Physical and unphysical solutions of random-phase approximation equation

Properties of solutions of the RPA equation is reanalyzed mathematically, which is defined as a generalized eigenvalue problem of the stability matrix $\mathsf{S}$ with the norm matrix $\mathsf{N}=\mathrm{diag.}(1,-1)$. As well as physical solutions, unphysical solutions are examined in detail, with taking the possibility of Jordan blocks of the matrix $\mathsf{N\,S}$ into consideration. Two types of duality of eigenvectors and basis vectors of the Jordan blocks are pointed out and explored, which disclose many basic properties of the RPA solutions.

1 Introduction remains to be a minimum for d.o.f. ignored in the calculation, as in a spherical HF calculation for a quadrupolarly deformed nucleus. Whereas stability of the RPA solutions in vicinity of the MF minimum was investigated in Refs. [8,9] by Thouless et al., more general arguments with rigorous mathematical treatment are desired for a variety of extensions of the RPA developed to date.
In this article, I reanalyze properties of RPA solutions mathematically, in terms of the linear algebra. Unphysical solutions as well as physical solutions are examined in some detail.
Although one may consider that unphysical solutions are just meaningless, they are useful for understanding properties of the RPA equation more profoundly. As a result, they help us to comprehend in what manner physical solutions and NG modes come out. The present analysis will be of practical significance as well in coding programs for numerical calculations in the RPA, because one often has to prepare for various situations in numerical studies, without knowing structure of the vector space around the MF state sufficiently.

RPA equation
The RPA equation is written as where α, β represent particle-hole bases on the HF solution or two quasiparticle bases on the HFB solution. The matrices A and B are obtained from the residual interaction and the HF s.p. (or the HFB q.p.) energies, and satisfy In order to cope with a variety of physical situations, I start discussions only from this structure of the RPA equation, without any further assumptions. The solution of Eq. (1) is comprised of ω ν and (X (ν) , Y (ν) ), which correspond to the energy and the wave function of the ν-th excited state. It is imposed that (X (ν) , Y (ν) ) obeys the normalization, 3 The dimension of the A and B matrices is denoted by D. D is finite in many practical calculations. Although D can be infinite in principle, the arguments here will cover infinite D as a limiting case. Properties of the RPA solutions are better argued in terms of the 2D × 2D matrices, For the matrices N and Σ x , N 2 = Σ 2 x = 1 and Σ x N + N Σ x = 0 hold. The matrix S is nothing but the curvature matrix at the HF (or HFB) solution, and is known as the stability matrix. With the matrices defined in Eq. (4), the RPA equation (1) is expressed as and the normalization condition (3) as The number of the solutions satisfying (6) matches the dimension D of the A, B matrix in ideal cases. However, it is not obvious whether there exist D solutions that satisfy the normalization condition (6).

Dualities and eigenvalues
The RPA equation (5) is equivalent to the eigenvalue problem of the matrix N S. An eigensolution is defined by a set of an eigenvalue and an eigenvector (ω ν , x ν ). At the same time, Eq. (5) reads the eigenvalue problem of S N for an eigensolution (ω ν , N x ν ). This duality is important to derive basic properties of the RPA solutions. The relevant duality will be established later, and is called LR-duality in this article, because it connects left and right basis vectors.
Another important ingredient is the symmetry of S. The structure of S given in Eq. (4) with (2) is characterized by It should be noted that, owing to the first equation of (7), an eigenvector of S N associated with an eigenvalue ω ν immediately gives a left eigenvector of N S corresponding to the eigenvalue ω * ν , because S N y ν = ω ν y ν is equivalent to y † ν N S = ω * ν y † ν .

Proposition 1.
If ω ν is an eigenvalue of N S, −ω ν is also an eigenvalue with equal degeneracy. So is ω * ν .
It is straightforward to show the following corollary from the second equation of (7).
is also a solution. This relation indicates another kind of duality, which is called UL-duality to distinguish from the LR-duality, since it is related to interchange (with taking complex conjugate) of the upper and the lower components of the basis vectors. With respect to the normalization of Eq. (6), the following relation is obtained, The second equation of (6) is interpreted as a relation between Σ x x * ν and x ν ′ , For an arbitrary vector In this respect the 'norm' is not positive-definite.
Indeed, if x † N x = r 2 (> 0), x/r is normalized in the respect of Eq. (6). As noted above, it is not guaranteed that an eigenvector x ν of N S can be normalized. The normalizability of an eigenvector x ν is a key to whether the RPA equation is solvable.
In arguments with respect to the RPA [8], the 'orthogonality' between two vectors x and y is sometimes defined when x † N y = 0, by regarding N as the metric. However, in this article I use the usual definition that x and y are orthogonal when x † y = 0.
The hermiticity of S leads to the relation between the solutions of Eq. (5), Equation (10) concludes: follows. The case of ν = ν ′ was argued in Ref. [8].
It is now possible to classify solutions of Eq. (5).
A solution belonging to Class (1) of Prop. 2 may be called physical solution, while a solution to one of (2) -(4) unphysical solution. Class (5) is closely connected to the NG mode, and a solution belonging to it will be called NG-mode solution. Focusing on solutions in vicinity of a MF minimum, Thouless did not discuss solutions of Classes (2) and (4) in Refs. [8,9].

