Symmetry broken and restored coupled-cluster theory I. Rotational symmetry and angular momentum

We extend coupled-cluster theory performed on top of a Slater determinant breaking rotational symmetry to allow for the exact restoration of the angular momentum at any truncation order. The main objective relates to the description of near-degenerate finite quantum systems with an open-shell character. As such, the newly developed many-body formalism offers a wealth of potential applications and further extensions dedicated to the ab initio description of, e.g., doubly open-shell atomic nuclei and molecule dissociation. The formalism, which encompasses both single-reference coupled cluster theory and projected Hartree-Fock theory as particular cases, permits the computation of usual sets of connected diagrams while consistently incorporating static correlations through the highly non-perturbative restoration of rotational symmetry. Interestingly, the yrast spectroscopy of the system, i.e. the lowest energy associated with each angular momentum, is accessed within a single calculation. A key difficulty presently overcome relates to the necessity to handle generalized energy {\it and} norm kernels for which naturally terminating coupled-cluster expansions could be eventually obtained. The present work focuses on $SU(2)$ but can be extended to any (locally) compact Lie group and to discrete groups, such as most point groups. In particular, the formalism will be soon generalized to $U(1)$ symmetry associated with particle number conservation. This is relevant to Bogoliubov coupled cluster theory that was recently formulated and applied to singly open-shell nuclei.