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

Abstract

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 openshell 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 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 applied to singly open-shell nuclei. 0 can be extracted as

In-Medium Similarity Renormalization Group Approach to the Nuclear Many-Body Problem

“…Because of this, the intrinsic HFB wave functions are superpositions of states with different particle numbers. The broken symmetries must eventually be restored by means of projection methods, which have a long history in nuclear many-body theory [117,[141][142][143][144][145][146][147][148][149][150][151]). …”

Ab initio-driven nuclear energy density functional method

“…[28] in such a way that the symmetry is exactly and consistently restored at any truncation order. While Ref.…”

“…It is only in the last five years that ab initio many-body methods combining self-consistent Green's function (SCGF) [11,13] or coupled-cluster (CC) [10,27,14] with the concept of symmetry breaking have been implemented to tackle open-shell nuclei. It is even more recently that methods adding the exact restoration of the broken symmetry have been formulated [28].…”

“…The proposal is fundamentally based on the novel many-body formalism, or more specifically on its simplified many-body perturbation theory (MBPT) version, proposed in Ref. [28]. As will be described below, this formalism consistently builds on the successive breaking and restoration of symmetries and is, as such, the first and only full-fledged many-body technique that can provide a constructive approach to off-diagonal EDF kernels.…”

Combining symmetry collective states with coupled-cluster theory: Lessons from the Agassi model Hamiltonian

“…Similarly, there has been a renaissance of the method in nuclear physics where high precision studies of medium-mass nuclei close to magic numbers were successfully performed in the last decade [6][7][8]. However, single-reference CC is only accurate when applied to systems characterized by weak correlations, a category that excludes systems exhibiting such important properties as superconductivity or superfluidity [9,10], or nuclear deformation [11], and such ubiquitous phenomena as bond breaking in quantum chemistry [12]. Under weak correlation, the underlying reference wave function that coupled cluster usually takes as a given, and which in single-reference CC must be a single determinant, is a qualitatively good approximation of the ground state eigenfunction.…”