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.
Many-body theory is largely based on self-consistent equations that are constructed in terms of the physical quantity of interest itself, for example the density. Therefore, the calculation of important properties such as total energies or photoemission spectra requires the solution of nonlinear equations that have unphysical and physical solutions. In this work we show in which circumstances one runs into an unphysical solution, and we indicate how one can overcome this problem. Moreover, we solve the puzzle of when and why the interacting Green's function does not unambiguously determine the underlying system, given in terms of its potential, or non-interacting Green's function. Our results are general since they originate from the fundamental structure of the equations. The absorption spectrum of lithium fluoride is shown as one illustration, and observations in the literature for some widely used models are explained by our approach. Our findings apply to both the weak and strong-correlation regimes. For the strong-correlation regime we show that one cannot use the expressions that are obtained from standard perturbation theory, and we suggest a different approach that is exact in the limit of strong interaction.
Understanding the physics of strongly correlated materials is one of the grand-challenges in condensed-matter physics.Simple approximations such as the local density approximation fail, due to the importance of the Coulomb repulsion between localized electrons.Instead we have to resort to non-perturbative many-body techniques.Such calculations are, however, only feasible for quite small model systems.This means that the full Hamiltonian of a real material has to be approximated by a model Hamiltonian comprising only the most important electronic degrees of freedom, while the effect of all other electrons can merely be included in an average way.Realistic calculations of strongly correlated materials need to include as many of the electronic degrees of freedom as possible.We use the Lanczos method for calculating ground state and dynamical properties of such model Hamiltonians.In the method we have to handle the full many-body state of the correlated system.The method is thus limited by the available main memory.The principal problem for a distributed-memory implementation is that the central routine of the code, the application of the Hamiltonian to the many-body state, leads, due to the kinetic energy term, to very non-local memory access.We describe a solution for this problem and show that the new algorithm scales very efficiently on BlueGene/L.Moreover, our approach is not restricted to correlated electrons but can also be used to simulate quantum spin systems relevant for quantum computing.