粒子数高阶恢复的Lipkin方法
Lipkin method of particle-number restoration to higher orders
摘要：
背景资料：在平均场水平上，通过Bogoliubov-Valatin变换将配对关联合并，从而将粒子自由度替换为准粒子。这种方法导致粒子数对称性的自发破缺和不同粒子数的状态混合。为了恢复粒子数，已经采用了各种方法，这些方法基于变化之前或之后的投影方法。近似投影后变分（VAP）方案，利用Lipkin方法，主要用于Lipkin-Nogami处方。目的：在不重复Lipkin-Nogami处方的情况下，使用规范空间中旋转的状态，我们导出了粒子数恢复到六阶的Lipkin方法，并测试了所获得的扩展的收敛性和准确性。研究方法：在超流核能-密度泛函理论的框架下，利用高阶Lipkin方法进行了粒子数对称性的自洽计算.我们还将Lipkin方法应用于一个示意性的精确可解的两能级配对模型。结果如下：在开壳层锡和铅同位素中进行的计算表明，Lipkin方法收敛于四阶，并令人满意地再现了VAP基态能量和能量核。在封闭壳层附近，高阶Lipkin方法不能应用，因为基态能量随粒子数的函数存在非解析扭结。结论：在开壳层核中，高阶Lipkin方法提供了一个很好的近似的精确VAP能量。该方法是计算成本低，使其特别适合，例如，未来的优化的核能密度泛函和同时恢复不同的对称性。
英文摘要：
Background: On the mean-field level, pairing correlations are incorporated through the Bogoliubov-Valatin transformation, whereupon the particle degrees of freedom are replaced by quasiparticles. This approach leads to a spontaneous breaking of the particle-number symmetry and mixing of states with different particle numbers. In order to restore the particle number, various methods have been employed, which are based on projection approaches before or after variation. Approximate variation-after-projection (VAP) schemes, utilizing the Lipkin method, have mostly been used within the Lipkin-Nogami prescription. Purpose: Without recurring to the Lipkin-Nogami prescription, and using instead states rotated in the gauge space, we derive the Lipkin method of particle-number restoration up to sixth order and we test the convergence and accuracy of the obtained expansion. Methods: We perform self-consistent calculations using the higher-order Lipkin method to restore the particle-number symmetry in the framework of superfluid nuclear energy-density functional theory. We also apply the Lipkin method to a schematic exactly solvable two-level pairing model. Results: Calculations performed in open-shell tin and lead isotopes show that the Lipkin method converges at fourth order and satisfactorily reproduces the VAP ground-state energies and energy kernels. Near closed shells, the higher-order Lipkin method cannot be applied because of a non-analytic kink in the ground-state energies in function of the particle number. Conclusions: In open-shell nuclei, the higher-order Lipkin method provides a good approximation to the exact VAP energies. The method is computationally inexpensive, making it particularly suitable, for example, for future optimizations of the nuclear energy-density functionals and simultaneous restoration of different symmetries.
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