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Machine-Learning Accelerated Calculations of Reduced Density Matrices

Awwab A. Azam, Lexu Zhao, Jiabin Yu

TL;DR

This work tackles the computational bottleneck of obtaining $n$-particle reduced density matrices ($n$-RDMs) in strongly correlated systems. It develops two NN architectures, a self-attention network and a Sinusoidal Representation Network (SIREN), to interpolate and predict large-size $n$-RDMs from small-size data by exploiting smooth momentum-space structure. The methods yield high predictive accuracy on benchmark models (e.g., $r_n$ up to $0.9189$ for Richardson model pair-pair correlators) and dramatically accelerate Hartree-Fock convergence (often by >$90\%$ iterations) in both translationally invariant and symmetry-broken settings. These results highlight a path toward efficiently exploring correlated phases and generalizing to broader classes of $2$-RDMs in condensed matter systems.

Abstract

$n$-particle reduced density matrices ($n$-RDMs) play a central role in understanding correlated phases of matter. Yet the calculation of $n$-RDMs is often computationally inefficient for strongly-correlated states, particularly when the system sizes are large. In this work, we propose to use neural network (NN) architectures to accelerate the calculation of, and even predict, the $n$-RDMs for large-size systems. The underlying intuition is that $n$-RDMs are often smooth functions over the Brillouin zone (BZ) (certainly true for gapped states) and are thus interpolable, allowing NNs trained on small-size $n$-RDMs to predict large-size ones. Building on this intuition, 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 in three 2D models: the pair-pair correlation functions of the Richardson model of superconductivity, the translationally-invariant 1-RDM in a four-band model with short-range repulsion, and the translation-breaking 1-RDM in the half-filled Hubbard model. We find that a SIREN trained on a $6\times 6$ momentum mesh can predict the $18\times 18$ pair-pair correlation function with a relative accuracy of $0.839$. The NNs trained on $6\times 6 \sim 8\times 8$ meshes can provide high-quality initial guesses for $50\times 50$ translation-invariant Hartree-Fock (HF) and $30\times 30$ fully translation-breaking-allowed HF, reducing the number of iterations required for convergence by up to $91.63\%$ and $92.78\%$, respectively, compared to random initializations. Our results illustrate the potential of using NN-based methods for interpolable $n$-RDMs, which might open a new avenue for future research on strongly correlated phases.

Machine-Learning Accelerated Calculations of Reduced Density Matrices

TL;DR

This work tackles the computational bottleneck of obtaining -particle reduced density matrices (-RDMs) in strongly correlated systems. It develops two NN architectures, a self-attention network and a Sinusoidal Representation Network (SIREN), to interpolate and predict large-size -RDMs from small-size data by exploiting smooth momentum-space structure. The methods yield high predictive accuracy on benchmark models (e.g., up to for Richardson model pair-pair correlators) and dramatically accelerate Hartree-Fock convergence (often by > iterations) in both translationally invariant and symmetry-broken settings. These results highlight a path toward efficiently exploring correlated phases and generalizing to broader classes of -RDMs in condensed matter systems.

Abstract

-particle reduced density matrices (-RDMs) play a central role in understanding correlated phases of matter. Yet the calculation of -RDMs is often computationally inefficient for strongly-correlated states, particularly when the system sizes are large. In this work, we propose to use neural network (NN) architectures to accelerate the calculation of, and even predict, the -RDMs for large-size systems. The underlying intuition is that -RDMs are often smooth functions over the Brillouin zone (BZ) (certainly true for gapped states) and are thus interpolable, allowing NNs trained on small-size -RDMs to predict large-size ones. Building on this intuition, 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 in three 2D models: the pair-pair correlation functions of the Richardson model of superconductivity, the translationally-invariant 1-RDM in a four-band model with short-range repulsion, and the translation-breaking 1-RDM in the half-filled Hubbard model. We find that a SIREN trained on a momentum mesh can predict the pair-pair correlation function with a relative accuracy of . The NNs trained on meshes can provide high-quality initial guesses for translation-invariant Hartree-Fock (HF) and fully translation-breaking-allowed HF, reducing the number of iterations required for convergence by up to and , respectively, compared to random initializations. Our results illustrate the potential of using NN-based methods for interpolable -RDMs, which might open a new avenue for future research on strongly correlated phases.

