应用于周期性二维和三维团簇以及分子，前提是它们的格点首先沿一条线进行枚举；这一步可称为“晶格编译”。本文讨论了三种用于寻找这种最优枚举的代理损失函数：图带宽 $B$（最大相互作用范围）、切割宽度 $C$（跨越切割的最大键数）以及平均相互作用范围 $R$。针对阻挫磁性中感兴趣的一大类团簇构造哈密顿量MPO（矩阵乘积算符），我发现 $C$ 决定了SU(2)海森堡MPO键维数的峰值，而 $R$ 决定了其平均值。按字典序优化 $(C,R)$ 可获得最佳能量。优化 $B$ 间接降低了 $C$，但不如直接优化 $C$ 高效。此外，$B$ 的值本身在很大程度上无关紧要，因为好的DMRG能量可以对应较大的 $B$。为了执行优化，经典启发式算法（例如反向Cuthill–McKee）即使对于小团簇也被证明不可靠，并且我发现QUBO公式化仅能略微改进它们。相反，我提出了一种基于布尔可满足性（SAT）和约束规划（CP）求解器的分阶段优化方法，主要是Google OR-Tools的CP-SAT，它结合了CP传播和SAT子句学习。这种方法产生了显著更好的排序以及严格的界限。相应的Lattice in Line代码可在https URL获取，并在设计过程中广泛使用了Fable 5大型语言模型。
英文摘要
The density-matrix renormalization group (DMRG) is a one-dimensional tensor-network technique, but it is not limited to one-dimensional systems: it can be applied to periodic 2D and 3D clusters and molecules, provided their sites are first enumerated along a line; a step one may call "lattice compilation". This paper discusses three proxy loss functions for finding this optimal enumeration: the graph bandwidth $B$ (maximum interaction range), the cutwidth $C$ (maximum number of bonds crossing a cut), and the average interaction range $R$. Constructing the Hamiltonian MPO (matrix-product operator) for a large set of clusters that are of interest in frustrated magnetism, I find that $C$ determines the peak SU(2) Heisenberg MPO bond dimension and $R$ the average one. Targeting $(C,R)$ lexicographically yields the best energies. Targeting $B$ indirectly reduces $C$, but not as efficiently as targeting $C$ directly. Otherwise, the value of $B$ itself is largely irrelevant in the sense that good DMRG energies can have large $B$. To perform the optimization, classic heuristics (e.g. reverse Cuthill--McKee) prove unreliable even for small clusters, and I find that a QUBO formulation improves them only marginally. Instead I propose a staged optimization built on Boolean satisfiability (SAT) and constraint-programming (CP) solvers, chiefly CP-SAT of Google's OR-Tools, a hybrid of CP propagation and SAT clause learning. This approach yields significantly better orderings together with rigorous bounds. The corresponding Lattice in Line code is available at https://github.com/spinflip/lattice_in_line and was designed with extensive use of the Fable 5 large language model.
