Lattice in Line: Optimized DMRG ordering for complex lattice geometries

Abstract

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.