The ground-state single-particle Green’s function describing hole propagation is calculated exactly for the 1/r2 quantum many-body system at integer coupling. The result is in agreement with a recent conjecture of Haldane.
We present an elementary and fairly detailed review of several Green’s function methods for treating nuclear and other many-body systems. We first treat the single-particle Green’s function, by way of which some details concerning linked diagram expansion, rules for evaluating Green’s function diagrams and solution of the Dyson’s integral equation for Green’s function are exhibited. The particle-particle hole-hole (pphh) Green’s function is then considered, and a specific time-blocking technique is discussed. This technique enables us to have a one-frequency Dyson’s equation for the pphh and similarly for other Green’s functions, thus considerably facilitating their calculation. A third type of Green’s function considered is the particle-hole Green’s function. RPA and high order RPA are treated, along with examples for setting up particle-hole RPA equations. A general method for deriving a model-space Dyson’s equation for Green’s functions is discussed. We also discuss a method for determining the normalization of Green’s function transition amplitudes based on its vertex function. Some applications of Green’s function methods to nuclear structure and recent deep inelastic lepton-nucleus scattering are addressed.
Abstract Starting with the equation of motion for the field operator ψ(x,t) of an interacting many-particle system, the n-particle Green’s function (Gn) equation of motion is developed, with interparticle interactions generating an infinite chain of equations coupling it to (n+1)- and (n−1)-particle Green’s functions (Gn+1 and Gn−1, respectively). Particularly important are the one-particle Green’s function equation with its coupling to the two-particle Green’s function and the two-particle Green’s function equation with its coupling to the three-particle Green’s function. To develop solutions, it is necessary to introduce non-correlation decoupling procedures involving the Hartree and Hartree-Fock approximations for G2 in the G1 equation; and a similar factorization “ansatz” for G3 in the G2 equation, resulting in the Sum of Ladder Diagrams integral equation for G2, with multiple Born iterates and finite collisional lifetimes. Similar treatment of the G11-equation for the joint propagation of one-electron and one-hole subject to mutual Coulomb attraction leads to bound electron-hole exciton states having a discrete hydrogen like spectrum of energy eigenstates. Its role in single-particle propagation is also discussed in terms of one-electron self-energy Σ and the T-matrix
An end-to-end strategy for hybrid quantum-classical computations of Green's functions in many-body systems is presented and applied to the pairing model. The scheme makes explicit use of the spectral representation of the Green's function, which entails the calculation of the N-body ground state as well as eigenstates and associated energies of the (N±1)-body neighbors. While the former is accessed via variational techniques, the latter are constructed by means of the quantum subspace expansion method. Different ansatzes for the ground-state wave function, originating from either classical or quantum approaches, are tested and compared to exact calculations. The resulting one-body Green's functions prove to be accurate approximations of the exact one for a large range of parameters, including across the normal-to-superfluid transition. As a byproduct, this approach yields a good description of odd systems provided that the starting even system is well reproduced by the variational ansatz.
Holes in a Mott insulator are represented by spinless fermions in the fermion-boson model introduced by Edwards. Although the physically interesting regime is for low to moderate fermion density, the model has interesting properties over the whole density range. It has previously been studied at half-filling in the one-dimensional (1D) case by numerical methods, in particular using exact diagonalization and the density matrix renormalization group (DMRG). In the present study the one-particle Green's function is calculated analytically by means of a decoupling scheme for the equations of motion, valid for arbitrary density in 1D, 2D and 3D with fairly large boson energy and zero boson relaxation parameter. The Green's function is used to compute some ground state properties, and the one-fermion spectral function, for fermion densities n = 0.1, 0.5 and 0.9 in the 1D case. The results are generally in good agreement with numerical results obtained using the DMRG and dynamical DMRG, and new light is shed on the nature of the ground state at different fillings. The Green's function approximation is sufficiently successful in 1D to justify future application to the 2D and 3D cases.
Exactly solvable quantum many-body models are rare, and even when their spectra are algebraically organized, dynamical responses generally remain difficult to obtain because they probe an extensive number of excited states. Here we show that quantum geometric nesting (QGN) models admit an unusually strong form of solvability rooted in \emph{Fock-space fragmentation}: excitations on top of the exact frustration-free ground states decouple into Krylov subspaces with a fixed number of particle and hole operators, and hence remain dynamically invariant. Exploiting this structure, we prove that the stiffness of the spontaneously broken continuous symmetry in QGN models is exactly equal to its variational value in the Gaussian manifold, confirming a conjecture from quantum many-body bootstrap~\cite{GaoHanKhalaf2026}. The proof shows that an infinitesimal phase twist couples the ground state only to the one-particle, one-hole fragment, which coincides with the tangent space of the ground state within the variational manifold, thereby making the variational curvature exact. More generally, perturbations whose action remains within a fixed Fock-space fragment have response functions determined exactly by the corresponding few-body sector, enabling exact access to quantities including static susceptibility, optical conductivity, dynamical structure factors, and single-particle Green's functions.
Recently we proposed a multichannel Dyson equation, in which we coupled the one- and the three-body Green's functions, to model photoemission spectra. In this work we extend this concept to higher-order Green's functions and to other spectroscopies. We show the general structure of the equations and how one can systematically approximate the corresponding multichannel self-energy. As a particular case, we focus on the coupling of the two-body and the four-body Green's functions in the electron-hole channel to describe neutral excitations. This formulation allows for the description of important many-body effects, such as biexcitons, in a natural way. We illustrate our approach by applying it to a two-level model system, which, in a one-particle picture, exhibits single and double excitations. Our method can correctly describe both kinds of excitation, unlike standard approaches, and in good agreement with the exact results.