Benchmarking Non-perturbative Many-Body Approaches in the Exactly Solvable Hatsugai-Kohmoto Model

Abstract

The accurate simulation of strongly correlated electron systems remains a central challenge in condensed matter physics, motivating the development of various non-perturbative many-body methods. Such methods are typically benchmarked against the numerical exact determinant quantum Monte Carlo (DQMC) in the Hubbard model; however, DQMC is limited by the fermionic sign problem and the uncertainties of numerical analytic continuation. To address these issues, we use the exactly solvable Hatsugai-Kohmoto (HK) model as a benchmarking platform to evaluate three many-body approximations: $GW$, $HGW$, and $SGW$. We compare the Green's functions, spectral functions, and response functions obtained from these approximations with the exact solutions. Our analysis shows that the $GW$ approximation, often considered insufficient for describing strong correlation, exhibits a previously unreported solution branch that accurately reproduces Mott physics in the HK model. In addition, using a covariant formalism, we find that $HGW$ provides an accurate description of charge response, while $SGW$ performs well for spin correlations. Overall, our work demonstrates that the HK model can effectively benchmark many-body approximations and helps refine the understanding of $GW$ methods in strongly correlated regimes.

Figures (7)

• Figure 1: Comparison of imaginary-time Green's functions at half-filling for the antinodal point $k_{\mathrm{AN}}=(\pi,0)$. Results are shown for the exact solution (black lines), $GW$ (blue), $HGW$ (red), and $SGW$ (green) methods at different parameters: (a) $\beta=2, U=2$; (b) $\beta=8, U=2$; (c) $\beta=2, U=4$; (d) $\beta=8, U=4$. Different line styles for a given color correspond to different branches calculated by the same method.
• Figure 2: Imaginary-time Green's functions at the antinodal point $k_{\mathrm{AN}}=(\pi,0)$ for finite doping, comparing the exact solution (black lines) with the $GW$ (blue), $HGW$ (red), and $SGW$ (green) methods under different parameters: (a) $\beta=2, U=2$; (b) $\beta=8, U=2$; (c) $\beta=2, U=4$; (d) $\beta=8, U=4$. Different line styles within the same color represent distinct branches from the same method.
• Figure 3: Spectral functions at the antinodal point $k_{\mathrm{AN}}=(\pi,0)$ under half-filling are shown for the exact solution, $GW$, $HGW$, and $SGW$ methods across different parameter sets: (a) $\beta=2, U=2$; (b) $\beta=8, U=2$; (c) $\beta=2, U=4$; (d) $\beta=8, U=4$. The black, blue, red, and green lines represent results from the exact formula, $GW$, $HGW$, and $SGW$ approximations, respectively. Notably, the $GW$ method yields a two-peak Mott insulating structure.
• Figure 4: Spectral functions at the antinodal point $k_{\mathrm{AN}}=(\pi,0)$ under finite doping, comparing the exact solution (black lines) with the $GW$ (blue), $HGW$ (red), and $SGW$ (green) methods. Results are shown for different parameter sets: (a) $\beta=2, U=2$; (b) $\beta=8, U=2$; (c) $\beta=2, U=4$; (d) $\beta=8, U=4$. Both the $GW$ and $SGW$ methods exhibit an additional spectral branch, which manifests as an extra peak associated with the Mott structure.
• Figure 5: Mott gap as a function of interaction strength $U$ at the antinodal point $k=(\pi,0)$ under half-filling. Panels (a) and (b) show results for $\beta=4$ and $\beta=8$, respectively. The exact solution is represented by a black line, while results from the $GW$ (blue), $SGW$ (green), and $HGW$ (red) methods are shown as discrete points. The $GW$ and $HGW$ data points are fitted with linear functions (dashed lines). For $\beta=4$ (a), the fitted slope is 1.11 ($R^2=0.998$) for $GW$ and 1.33 ($R^2=0.998$) for $HGW$. For $\beta=8$ (b), the fitted slope is 1.07 ($R^2=0.999$) for $GW$ and 1.06 ($R^2=0.987$) for $HGW$.