Physical and unphysical solutions of the random-phase approximation equation
Hitoshi Nakada · Progress of Theoretical and Experimental Physics · 2016
Addendum: Prog. Theor. Exp. Phys. 2016, 063D02 (2016) As an addendum to the previous paper [1], it is mathematically proven that, if the stability matrix S is positive-semidefinite, solutions of the random-phase approximation (RPA) equation are all physical or belong to Nambu–Goldstone (NG) modes, and the NG-mode solutions may form Jordan blocks of N S (N is the norm matrix) but their dimension is not more than two. This guarantees that the NG modes in the RPA can be separated out as canonically conjugate variables. The random-phase approximation (RPA) is widely used as describing excitation properties on top of mean-field (MF) solutions. In Ref. [1], I mathematically argued properties of solutions of the RPA equation based on two types of dualities: UL- and LR-dualities. The solutions were classified into five categories, as disclosed by the dualities, in Prop. 2 of Ref. [1]. It was also reconfirmed that, if the stability matrix S is positive-definite, the solutions are all physical, belonging to Class (1) of Prop. 2, which has already been verified in Ref. [2]. Its opposite was also proven in Ref. [1]. However, spontaneous symmetry breakdown (SSB) necessarily occurs for the MF solution in localized self-bound systems like atomic nuclei [3]. Individual SSB leads to a Nambu–Goldstone (NG) mode, and therefore S is quite generally positive-semidefinite in physical cases, rather than positive-definite. A method to handle NG modes has been established [2,4,5], which seems valid as long as the NG mode corresponds to physical degrees of freedom (d.o.f.). This method presumes that each NG mode forms a two-dimensional Jordan block of N S, where N is the norm matrix. On the contrary, the dualities do not limit the dimension of the Jordan blocks, as exemplified in Appendix C.5 of Ref. [1]. To the best of my knowledge, there have been no rigorous arguments that elucidate the dimensions of Jordan blocks for NG modes, except for restricted cases [6]. In this addendum, I shall prove what dimensionality is possible for Jordan blocks associated with NG-mode solutions (i.e., Class (5) in Prop. 2 of Ref. [1]) when S is positive-semidefinite. The section, appendix, and proposition numbers of Ref. [1] will be referred to directly in the text. If the stability matrix S is positive-semidefinite, solutions of the RPA equation are constrained to those of Classes (1) and (5) in Prop. 2, and the dimensionality of the Jordan blocks for the NG-mode solutions does not exceed two. While the above proposition does not exclude the possibility of a pair of dν=1 NG modes, they result in a trivial case in which both the canonically conjugate d.o.f. are not included in the RPA Hamiltonian (i.e., S). Such an example is given by the 2 × 2 model of Appendix C.1 (the case of a = b = 0), and by the d.o.f. corresponding to the center-of-mass coordinate R and the total momentum P, when the original Hamiltonian H satisfies [H,R]=[H,P]=0. If we focus on the dν=2 cases, Prop. 6 ensures that the two-dimensional Jordan blocks for the NG-mode solutions can be made doubly self dual. Notice that, for self UL-dual basis vectors, “canonical conjugacy” corresponds to LR-duality.1 The relevant subspace Wν(=W[ν]; see Sect. 4 for the notation) is spanned by the canonically conjugate variables. Via the decomposition described in Sect. 4, these variables are projected out of the RPA Hamiltonian. Thus, as long as S is positive-semidefinite, it is mathematically guaranteed that the prescription to separate out the NG modes in terms of canonically conjugate variables [2,4,5,7] is applicable. It is mentioned that Eqs. (4) and (5) above are identical to Eqs. (8.104) and (8.106) of Ref. [5] respectively, and that Eq. (6) validates (8.107) of Ref. [5], as is clear by recalling the self UL-dual condition Σxξk(ν)∗=−ξk(ν)=(Ξ(ν,k)−Ξ(ν,k)∗) (see Sect. 5.2). The so-called mass parameter is defined by 1/c1(ν), if the norm xν† xν is properly chosen. After removing all the NG modes, the remaining space [i.e., the space complementary to Ker(S˜)] is positive-definite. The author is grateful to K. Neergård for drawing attention to the subject of this addendum. This work is financially supported in part by JSPS KAKENHI Grant Numbers 24105008 and 16K05342.