We extend coupled-cluster (CC) 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 CC 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 CC 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 CC theory that was recently applied to singly open-shell nuclei.
Abstract We have recently extended many-body perturbation theory (MBPT) and coupled-cluster theory performed on top of a Slater determinant breaking rotational symmetry to allow for the restoration of the angular momentum at any truncation order (Duguet 2015 J. Phys. G: Nucl. Part. Phys. 42 025107 ). Following a similar route, we presently extend Bogoliubov MBPT and Bogoliubov coupled cluster theory performed on top of a Bogoliubov reference state breaking global gauge symmetry to allow for the restoration of the particle number at any truncation order. Eventually, formalisms can be merged to handle SU (2) and U (1) symmetries at the same time. The long-term goal relates to the ab initio description of near-degenerate finite quantum systems with an open-shell character.
Background: Ab initio many-body methods have been developed over the past 10 yr to address closed-shell nuclei up to mass $A\ensuremath{\approx}130$ on the basis of realistic two- and three-nucleon interactions. A current frontier relates to the extension of those many-body methods to the description of open-shell nuclei. Several routes to address open-shell nuclei are currently under investigation, including ideas that exploit spontaneous symmetry breaking.Purpose: Singly open-shell nuclei can be efficiently described via the sole breaking of U(1) gauge symmetry associated with particle-number conservation as a way to account for their superfluid character. While this route was recently followed within the framework of self-consistent Green's function theory, the goal of the present work is to formulate a similar extension within the framework of coupled cluster theory.Methods: We formulate and apply Bogoliubov coupled cluster (BCC) theory, which consists of representing the exact ground-state wave function of the system as the exponential of a quasiparticle excitation cluster operator acting on a Bogoliubov reference state. Equations for the ground-state energy and the cluster amplitudes are derived at the singles and doubles level (BCCSD) both algebraically and diagrammatically. The formalism includes three-nucleon forces at the normal-ordered two-body level. The first BCC code is implemented in
m scheme, which will permit the treatment of doubly open-shell nuclei via the further breaking of SU(2) symmetry associated with angular momentum conservation.Results: Proof-of-principle calculations in an
Nmax=6 spherical harmonic oscillator basis for
16,18O and
18Ne in the BCCD approximation are in good agreement with standard coupled cluster results with the same chiral two-nucleon interaction, while
20O and
20Mg display underbinding relative to experiment. The breaking of U(1) symmetry, monitored by computing the variance associated with the particle-number operator, is relatively constant for all five nuclei, in both the Hartree-Fock-Bogoliubov and BCCD approximations.Conclusions: The newly developed many-body formalism increases the potential span of ab initio calculations based on single-reference coupled cluster techniques tremendously, i.e., potentially to reach several hundred additional midmass nuclei. The new formalism offers a wealth of potential applications and further extensions dedicated to the description of ground and excited states of open-shell nuclei. Short-term goals include the implementation of three-nucleon forces at the normal-ordered two-body level. Midterm extensions include the approximate treatment of triples corrections and the development of the equation-of-motion methodology to treat both excited states and odd nuclei. Long-term extensions include exact restoration of U(1) and SU(2) symmetries.
Background: Ab initio many-body methods whose numerical cost scales polynomially with the number of particles have been developed over the past fifteen years to tackle closed-shell mid-mass nuclei. Open-shell nuclei have been further addressed by implementing variants based on the concept of spontaneous symmetry breaking (and restoration). These methods typically access ground states properties and some restricted aspects of spectroscopy.Purpose: In order to access the spectroscopy of open-shell nuclei more systematically while controlling the numerical cost, we design a novel many-body method that combines the merit of breaking and restoring symmetries with those brought about by low-rank individual excitations.Methods: The recently proposed truncated configuration-interaction method based on optimized symmetry-broken and -restored states is extended to the
SU(2) group associated with total angular momentum. Dealing more specifically with the Lipkin Hamiltonian, the present study focuses on the breaking and the restoration of the
z-signature symmetry associated with a discrete subgroup of
SU(2). The highly-truncated
N-body Hilbert subspace within which the Hamiltonian is diagonalized is spanned by a
z-signature broken and restored Slater determinant vacuum and associated low-rank, e.g., one-particle/one-hole and two-particle/two-hole, excitations. Furthermore, the extent by which the symmetry-unrestricted vacuum breaks
z-signature symmetry is optimized in the presence of projected low-rank particle-hole excitations. The quality of the method is gauged against exact ground- and excited-state eigenenergies for a large range of values of the two-body interaction strength. Furthermore, results are compared to those obtained from the generator coordinate method, the random-phase approximation, and the self-consistent (second) random-phase approximation.Results: The proposed method provides an excellent reproduction of the ground-state energy and of low-lying excitation energies of various
z signatures and total angular momenta across the full range of internucleon coupling defining the Lipkin Hamiltonian and driving the normal-to-deformed quantum phase transition. In doing so, the successive benefits of (i) breaking the symmetry, (ii) restoring the symmetry, (iii) including low-rank particle-hole excitations, and (iv) optimizing the amount by which the underlying vacuum breaks the symmetry are illustrated. While the generator coordinate method, built on the same deformed vacua, provides results of similar quality, this is not the case for the symmetry-restricted random-phase and self-consistent (second) random-phase approximations in the strongly interacting regime.Conclusions: The numerical cost of the newly designed variational method is polynomial with respect to the system size. It achieves a good accuracy on the ground-state energy and the low-lying spectroscopy for both weakly and strongly interacting systems. The present study confirms the results obtained previously for the attractive pairing Hamiltonian in connection with the breaking and restoration of
U(1) global gauge symmetry. These two studies constitute a strong motivation to apply this method to realistic nuclear Hamiltonians in view of providing a complementary, accurate and versatile ab initio description of mid-mass open-shell nuclei.
