The quantum-mechanical many-body problem is studied using the extended coupled-cluster method (ECCM) as a convenient parametrization of the Hilbert space. A systematic development of the formalism is given, and the quantum-mechanical problem is cast into the form of a classical Hamiltonian field theory in a complex symplectic phase space. The equations of motion of the basic ECCM amplitudes are derived both from a dynamical action principle using the ECCM average-value functional, and directly from the Schrödinger equation using the ECCM double similarity transformation. Rules are given for the construction of the Hamiltonian and other observables as well as their products. In particular, commutators are shown to be mapped into generalized classical Poisson brackets. The description is interpreted as an exact bosonization of the quantum theory in which the concept of bosonization is carried to the logical extreme, namely, the resulting generalized coherent bosons are identifiable with classical fields. The quantum-mechanical states of the system are points in the ECCM phase space, and their time evolution, or trajectories, are controlled by a classical Hamiltonian. The bosonization has a perturbation-theoretical basis in terms of maximally linked generalized tree diagrams. The increased degree of locality of the basic amplitudes allows applications to topological excitations and cases with spontaneous symmetry breaking.

The coupled-cluster or $\exp \mathrm{}\left(S\right)\mathrm{}$ formalism is applied to the problem of correlations in a many-electron system. In this first work in a series of papers we concentrate on exact results and restrict ourselves to various ring approximations applicable essentially to the high-density regime. It is shown that the well-known random-phase approximation (RPA) can be formulated as a nonlinear three-dimensional integral equation for the four-point function $S_2$ which provides a measure of the two-particle—hole pair component in the true ground-state wave function. Exact analytic solutions are presented for both $S_2$ and the three-point particle-hole vertex function in the RPA, and their properties are discussed. The Tamm-Dancoff approximation to the ring summation is also formulated, and the analagous exact solutions are presented for the first time. It is demonstrated that both solutions are unique, and a comparison of the two approximations is presented in terms of the correlation energy.

Liquids and solids, atoms and molecules, nuclei—all these clearly are interacting many-body systems. Even a nucleon may be regarded as a manyparticle system, not just because it is now known to consist of three quarks interacting via gluons, but because of the possibility in quantum field theory of virtual excitation of many particles from the vacuum. in the table page 56 we list some of the many-particle systems we encounter in the physical world at length scales that range from a few centimeters to a few fermis. Many of these systems exhibit phenomena—superconductivity in solids and fission in nuclei, for example—whose understanding does not follow immediately from knowledge of the constituents of the system and the interactions among the constituents, but requires new concepts and ideas. Many-body physics is the branch of theoretical physics that studies the new phenomena or “emergent properties” that arise from interactions among “elementary” constituents of a many-particle system and provides means and methods for carrying out precise calculations of such characteristic properties of these systems as may be compared with experimental results to verify hypotheses about the nature of the constituents and their interactions.

In this second work in a series of papers the coupled-cluster or $\exp \mathrm{}\left(S\right)\mathrm{}$ formalism is applied to the problem of ground-state correlations in the one-component Fermi plasma. The main approximation employed is the so-called SUB2 approximation for the two-body subsystem correlation operator $S_2$ which provides a measure of the two-particle—two-hole component in the true ground-state wave function, and in which the coupling to larger subsystems is neglected. The SUB2 equations for Fermi systems are brought into tractable form by a "state-averaging" procedure which we develop by analogy with the comparable Bose equations, and which is shown by comparison with earlier exact results to be accurate in the metallic density regime at about the 1% level. Our final results for metallic densities include the completely integrated and self-consistent effects of the terms which by themselves generate (i) the random-phase approximation (RPA) and its associated exact long-range screening effects, (ii) the extra random-phase approximation exchange terms necessary to keep the RPA explicitly antisymmetric, (iii) the self-consistent particle-particle ladders (LAD) that describe two-particle scattering within the many-body medium and which describe the exact short-range behavior, (iv) a class of particle-hole ladder terms, and (v) the self-consistent hole-potential terms. Particular attention is paid to the important effects caused by the interference at intermediate separations of the long-range RPA and the short-range LAD effects. By comparison with recent and essentially exact stochastic simulations of the many-body Schrödinger equation, our results are seen to be accurate to about 1% for metallic densities, and hence to provide what is probably the currently best microscopic description available for this system. We show also that even in the low-density limit the SUB2 approximation provides a good "translationally-invariant-solid" description of both charged Fermi and Bose systems in this exact Wigner crystal regime.

We use the coupled cluster method (CCM) to study the zero-temperature ground-state (GS) properties of a spin-$\frac{1}{2}$ $J_1-J_2$ Heisenberg antiferromagnet on a triangular lattice with competing nearest-neighbor and next-nearest-neighbor exchange couplings $J_1>0$ and $J_2\equiv \kappa J_1>0$, respectively, in the window $0\le \kappa <1$. The classical version of the model has a single GS phase transition at ${\kappa }^{\mathrm{cl}}=\frac{1}{8}$ in this window from a phase with 3-sublattice antiferromagnetic (AFM) ${120}^{\circ }$ Néel order for $\kappa <{\kappa }^{\mathrm{cl}}$ to an infinitely degenerate family of 4-sublattice AFM Néel phases for $\kappa >{\kappa }^{\mathrm{cl}}$. This classical accidental degeneracy is lifted by quantum fluctuations, which favor a 2-sublattice AFM striped phase. For the quantum model we work directly in the thermodynamic limit of an infinite number of spins, with no consequent need for any finite-size scaling analysis of our results. We perform high-order CCM calculations within a well-controlled hierarchy of approximations, which we show how to extrapolate to the exact limit. In this way we find results for the case $\kappa =0$ of the spin-$\frac{1}{2}$ model for the GS energy per spin, $E/N=-0.5521\left(2\right)J_1$, and the GS magnetic order parameter, $M=0.198\left(5\right)$ (in units where the classical value is $M^{\mathrm{cl}}=\frac{1}{2}$), which are among the best available. For the spin-$\frac{1}{2}$ $J_1-J_2$ model we find that the classical transition at $\kappa ={\kappa }^{\mathrm{cl}}$ is split into two quantum phase transitions at ${\kappa }_1^c=0.060\left(10\right)$ and ${\kappa }_2^c=0.165\left(5\right)$. The two quasiclassical AFM states (viz., the ${120}^{\circ }$ Néel state and the striped state) are found to be the stable GS phases in the regime $\kappa <{\kappa }_1^c$ and $\kappa >{\kappa }_2^c$, respectively, while in the intermediate regimes ${\kappa }_1^c<\kappa <{\kappa }_2^c$ the stable GS phase has no evident long-range magnetic order.