Particle-number restoration within the energy density functional formalism: Nonviability of terms depending on noninteger powers of the density matrices

We discuss the origin of pathological behaviors that have been recently identified in particle-number-restoration calculations performed within the nuclear energy density functional framework. A regularization method that removes the problematic terms from the multi reference energy density functional and that applies (i) to any symmetry-restoration-and/or generator-coordinate-method-based configuration mixing calculation and (ii) to energy density functionals depending only on integer powers of the density matrices was proposed in [D. Lacroix, T. Duguet, and M. Bender, Phys. Rev. C 79, 044318 (2009)] and implemented for particle-number-restoration calculations in [M. Bender, T. Duguet, and D. Lacroix, Phys. Rev. C 79, 044319 (2009)]. In the present article, we address the viability of noninteger powers of the density matrices in the nuclear energy density functional. Our discussion builds on the analysis already carried out in [J. Dobaczewski et al., Phys. Rev. C 76, 054315 (2007)]. First, we propose to reduce the pathological nature of terms depending on a noninteger power of the density matrices by regularizing the fraction that relates to the integer part of the exponent using the method proposed in [D. Lacroix, T. Duguet, and M. Bender, Phys. Rev. C 79, 044318 (2009)]. Then, we discuss the spurious features brought about by the remaining fractional power. Finally, we conclude that noninteger powers of the density matrices are not viable and should be avoided in the first place when constructing nuclear energy density functionals that are eventually meant to be used in multi reference calculations.