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arXiv:cond-mat/0511430 · Physical Review A

Coupled-cluster theory for systems of bosons in external traps

L. S. Cederbaum, Ofir E. Alon, Alexej I. Streltsov

Published Apr 17, 2006• 35 citations
arXivHTMLPDF

Abstract

A coupled-cluster approach for systems of $N$ bosons in external traps is developed. In the coupled-cluster approach the exact many-body wave function is obtained by applying an exponential operator $\mathrm{exp}{T}$ to the ground configuration $\ensuremath{\mid}{\ensuremath{\phi}}_{0}⟩$. The natural ground configuration for bosons is, of course, when all reside in a single orbital. Because of this simple structure of $\ensuremath{\mid}{\ensuremath{\phi}}_{0}⟩$, the appearance of excitation operators $T={\ensuremath{\sum}}_{n=1}^{N}{T}_{n}$ for bosons is much simpler than for fermions. We can treat very large numbers of bosons with coupled-cluster expansions. In a substantial part of this work, we address the issue of size consistency for bosons and enquire whether truncated coupled-cluster expansions are size consistent. We show that, in contrast to the familiar situation for fermions for which coupled-cluster expansions are size consistent, for bosons the answer to this question depends on the choice of ground configuration. Utilizing the natural ground configuration, working equations for the truncated coupled-cluster with $T={T}_{1}+{T}_{2}$, i.e., coupled-cluster singles doubles are explicitly derived. Finally, an illustrative numerical example for a condensate with up to $N=10\phantom{\rule{0.2em}{0ex}}000$ bosons in an harmonic trap is provided and analyzed. The results are highly promising.

DOI
10.1103/physreva.73.043609

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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 ${N}_{\text{max}}=6$ spherical harmonic oscillator basis for $^{16,18}\mathrm{O}$ and $^{18}\mathrm{Ne}$ in the BCCD approximation are in good agreement with standard coupled cluster results with the same chiral two-nucleon interaction, while $^{20}\mathrm{O}$ and $^{20}\mathrm{Mg}$ 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.

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The ground-state pairing correlations in finite fermionic systems are described with a high degree of accuracy within a variational approach based on a combined coupled-cluster and particle-number-projected BCS ansatz. The flexibility of this symmetry-preserving wave function enables a unified picture valid from weak to strong coupling, both in small and large systems. The present variational approach consistently yields an energy upper bound while operating at the same level of precision as the nonvariational particle-number-projected Bogoliubov-coupled-cluster theory [Phys. Rev. C 99, 044301 (2019)].

