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Journal of Physics Condensed Matter

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

Hui Li, Ziyu Li, Run-Yu Chen

Published Mar 10, 2026• 0 citations
Publisher

Abstract

Abstract 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 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.

DOI
10.1088/1361-648x/ae5010

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Abstract 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 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.
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: GWGWGW, HGWHGWHGW, and SGWSGWSGW. We compare the Green's functions, spectral functions, and response functions obtained from these approximations with the exact solutions. Our analysis shows that the GWGWGW 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 HGWHGWHGW provides an accurate description of charge response, while SGWSGWSGW 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 GWGWGW methods in strongly correlated regimes.
This pedagogic review aims to give a gentle introduction to an exactly solvable model, the Hatsugai-Kohmoto (HK) model, which has infinite-ranged interaction but conserves the center of mass. Although this model is invented in 1992, intensive studies on its properties ranging from unconventional superconductivity, topological ordered states to non-Fermi liquid behaviors are made since 2020. We focus on its emergent non-Fermi liquid behavior and provide discussion on its thermodynamics, single-particle and two-particle correlation functions. Perturbation around solvable limit has also been explored with the help of perturbation theory, renormalization group and exact diagonalization calculation. We hope the present review will be helpful for graduate students or researchers interested in HK-like models or more generic strongly correlated electron systems.
Motivated by the non-Fermi liquid (NFL) phase in solvable Hatsugai–Kohmoto (HK) model and ubiquitous quantum oscillation (QO) phenomena observed in strongly correlated electron systems, we have studied the QO in HK model in terms of a combination of analytical and numerical calculation. In the continuum limit, the analytical results indicate the existence of QO in NFL state and its properties can be described by Lifshitz–Kosevich-like formula. Furthermore, numerical calculations with Luttinger’s approximation on magnetic-field-dependent density of state, magnetization and particle’s density agree with the findings of analytical treatment. Although numerical simulation from exact diagonalization exhibits certain oscillation behavior, it is difficult to extract its oscillation period and amplitude. Therefore, more work (especially the large-scale numerical simulation) on this interesting issue is highly desirable and we expect the current HK model study to be helpful in understanding generic QO in correlated electron materials.
The study of strongly correlated quantum many-body systems is an outstanding challenge. Highly accurate results are needed for the understanding of practical and fundamental problems in condensed-matter physics, high energy physics, material science, quantum chemistry and so on. Our familiar mean-field or perturbative methods tend to be ineffective. Numerical simulations provide a promising approach for studying such systems. The fundamental difficulty of numerical simulation is that the dimension of the Hilbert space needed to describe interacting systems increases exponentially with the system size. Quantum Monte Carlo (QMC) methods are one of the best approaches to tackle the problem of enormous Hilbert space. They have been highly successful for boson systems and unfrustrated spin models. For systems with fermions, the exchange symmetry in general causes the infamous sign problem, making the statistical noise in the computed results grow exponentially with the system size. This hinders our understanding of interesting physics such as high-temperature superconductivity, metal-insulator phase transition. In this thesis, we present a variety of new developments in the auxiliary-field quantum Monte Carlo (AFQMC) methods, including the incorporation of symmetry in both the trial wave function and the projector, developing the constraint release method, using the force-bias to drastically improve the efficiency in Metropolis framework, identifying and solving the infinite variance problem, and sampling Hartree-Fock-Bogoliubov wave function. With these developments, some of the most challenging many-electron problems are now under control. We obtain an exact numerical solution of two-dimensional strongly interacting Fermi atomic gas, determine the ground state properties of the 2D Fermi gas with Rashba spin-orbit coupling, provide benchmark results for the ground state of the two-dimensional Hubbard model, and establish that the Hubbard model has a stripe order in the underdoped region.
A modified GWGWGW approximation to many-body systems is developed. The approximation has the same computational complexity as the traditional GWGWGW approach, but uses a different truncation scheme. This scheme neglects the high-order connected correlation functions. A covariant (preserving the Ward identities due to the charge conservation) scheme for the two-body correlators is employed, which holds the relation between the charge correlator and the charge susceptibility. The method is tested on the two-dimensional one-band Hubbard model. The results are compared with exact diagonalization, the GWGWGW approximation, the fluctuation-exchange (FLEX) theory, and determinantal Monte Carlo approach. The comparison for the (one-body) Green's function demonstrates that it is more precise in the strong-coupling regime (especially away from half filling) than the GWGWGW and FLEX approximations, which have a similar complexity. More importantly, this method indicates a Mott-Hubbard gap as the Hubbard UUU increases, whereas the GWGWGW and FLEX methods fail. In addition, the charge correlator obtained from the covariant scheme not only holds the consistency of the static charge susceptibility, but also makes a significant improvement over the random phase approximation calculations.
Describing correlated electron systems has been a major challenge in computational condensed-matter physics. Quantum Monte Carlo, a powerful computational tool for the study of correlated systems, solves electron correlation problems explicitly. It has been taken as a benchmark method for understanding the correlated systems. Instead of making approximations to Hamiltonian, QMC methods work with the wave functions, and the computational cost scales well with the system size. With the development of parallel computing, QMC calculations on large systems are becoming more and more feasible. We have investigated two correlated systems with highly accurate fixed node QMC techniques. The first system is a correlated hydrogen model system near the metal to insulator transition. We have successfully identified the transition point by calculating spin and charge properties and analyzing the low energy Hilbert space. The second one is a strongly correlated Fe/O system. Calculations on the Fe atoms, O atoms, and FeO molecules are conducted with multiple highly accurate many-body techniques. The source of errors has been disentangled by comparing the results of the many body techniques with the experimental results. For the Fe and O atoms, the calculated properties coincide well with previous experimental results. For the basis-based techniques, the performance is mainly limited by the basis set. The calculated equilibrium bond length, excitation energy and vibrational frequency of the FeO molecules are also in close agreement with the known values from previous experiments.