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arXiv:1501.03684 · Physical Review C

Accuracy of the new pairing theory and its improvement

Lu Jia

Published Dec 19, 2013• 7 citations
arXivHTMLPDF

Abstract

Recently I proposed [Jia, Phys. Rev. C 88, 044303 (2013)] a new method for solving the pairing Hamiltonian with the pair-condensate wave-function ansatz based on the Heisenberg equations of motion for the density matrix operators. In this work an improved version is given by deriving the relevant equations more carefully. I evaluate both versions in a large ensemble with random interactions, and the accuracy of the methods is given statistically in terms of root-mean-square derivations from the exact results. The widely used variational calculation is also done and the results and computing-time costs are compared.

DOI
10.1103/physrevc.88.064321

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Recently I proposed [Jia, Phys. Rev. C 88, 044303 (2013)] a new method for solving the pairing Hamiltonian with the pair-condensate wave-function ansatz based on the Heisenberg equations of motion for the density matrix operators. In this work an improved version is given by deriving the relevant equations more carefully. I evaluate both versions in a large ensemble with random interactions, and the accuracy of the methods is given statistically in terms of root-mean-square derivations from the exact results. The widely used variational calculation is also done and the results and computing-time costs are compared.
A new microscopic theory for nuclear pairing is proposed through the generalized density matrix formalism. Instead of the variation principle, we start from the Heisenberg equations of motion for density matrix operators ${K}_{12}\ensuremath{\equiv}{a}_{2}{a}_{1}$. The theory generalizes the well-known particle-particle random-phase approximation. Its analytical equations are as simple as that of the BCS theory and can be solved within a similar computer time if the model space is not too large. The current theory conserves the exact particle number and is valid at arbitrary pairing strength (including those below the BCS critical strength). It is also of interest to other mesoscopic systems such as ultrasmall metallic grains.
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A new technique is developed in this paper for solving Hamiltonians of the form $H=\ensuremath{\Sigma}\stackrel{}{k>0}{\ensuremath{\epsilon}}_{k}({{a}_{k}}^{+}{a}_{k}+{{a}_{\ensuremath{-}k}}^{+}{a}_{\ensuremath{-}k})\ensuremath{-}\ensuremath{\Sigma}\stackrel{}{k,l>0}G(k,l){{a}_{k}}^{+}{{a}_{\ensuremath{-}k}}^{+}{a}_{\ensuremath{-}l}{a}_{l},$ where G(k,l)G(k, l)G(k,l) may have the forms: G(k,l)=GG(k, l)=GG(k,l)=G (independent of kkk and lll), G(k,l)=GkGlG(k, l)={G}_{k}{G}_{l}G(k,l)=Gk​Gl​, or G(k,l)=G(k,l)G(k, l)=G(k, l)G(k,l)=G(k,l) (nonseparable in kkk and lll). The eigenvalues obtained with this technique are compared with the exact solutions of Kerman, Lawson, and Macfarlane. The agreement with the exact solutions is fairly good.
Variational methods are of fundamental importance and widely used in theoretical physics, especially for strongly interacting systems. In this work, we present a set of variational equations of state (VES) for pure states of an interacting quantum Hamiltonian. The VES can be expressed in terms of the variation of the density operators or static correlation functions. We derive the algebraic relationship between a known pure state density matrix and its variation, and obtain the VES by applying this relation to the averaged Heisenberg-equations-of-motion for the exact density matrix. Additionally, we provide a direct expression of the VES in terms of correlation functions to make it computable. We present three nontrivial applications of the VES: a perturbation calculation of correlation functions of the transverse field Ising model in arbitrary spatial dimensions, a study of a longitudinal field perturbation to the one-dimensional transverse field Ising model at the critical point and variational calculation of magnetization and ground state energy of the two-dimensional spin-1/2 Heisenberg model on a square lattice. For the second one, our results not only recover the scaling limit, but also indicate the possibility of continuous tuning of the critical exponents by adjusting the longitudinal fields differently from the scaling limit. For the Heisenberg model, we obtained results numerically comparable to established results with simple calculations. The VES approach provides a powerful and versatile tool for studying interacting quantum systems.
L. Y. Jia recently proposed an algorithm that applies the variational method directly to coherent-pair condensates (VDPC) [L. Jia, Phys. Rev. C (2019).] Now I’ll make changes to the VDPC algorithm that applies it to the Constraint Hatree–Fock theory. I derived expressions of the mean energy extremum with linear constraints and the analytic form of the Lagrangian multiplier. In addition, I acquire an angle through the inverse tangent of the energy minimum to the electric quadrupole moment in Fig. 3, which describes the unconstrained energy under an equivalent rotating system. I demonstrate the new algorithm in a semirealistic example using the realistic [Formula: see text] interaction in the large model spaces (up to 15 harmonic-oscillator major shells). In this work, I am interested in a wave function [Formula: see text], which minimizes the total energy under the constraint that has a fixed expectation [Formula: see text]. I graphed the results as just an energy extreme versus electric quadrupole moment graph. The result shows that Fig. 3 is similar to Fig. 7.2 in [P. Ring and P. Schuck, The Nuclear Many-body Problem (Springer Science & Business Media, 2004)].