Machine Learning Green's Functions of Strongly Correlated Hubbard Models

TL;DR

The paper tackles the challenge of predicting spectral properties in strongly correlated electrons described by the 1D Hubbard Hamiltonian. It introduces a kernel ridge regression approach that learns the imaginary-frequency self-energy from mean-field inputs (HF and GW) on a compact frequency grid, and then obtains real-frequency Green's functions via Dyson's equation and Padé analytic continuation. The authors demonstrate accurate self-energy and DOS predictions for both nearest-neighbor and long-range hopping ($t', t'', t'''$) across a broad range of $U/t$, with ARD indicating low error for small to intermediate $U$ and controlled errors at large $U$; the main remaining limitation stems from the analytic continuation step. Together, these results establish a transparent, data-efficient route to connect mean-field calculations with many-body spectral observables, offering a scalable bridge to more advanced MB methods and potential extensions to larger systems, different doping levels, and higher dimensions.

Abstract

We demonstrate that a machine learning framework based on kernel ridge regression can encode and predict the self-energy of one-dimensional Hubbard models using only mean-field features such as static and dynamic Hartree-Fock quantities and first-order GW calculations. This approach is applicable across a wide range of on-site Coulomb interaction strengths $U/t$, ranging from weakly interacting systems ($U/t \ll 1$) to strong correlations ($U/t > 8$). The predicted self-energy is transformed via Dyson's equation and analytic continuation to obtain the real-frequency Green's function, which allows access to the spectral function and density of states. This method can be used for nearest-neighbor interactions $t$ and long-range hopping terms $t'$, $t''$, and $t'''$.

Figures (13)

• Figure 1: One-dimensional Hubbard model with on-site Coulomb repulsion $U$. (a) Nearest-neighbor hopping $t$. (b) Zig-zag ladder with next-neighbor hopping $t'$. (c) Zig-zag ladder with diagonal hopping term $t"$. (d) Additional hopping $t"'$ between next-nearest-neighbor atoms on both legs of the zig-zag ladder.
• Figure 2: Self-energy and DOS for unseen test data of a Hubbard model with nearest-neighbor interactions $t=1$. (a)-(c) Local self-energy on leftmost site $\Sigma_{1,1}(\mathrm{i}\omega)$ versus frequency for $U = 1$, $U = 2$, and $U = 8$, respectively. Machine-learned prediction (ML, orange, small circles) versus exact solution (FCI, blue, large circles). Solid lines: real part. Dashed line: imaginary part. (d)-(f) DOS from FCI (green line) compared to ML prediction (orange filled area) around the Fermi energy $E_F = U/2$.
• Figure 3: Same as fig. \ref{['fig:fig1']}, but for training data. (a)-(c) Local self-energy $\Sigma_{1,1}(\mathrm{i}\omega)$ in normalized units versus frequency for $U = 1.0625$, $U = 2.046875$, and $U = 8.109375$, respectively, and (d)-(f) DOS around the Fermi energy for the same $U$-values.
• Figure 4: Overview of absolute relative difference (ARD) between exact and ML-predicted self-energy of the Hubbard model with nearest-neighbor hopping $t=1$. (a)-(d) show results for test data, and (e)-(h) for training data. (a) ARD(U)$\vert_{\text{on,off}}$ for all on- and off-diagonal matrix elements, showing that systems with large $U$ can be predicted less accurately. (b) ARD(U)$\vert_{\omega}$ for individual frequencies from smallest $\omega_1$ (purple) to largest $\omega_{30}$ (yellow) shows reduced accuracy at small frequencies. (c) ARD($\omega$)$\vert_{\text{on,off}}$ as a function of frequency over all $U$ shows nearly constant error across most frequencies. (d) ARD($\omega$)$\vert_{U}$ for different interaction strengths from $U=0.25$ (purple) to $U = 10$ (yellow) as a function of frequency index. Note that no error bars are plotted for the bottom plots. (e)–(h) Corresponding quantities for the training data.
• Figure 5: Test data for a Hubbard model with $t = 1$, $t' = 0.25$, and $t"=0.1$. Comparison of ML prediction to FCI for (a) and (b) DOS at $U = 1$ and $U = 6$, respectively, and (c) local self-energy $\Sigma_{1,1}(\mathrm{i}\omega)$ versus frequency for $U=6$. (d) and (e) DOS for a system with additional interaction $t"' = 0.1$ for $U = 1$ and $U = 6$, respectively, and (f) local self-energy for $U=6$.