Given elementary inter-nucleon interactions,the resolution of the A-body Schrödinger equation gives access to the properties of the nuclear eigenstates. Many-body perturbation theory (MBPT) based on a Hartree-Fock mean field allows one to treat the dynamical correlations at play in doubly-closed-shell nuclei. Beyond shell closures,breaking U(1) symmetry associated with the conservation of neutron and proton numbers further permits to capture static correlations at the meanfield level and to formulate the well-behaved Bogoliubov many-body perturbation theory (BMBPT).Nevertheless, BMBPT results show contaminations associated with the broken symmetry that is only emergent on finite quantum system such as theatomic nucleus. Thus, the restoration of U(1) symmetry beyond the mean field is necessary for acorrect description and gives rise to the projected BMBPT method (PBMBPT). The goal is to implement PBMBPT in order to perform ab initio calculations of singly-open-shell mid-mass nuclei.The present work provides systematic solutions to the formal and technical problems arising during the implementation of PBMBPT: an automatic and safe method to generate PBMBPT diagrams and associated expressions, a framework to calculate other observables than the energy, an extension of the normal ordering approximation to symmetry breaking methods as well as an automatic and safe tool for spherical reduction of tensor networks.
Coupled cluster theory is the method of choice for weakly correlated systems. But in the strongly correlated regime, it faces a symmetry dilemma, where it either completely fails to describe the system or has to artificially break certain symmetries. On the other hand, projected Hartree-Fock theory captures the essential physics of many kinds of strong correlations via symmetry breaking and restoration. In this work, we combine and try to retain the merits of these two methods by applying symmetry projection to broken symmetry coupled cluster wave functions. The non-orthogonal nature of states resulting from the application of symmetry projection operators furnishes particle-hole excitations to all orders, thus creating an obstacle for the exact evaluation of overlaps. Here we provide a solution via a disentanglement framework theory that can be approximated rigorously and systematically. Results of projected coupled cluster theory are presented for molecules and the Hubbard model, showing that spin projection significantly improves unrestricted coupled cluster theory while restoring good quantum numbers. The energy of projected coupled cluster theory reduces to the unprojected one in the thermodynamic limit, albeit at a much slower rate than projected Hartree-Fock.
The goal of electronic structure theory is to solve the time-independent Schrodinger equation with a method that scales polynomially with the system size. The problem has been solved in the weakly-correlated regime, where coupled cluster theory (CC) with singles, doubles and perturbative triples (CCSD(T)) can achieve chemical accuracy. However, in the strongly-correlated regime, where the two-body interaction dominates, there are still no methods that can achieve a comparable accomplishment. Strong correlation always comes hand in hand with spontaneous symmetry breaking in mean-field theory. For example, coupled cluster theory faces a symmetry dilemma in the strongly correlated regime, where it either completely fails to describe the system if symmetry adapted, or has to artificially break certain symmetries in order to produce reasonable energies. It is plausible that the problem of strong correlation can be solved by curing the symptoms, i.e., fixing the symmetries. Symmetry projection has been carried out at the mean-field level, but it was unclear until this work how to incorporate correlation on top of it because of the language barrier between symmetry projection and coupled cluster theory. This thesis is devoted to combining CC and symmetry projection, focusing on spin and number symmetry. Different flavours of theories are obtained depending on whether a restricted or unrestricted reference is used. On the restricted side, the spin projected unrestricted Hartree Fock (SUHF) state can be rewritten as a polynomial of particle-hole excitations, acting on the restricted reference. Thus, this formalism opens up many ways to combine SUHF with CC. One the unrestricted side, the concept of disentangled cluster is introduced to describe the CC wavefunction transformed by symmetry rotation. A scheme basing on differential equations is devised to approximate these disentangled clusters, which scales polynomially and is systematically improvable. By combining symmetry projection and coupled cluster theory, the symmetry dilemma can be solved to a good extent.