Benchmarking non-perturbative many-body approaches in the exactly solvable Hatsugai-Kohmoto model

The accurate simulation of strongly correlated electron systems, especially non-Fermi liquid states, remains a central challenge in condensed matter physics, motivating the development of various non-perturbative many-body methods. Such methods are typically benchmarked against determinant quantum Monte Carlo (DQMC) in the Hubbard model, which 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 rigorous testing platform to evaluate three many-body approximations: GW, HGW, and SGW. By comparing Green's functions, spectral properties, and response functions with exact solutions, we show that the GW approximation, often considered insufficient for describing strong correlation, exhibits a previously unreported solution branch that accurately reproduces the gap structure 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. Our results establish the HK model as a highly effective benchmark system and help refine the understanding of GW methods in strongly correlated regimes.