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Physical Review C

Coherent correlated pair method versus Hartree-Fock with a soluble SU(3) Hamiltonian

A. Klar, Frank James Margetan, James P. Vary, Angelo Plastino

Published Dec 1, 1983• 5 citations
Publisher

Abstract

We test the coherent correlated pair method with the soluble SU(3) Hamiltonian of Li, Klein, and Dreizler. We compare the results for the ground state energy with the exact solution and with the Hartree-Fock results as a function of coupling parameters and number of particles. For all cases the coherent correlated pair method yields the better upper bound to the ground state energy.NUCLEAR STRUCTURE Correlated pairs; coherent states; upper bounds to ground state energies; applications to exactly soluble SU(3) $N$-fermion model.

DOI
10.1103/physrevc.28.2499

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We develop a method for treating the NNN-fermion problem as a system of correlated pairs of particles. Coherent pair states are introduced and a variational as well as a secular method are discussed for providing upper bounds to the exact ground state and approximations to the low lying spectra. Specific applications to two soluble models are presented. For these models the coherent correlated pair method is superior to the Bardeen-Cooper-Schrieffer approximation and to the Hartree-Fock approximation and in fact yields the exact solutions for all particle numbers and sufficiently large coupling constants.NUCLEAR STRUCTURE Correlated pairs; coherent states; upper bound to ground state energy; applications to soluble NNN-fermion models.
The method of coherent correlated pairs is applied with a realistic effective Hamiltonian to the four-nucleon system. Three methods of solution are compared for ease of implementation and convergence rates. A Monte Carlo method yields the best upper bound within the coherent correlated pair space. The full variational treatment of the four-nucleon coherent correlated pair problem yields equations which have poor stability properties. An ansatz variational method proves to be rapidly convergent and yields results very close to those of the Monte Carlo method. The results for binding energy and the one body density distribution are compared with Hartree-Fock results using the same effective Hamiltonian in the same model space. As in previous soluble model applications the coherent correlated pair method provides a superior upper bound to the ground state energy.
Upper bounds, lower bounds, and estimates based on moments are compared for the ground state energy of a soluble Hamiltonian due to Lipkin. Several bounding methods based on variational approaches and one based on the method of moments are employed. We show that lower bounds are of comparable quality with upper bounds and are no more difficult to obtain. No single bounding method is found superior over the entire range of parameters we study for the soluble Hamiltonian. However, in one region we find an upper bound from one method which coincides with the lower bound of another method, thus yielding the exact ground state energy. We also show that estimation methods based on the easily calculated first and second moments of the eigenvalue distribution are surprisingly accurate for a wide range of coupling parameters and number of active particles.NUCLEAR STRUCTURE Upper and lower bounds to ground state energy with soluble model; spectral properties of Lipkin model; variational methods; moment methods.
Quantum computing is increasingly offering concrete solutions toward the simulation of nuclear structure, with the potential to overcome the exponential scaling that limits classical diagonalization methods in large spaces. A particularly efficient encoding scheme, based on collective like-nucleon pairing modes, reduces the qubit requirements by half and avoids the non-local operator strings of standard fermion-to-qubit mappings. While this quasiparticle framework provides accurate results for semimagic nuclei, it does not adequately describe open-shell systems where proton-neutron correlations become important. In this work, we apply Brillouin-Wigner perturbation theory to systematically improve the quasiparticle description of open-shell nuclei in the sdsdsd shell, reaching an energy relative error below 0.2%0.2\%0.2% compared to the nuclear shell model. Furthermore, to make the effective Hamiltonian suitable for quantum simulation, we introduce a mean-field Hartree-Fock approximation of the non-quasiparticle resolvent, achieving ground-state energies typically within 2%2\%2% of the exact shell-model result. This represents a systematic improvement over the bare quasiparticle Hamiltonian while remaining within the reach of near-term quantum devices.
Using a Hartree-Fock-like partitioning of the two-matrix in terms of the true correlated one-matrix, the corresponding approximate ground-state energy E0{E}_{0}E0​ is calculated for an exactly solvable closed-shell system proposed earlier by Moshinsky [Am. J. Phys. 36, 52 (1968)]. Comparisons of E0{E}_{0}E0​ with the Hartree-Fock ground-state energy EHF{E}_{\mathrm{HF}}EHF​ and the exact ground-state energy Eex{E}_{\mathrm{ex}}Eex​ are made. These comparisons provide valuable insight into the usefulness of a Hartree-Fock-like partitioning of the two-matrix with repulsive interparticle potentials. In particular, an explicit quantification for the applicability limit of Lieb's [Phys. Rev. Lett. 46, 457 (1981)] bound ${E}_{0}\ensuremath{\geqslant}{E}_{\mathrm{HF}}$ with unbounded oscillator potentials is given.