We introduce $\mathrm{\Sigma }$-Attention, a transformer-based operator-learning framework for approximating the self-energy operator of strongly correlated electronic systems. By creating a batched dataset that combines results from three complementary approaches, i.e., many-body perturbation theory, strong-coupling expansion, and exact diagonalization, each effective in specific parameter regimes, $\mathrm{\Sigma }$-Attention is applied to learn an accurate approximation for the self-energy operator that is valid across a wide range of parameter regimes. This hybrid strategy leverages the strengths of existing methods while relying on the transformer's ability to generalize beyond individual limitations. More importantly, the scalability of the transformer architecture allows the learned self-energy to be extended to systems with larger sizes, leading to much improved computational scaling. Using the one-dimensional Hubbard model, we demonstrate that $\mathrm{\Sigma }$-Attention can accurately predict the Matsubara Green's function of large systems with a wide range of coupling strength. Our framework offers a promising and scalable pathway for studying strongly correlated systems with many possible generalizations.

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• Strongly correlated systems
• Hubbard model
• Machine learning
• Neural network simulations
• Quantum Monte Carlo