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.
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BibTeX
@misc{rausch2026lattice,
title = {{Lattice in Line: Optimized DMRG ordering for complex lattice geometries}},
author = {Roman Rausch},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25384},
eprint = {2609.25384},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25384},
}
RIS
TY - UNPB
TI - Lattice in Line: Optimized DMRG ordering for complex lattice geometries
AU - Roman Rausch
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.25384
UR - https://arxiv.org/abs/2609.25384
AB - 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.
ER -