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arXiv:1011.5146

The extended coupled cluster method and the pairing problem

Chris Snape, Niels R. Walet

Published Nov 23, 2010• 0 citations
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Abstract

We study the application of various forms of the coupled cluster method to systems with paired fermions. The novel element of the analysis is the study of the breaking and eventual restoration of particle number in the CCM variants. We specifically include Arponen's extended coupled cluster method, which describes the normal Hartree-Fock-Bogoliubov mean field at lowest level of truncation. We show that all methods converge to the exact results as we increase the order of truncation, but that the breaking of particle number at an intermediate level means that this convergence occurs in a surprising way. We argue that the most straightforward form of the method seems to be the most stable approach to implement for realistic (large number of particles) pairing problems

DOI
10.48550/arxiv.1011.5146

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We study the application of various forms of the coupled cluster method to systems with paired fermions. The novel element of the analysis is the study of the breaking and eventual restoration of particle number in the CCM variants. We specifically include Arponen's extended coupled cluster method, which describes the normal Hartree-Fock-Bogoliubov mean field at lowest level of truncation. We show that all methods converge to the exact results as we increase the order of truncation, but that the breaking of particle number at an intermediate level means that this convergence occurs in a surprising way. We argue that the most straightforward form of the method seems to be the most stable approach to implement for realistic (large number of particles) pairing problems
The phenomenon of pairing in atomic and nuclear many-body systems gives rise to a great number of different physical properties of matter, from areas as seemingly diverse as the shape of stable nuclei to superconductivity in metals and superfluidity in neutron stars. With the experimental realisation of the long sought BCS-BEC crossover observed in trapped atomic gases - where it is possible to fine tune the s-wave scattering length a of a many-fermion system between a dilute, correlated BCS-like superfluid of Cooper pairs and a densely packed BEC of composite bosons - pairing problems in atomic physics have found renewed interest in recent years. Given the high precision techniques involved in producing these trapped gas condensates, we would like to employ a suitably accurate many-body method to study such systems, preferably one which goes beyond the simple mean-field picture.The Coupled Cluster Method (CCM) is a widely applied and highly successful ab initio method in the realm of quantum many-body physics and quantum chemistry, known to be capable of producing extremely accurate results for a wide variety of different many-body systems. It has not found many applications in pairing problems however, at least not in a general sense. Our aim, therefore, is to study various models of pairing using a variety of CCM techniques - we are interested in studying the generic features of pairing problems and in particular, we are especially interested in probing the collective modes of a system which exhibits the BCS-BEC crossover, in either the BCS or BEC limit. The CCM seems a rather good candidate for the job, given the high precision results it can produce.
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.
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.
The BCS and/or Hartree-Fock-Bogoliubov theories are extended by treating the effect of four quasiparticle states perturbatively. The approach is tested on the pairing Hamiltonian, showing that it combines the advantage of standard perturbation theory, valid at low pairing strength, and of nonperturbative approaches breaking the particle number, valid at higher pairing strength. Including the restoration of particle number further improves the description of pairing correlation. In the presented test, the agreement between the exact solution and the combined perturbative plus projection is almost perfect. The proposed method scales friendly when the number of particles increases and provides a simple alternative to other more complicated approaches.
The Hartree-Fock-Bogoliubov approximation is very useful for treating both long- and short-range correlations in finite quantum fermion systems, but it must be extended in order to describe detailed spectroscopic properties. One problem is the symmetry-breaking character of the HFB approximation. We present a general and systematic way to restore symmetries and to extend the configuration space using pfaffian formulas for the many-body matrix elements. The advantage of those formulas is that the sign of the matrix elements is unambiguously determined. It is also helpful to extend the space of configurations by constraining the HFB solutions in some way. A powerful method for finding these constrained solutions is the gradient method, based on the generalized Thouless transformation. The gradient method also preserves the number parity of the Bogoliubov transformation, which facilitates the application of the theory to systems with odd particle number.