Dmrg Achieves Lowest Energy & Error with Optimal 2D Lattice Layouts

Scientists are continually seeking ways to improve the efficiency and accuracy of computational methods used to model complex physical systems. A new study, led by A. Scardicchio, investigates the optimal arrangement of lattice sites for the Density Matrix Renormalization Group (DMRG) , a powerful technique for simulating two-dimensional spin models. Scardicchio et al. demonstrate that strategically numbering these sites, specifically favouring a Hamiltonian path, can significantly reduce computational errors and accelerate convergence, yielding more reliable results. This research, focusing on antiferromagnetic and spin glass models on square and triangular lattices, offers a practical solution to enhance DMRG performance and unlock deeper insights into the behaviour of these materials.

Scientists are continually seeking ways to improve the efficiency and accuracy of computational methods used to model complex physical systems.

DMRG Layout Optimisation via Geometric Cost Function

The study reveals a novel approach to mapping higher-dimensional lattices onto one-dimensional chains, a critical step in adapting DMRG, naturally suited for one-dimensional systems, to more complex two-dimensional models. Researchers tackled the challenging combinatorial problem of finding the optimal layout, recognizing its relevance across diverse fields like circuit design, bioinformatics, and network optimization. This work builds upon previous heuristic solutions for DMRG layouts, extending the idea of fractal paths that explore local sites before leaping to neighbouring regions. Experiments show the team defined a layout distance between vertices and formulated a linear arrangement cost function, LA, representing the sum of distances between connected vertices in the chosen layout.

Minimizing this function, originally explored for hypercubes, became central to their investigation. They then generalized this cost function to LAq, introducing a parameter ‘q’ to explore different optimization strategies. The core innovation lies in identifying LA1/2 as the geometric cost function most strongly correlated with DMRG performance, building on prior recognition of its significance. This discovery allows for efficient evaluation of potential layouts, significantly accelerating the search for optimal configurations. The research establishes a strong link between geometric properties of the lattice layout and the efficiency of DMRG calculations, offering a powerful tool for enhancing simulations of complex quantum systems.

This breakthrough opens avenues for more accurate and faster simulations of two-dimensional quantum lattice models, impacting fields like condensed matter physics and materials science, and. The work defines a graph G(V, E) where V represents the set of vertices and E the edges, and a layout as a bijection φ mapping vertices to a sequence from 1 to N. The layout distance between vertices u and v is defined as λ(u, v) = |φ(u)−φ(v)|, representing the distance between their positions in the layout. The study systematically explored the relationship between geometric cost functions and DMRG performance, aiming to identify a proxy for evaluating layout optimality0.422 for a specific construction, but subsequent work utilising generalised Hilbert curves, dubbed “Gilbert paths”, yielded an improved constant of between 1.36 and 1.37. Supporting data, detailed in Figure 5, demonstrates the effectiveness of this optimisation search.

Results demonstrate that these Hilbert curves are excellent choices for DMRG performance, consistently providing good minima for the linear arrangement problem, even if not always the absolute best. The breakthrough delivers a method for efficiently arranging lattice sites to enhance DMRG calculations. For the antiferromagnetic model on the square lattice, researchers implemented a simulated annealing algorithm, sampling the Boltzmann distribution to optimise Hamiltonian paths starting and ending at corners of the lattice. The C++ code, aided by Gemini Pro, completes a single restart of the algorithm for a 64×64 lattice in just one second on a MacBook Pro M1 laptop.

The path optimisation algorithm employs a local topological update, iteratively perturbing the site ordering through a “back-bite” move involving split and mend operations. Data shows that for lattices of size L ≃10, the optimal path provides a gain in energy, maximum entropy, and truncation error. Measurements confirm that for a lattice of size L = 12, the optimal path requires approximately half the bond dimension to achieve the same accuracy as the Hilbert path. Given that computational cost scales with χ3, this translates to a substantial speed-up factor of approximately 10. The team also applied this geometric optimisation to disordered systems, including spin glass models and models on triangular lattices. Results for the spin glass model show a performance gain similar to that observed in the antiferromagnetic Heisenberg model. However, the optimisation on triangular lattices proved more complex, requiring further investigation to fully capture the dependence of the ground state on parameters like J2/J1.

Hamiltonian Paths Optimise DMRG Lattice Layouts for efficient

Specifically, the optimised paths consistently outperform the traditional snake path and even more sophisticated heuristics, sometimes achieving comparable precision with half the bond dimension. While the method works well on square lattices, the optimisation process on triangular lattices presents a more rugged landscape requiring further investigation. The authors acknowledge this limitation and suggest future work should explore different lattices and cost functions tailored to specific Hamiltonian characteristics, potentially moving beyond purely geometric considerations for even greater performance gains. They hope these observations will be useful for advancing numerical classical simulations in the field.