Paper Structure

This paper contains 45 sections, 225 equations, 10 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Interpolability of $n$-RDM
  3. NN Architectures
  4. Self-Attention NN
  5. SIREN
  6. Beyond 1-RDM
  7. Pair-Pair Correlation Functions
  8. Translationally-Invariant HF
  9. HF Allowing Translational Breaking
  10. Conclusion and Discussion
  11. Acknowledgments
  12. NN Architecture
  13. Self-Attention NN Architecture for the 1-RDM
  14. physical set-up
  15. Preprocessing (transform 1-RDM to NN input)
  16. ...and 30 more sections

Figures (10)

  • Figure 1: Interpolability of $n$-RDMs in the BZ. Momentum meshes for large systems can be generated by inserting new $\boldsymbol{k}$-points between those of the small momentum meshes. Therefore, $n$-RDMs in momentum space are often interpolable.
  • Figure 2: Schematics of our self-attention NN. (a) An initial 1–RDM on an $L\times L$ BZ is preprocessed to $\tilde{X}_{\mathrm{init}}$, transformed by the network to $\tilde{X}_{\mathrm{out}}$, and decoded to the predicted final 1–RDM. (b) Each layer applies an attention block and a FFN with Dropout and LayerNorm. (c) The attention block includes $N_{\text{head}}$ parallel heads. (d) Each head uses $Q,K,V$ with a learned periodic relative-position bias and a scalable softmax temperature (e.g., $\tau\propto \log L^{2}$).
  • Figure 3: Schematic of our SIREN. (a) Normalize BZ coordinates to $\boldsymbol{v}(\boldsymbol{k}) \in [-1,1]^2$, then pass $\boldsymbol{v}$ through layers of linear and sine functions to predict one component of the final $n$-RDM (e.g., a specific $\alpha,\alpha'$ component for 1-RDM); aggregating components reconstructs the predicted final $n$-RDM. (b) We train the NN using a total loss that contains: (i) MSE on the small $L_1 \times L_1$ mesh, and (ii) a loss computed by evaluating SIREN on a larger mesh $L_2 \times L_2$, symmetrizing, downsampling back to $L_1 \times L_1$ and comparing to the true final RDMs on the $L_1\times L_1$ mesh. At each step we evaluate a statistical prior ($\mathrm{TV}$ or $\mathrm{FH}$) on randomly chosen coordinates $\boldsymbol{u}_i\in[-1,1]^2$ to regularize the SIREN. Therefore, the training only invariants true final RDMs on the small $L_1\times L_1$ mesh.
  • Figure 4: Comparison between (a) the predicted final $L=18$$C_{kk'}$ given by a SIREN trained on $L=6$ and (b) the true final $C_{kk'}$ for $L=18$. Here the indices $k,k'\in\{0,\dots,L^{2}-1\}$ enumerate the $L\times L$ BZ points by flattening $(\ell_{1},\ell_{2})$ with $\ell_{1,2}\in\{0,\dots,L-1\}$ via $k=\ell_{1}+L\,\ell_{2}$; the corresponding momentum vector is $\boldsymbol{k}$ given in \ref{['eq:BZ_kpoints_main']}, and similarly for $k'$.
  • Figure 5: NN for translation-invariant HF. Here, " % Reduction" is defined in \ref{['eq:percent_reduction_maintext']}, (i.e. the percent difference between the number of HF iterations using a predicted final 1-RDM from the self-attention NN as the initial 1-RDM as compared to using a random initial 1-RDM), and is plotted as a function of the system size $L$ for the four-band model described in \ref{['eq:toyModel_phys_eq1', 'eq:real-VR']}. See \ref{['tab:toyModel_iters_summary']} in \ref{['sec:toymodelpart']} for the exact values. The HF calculation is forced to preserve the lattice translation symmetry.
  • ...and 5 more figures