Reduced Density Matrices Through Machine Learning

$n$-particle reduced density matrices ($n$-RDMs) play a central role in understanding correlated phases of matter, but their calculation is often computationally inefficient for strongly-correlated states at large system sizes. In this work, we use neural network (NN) architectures to accelerate and even predict $n$-RDMs for large systems. Our underlying intuition is that, for gapped states, $n$-RDMs are often smooth functions over the Brillouin zone (BZ) and are therefore interpolable, allowing NNs trained on small-size systems to predict large-size ones. Building on this, we devise two NNs: (i) a self-attention NN that maps random RDMs to physical ones, and (ii) a Sinusoidal Representation Network (SIREN) that directly maps momentum-space coordinates to RDM values. We test the NNs on RDMs in three 2D models: the pair-pair correlation functions of the Richardson model of superconductivity, the translationally-invariant Hartree-Fock (HF) 1-RDM in a four-band repulsive model, and the translation-breaking HF 1-RDM in the half-filled Hubbard model. We find that a SIREN trained on a $6\times 6$ momentum mesh and a SIREN trained on $4$ tilted meshes (each of which has $12$ momentum points) can predict the $18\times 18$ pair-pair correlation function with a relative accuracy of $94.29\%$ and $93.77\%$, respectively. NNs trained on $6\times 6$ and $8\times 8$ meshes provide high-quality initial guesses for $50\times 50$ translation-invariant HF and $30\times 30$ fully translation-breaking-allowed HF, reducing the required number of iterations by up to $91.63\%$ and $92.78\%$, respectively, compared to random initializations. Our results illustrate the potential of NN-based methods for interpolable $n$-RDMs, which might open a new avenue for future research on strongly correlated phases.