arXiv:2102.08233•Physical Review C•May 24, 2021•10 citations
A coupled-cluster approach for systems of NNN bosons in external traps is developed. In the coupled-cluster approach the exact many-body wave function is obtained by applying an exponential operator expT\mathrm{exp}{T}expT to the ground configuration \ensuremath∣\ensuremathϕ0⟩\ensuremath{\mid}{\ensuremath{\phi}}_{0}⟩\ensuremath∣\ensuremathϕ0​⟩. The natural ground configuration for bosons is, of course, when all reside in a single orbital. Because of this simple structure of \ensuremath∣\ensuremathϕ0⟩\ensuremath{\mid}{\ensuremath{\phi}}_{0}⟩\ensuremath∣\ensuremathϕ0​⟩, the appearance of excitation operators T=\ensuremath∑n=1NTnT={\ensuremath{\sum}}_{n=1}^{N}{T}_{n}T=\ensuremath∑n=1N​Tn​ for bosons is much simpler than for fermions. We can treat very large numbers of bosons with coupled-cluster expansions. In a substantial part of this work, we address the issue of size consistency for bosons and enquire whether truncated coupled-cluster expansions are size consistent. We show that, in contrast to the familiar situation for fermions for which coupled-cluster expansions are size consistent, for bosons the answer to this question depends on the choice of ground configuration. Utilizing the natural ground configuration, working equations for the truncated coupled-cluster with T=T1+T2T={T}_{1}+{T}_{2}T=T1​+T2​, i.e., coupled-cluster singles doubles are explicitly derived. Finally, an illustrative numerical example for a condensate with up to N=10000N=10\phantom{\rule{0.2em}{0ex}}000N=10000 bosons in an harmonic trap is provided and analyzed. The results are highly promising.
We develop a coupled-cluster theory for bosonic mixtures of binary species in external traps, providing a promising theoretical approach to demonstrate highly accurately the many-body physics of mixtures of Bose-Einstein condensates. The coupled-cluster wavefunction for the binary species is obtained when an exponential cluster operator eT, where T = T(1) + T(2) + T(12) and T(1) accounts for excitations in species-1, T(2) for excitations in species-2, and T(12) for combined excitations in both species, acts on the ground state configuration prepared by accumulating all bosons in a single orbital for each species. We have explicitly derived the working equations for bosonic mixtures by truncating the cluster operator up to the single and double excitations and using arbitrary sets of orthonormal orbitals for each of the species. Furthermore, the comparatively simplified version of the working equations are formulated using the Fock-like operators. Finally, using an exactly solvable many-body model for bosonic mixtures that exists in the literature allows us to implement and test the performance and accuracy of the coupled-cluster theory for situations with balanced as well as imbalanced boson numbers and for weak to moderately strong intra- and interspecies interaction strengths. The comparison between our computed results using coupled-cluster theory with the respective analytical exact results displays remarkable agreement exhibiting excellent success of the coupled-cluster theory for bosonic mixtures. All in all, the correlation exhaustive coupled-cluster theory shows encouraging results and could be a promising approach in paving the way for high-accuracy modeling of various bosonic mixture systems.
We study many-body correlations in the ground state of a general quantum system of bosons or fermions by including an additional Jastrow function in our recently proposed variational coupled-cluster method. Our approach combines the advantages of state-dependent correlations in the coupled-cluster theory and of the strong, short-ranged correlations of the Jastrow function. We apply a generalized linked-cluster expansion for the Jastrow wave function and provide a detailed analysis for practical evaluation of the Hamiltonian expectation value as an energy functional of the Jastrow function and the bare density-distribution functions introduced and calculated in our earlier publications; a simple, first-order energy functional is derived and detailed formulas for the higher-order contributions are provided. Our energy functional does not suffer the divergence as most coupled-cluster calculations often do when applying to Hamiltonians with hardcore potentials. We also discuss possible applications of our technique, including applications to strongly correlated fermion systems.
We study a system of AAA identical interacting bosons trapped by an external field by solving ab initio the many-body Schr\"odinger equation. A complete solution by using, for example, the traditional hyperspherical harmonics (HH) basis develops serious practical problems due to the large degeneracy of HH basis. Symmetrization of the wave function, calculation of the matrix elements, etc., become an immensely formidable task as AAA increases. Instead of the HH basis, here we use a new basis, called ``potential harmonics'' (PH) basis, which is a subset of HH basis. We assume that the contribution to the orbital and grand orbital [in $3(A\ensuremath{-}1)$-dimensional space of the reduced motion] quantum numbers comes only from the interacting pair. This implies inclusion of two-body correlations only and disregard of all higher-body correlations. Such an assumption is ideally suited for the Bose-Einstein condensate (BEC), which is required, for experimental realization of BEC, to be extremely dilute. Hence three and higher-body collisions are almost totally absent. Unlike the $(3A\ensuremath{-}4)$ hyperspherical variables in HH basis, the PH basis involves only three active variables, corresponding to three quantum numbers---the orbital lll, azimuthal mmm, and the grand orbital 2K+l2K+l2K+l quantum numbers for any arbitrary AAA. It drastically reduces the number of coupled equations and calculation of the potential matrix becomes tremendously simplified, as it involves integrals over only three variables for any AAA. One can easily incorporate realistic atom-atom interactions in a straightforward manner. We study the ground and excited state properties of the condensate for both attractive and repulsive interactions for various particle number. The ground state properties are compared with those calculated from the Gross-Pitaevskii equation. We notice that our many-body results converge towards the mean field results as the particle number increases.
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 mmm 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{N}_{\text{max}}=6Nmax​=6 spherical harmonic oscillator basis for 16,18O^{16,18}\mathrm{O}16,18O and 18Ne^{18}\mathrm{Ne}18Ne in the BCCD approximation are in good agreement with standard coupled cluster results with the same chiral two-nucleon interaction, while 20O^{20}\mathrm{O}20O and 20Mg^{20}\mathrm{Mg}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.
The workhorse method of computational materials science is undeniably density functional theory in the Kohn-Sham framework of approximate exchange and correlation energy functionals. However, the need for highly accurate predictions of ground and excited state properties in materials science motivates the further development and exploration of alternative as well as complementary techniques. Among these alternative approaches, quantum chemical wavefunction based theories and in particular coupled cluster theory hold the promise to fill a gap in the toolbox of computational materials scientists. Coupled cluster (CC) theory provides a compelling framework of approximate infinite-order perturbation theory in the form of an exponential of cluster operators describing the true quantum many-body effects of the electronic wave function at a computational cost that, despite being significantly more expensive than DFT, scales polynomially with system size. The hierarchy of size-extensive approximate methods established in the framework of CC theory achieves systematic improvability for many materials properties. This is in contrast to currently available density functionals that often suffer from uncontrolled approximations that limit the accuracy in the prediction of materials properties. In this tutorial-style review we will introduce basic concepts of coupled cluster theory and recent developments that increase its computational efficiency for calculations of molecules, solids and materials in general. We will touch upon the connection between coupled cluster theory and the random-phase approximation to bridge the gap between traditional quantum chemistry and many-body Green’s function theories that are widely-used in the field of solid state physics. We will discuss various approaches to improve the computational performance without compromising accuracy. These approaches include large-scale parallel design as well as techniques that reduce the prefactor of the computational complexity. A central part of this article discusses the convergence of calculated properties to the thermodynamic limit which is of significant importance for reliable predictions of materials properties and constitutes an additional challenge compared to calculations of large molecules. Furthermore we mention technical aspects of computer code implementations of periodic coupled cluster theories in different numerical frameworks of the one-electron orbital basis; the projector-augmented-wave formalism using a plane wave basis set and the numeric atom-centered-orbital with resolution-of-identity.
The ground-state pairing correlations in finite fermionic systems are described with a high degree of accuracy within a variational approach based on a combined coupled-cluster and particle-number-projected BCS ansatz. The flexibility of this symmetry-preserving wave function enables a unified picture valid from weak to strong coupling, both in small and large systems. The present variational approach consistently yields an energy upper bound while operating at the same level of precision as the nonvariational particle-number-projected Bogoliubov-coupled-cluster theory [Phys. Rev. C 99, 044301 (2019)].