Basis vectors in Jordan blocks
Several eigenvalues of N S could be degenerate, and degenerate eigenvalues may give rise to Jordan blocks. This possibility is examined in this subsection. Suppose that a Jordan block is generated from an eigenvector x ν . A basis vector of the Jordan block is denoted by The constant c (ν) k is subject to normalization and to relative phase of ξ      Most of the following arguments will cover the case of d ν = 1; i.e., the case that an eigenvector x ν does not generate a Jordan block.

Lemma 2.
It there is a Jordan block for an eigenvalue ω ν with dimension d ν , so are for −ω ν and ω * ν with equal dimension d ν .
Proof. A Jordan block for the eigenvalue ω ν of S N directly corresponds to a Jordan block for ω * ν of N S, by regarding the right basis vectors of S N as the left basis vectors of N S. Therefore the dimensions corresponding to ω ν and ω * ν must be equal. It follows from Eqs. (7) and (11) that, verifying that there exists a Jordan block for −ω * ν with equal dimension. Then, so is it for −ω ν .
In general, a single eigenvalue ω ν may give plural Jordan blocks. I hereafter reserve the subscript ν for representing individual Jordan block, which is connected to a single eigenvector x ν of N S, rather than the eigenvalue. The following proposition and lemmas are closely connected to the structure of the Jordan block, which is represented with the left and right basis vectors that form the inverse matrix of each other apart from normalization. Explicit proofs of Lemma 3 to Prop. 3 are given in Appendix A, which are instructive and useful to confirm their compatibility with later propositions and lemmas.
Lemma 3. Unless eigenvalues ω * ν and ω ν ′ are equal, any basis vector of N S associated with ω ν (either an eigenvector or a basis vector belonging to a Jordan block) is orthogonal to any basis vector of S N associated with ω ν ′ .
The above lemma suggests that a Jordan block of N S for a specific ω ν and a Jordan block of S N for ω * ν are paired.

Lemma 4.
It is possible to take so that each basis vector in a Jordan block of N S could overlap with no more than one basis vector that contained in a Jordan block of S N. In order for the overlap not to vanish, dimensions of these Jordan blocks of N S and S N must be equal.
The next corollary follows from the proof of Lemma 4 given in Appendix A.2:

Proposition 3. It is possible to produce a complete set of basis vectors of N S, by a proper
transformation if necessary, so that each of them could overlap with only one basis vector of S N. One-to-one correspondence is established between Jordan blocks of N S and S N that contain basis vectors having non-vanishing overlaps. The correspondence between basis vectors of N S and S N is also one to one.
This proposition is an expression that the left and the right basis vectors giving the Jordan representation constitute the inverse matrix of each other.
Proposition 3 enables us to define a basis vector of S N that is dual of individual basis vector of N S. Let us denote the basis vector dual to ξ k ′ does not vanish only for ν = ν ′ and k = k ′ . This is a realization of the Jordan block of (13).ξ (ν) k is the (d ν + 1 − k)-th basis vector of the Jordan block for an eigenvectorx ν :=ξ dν , which is associated with the eigenvalue ω * ν . Inverting Eq. (11), we obtain as confirmed in Appendix A.4. This establishes the LR-duality of basis vectors.

Definition 4. The basis vectors ξ
The normalizability in Def. 2 is now perceived as a part of the self LRduality.

Redefining eigenvalue problem
Because S is hermitian, it is diagonalizable with an appropriate unitary matrix U, By using this expression, S 1/2 can be taken as Though λ Obviously,S is hermitian only when λ i ≥ 0 for any i (= 1, · · · , 2D).
Lemma 5. All the eigenvalues and eigenvectors of N S correspond to those ofS, and vice versa.
Thus the RPA equation is equivalent to the eigenvalue problem ofS;S x ν = ω ν x ν . This redefinition is applied to investigate solvability of the RPA equation in Sec. 5.1.

Decomposition of vector space
The whole vector space V, in which the stability matrix S is defined, can be decomposed via the basis vectors produced by N S or those by S N. This furnishes further discussion on properties of the RPA solutions.
Recalling the LR-duality explored in Sec. 3, we obtain the projector which separates out the direction along a certain basis vector of N S, as in Ref. [9], The projector separating out the subspace corresponding to the Jordan block (including the d ν = 1 case) generated from the eigenvector x ν is obtained by The projector Λ ν defines a subspace W ν , for which Λ ν W ν = W ν and (1 − Λ ν )W ν = ∅ (empty set) are satisfied. The completeness is expressed as In association with the LR-duality, one may consider The projector corresponding toW ν is given by N Λ † ν N. In order for the arguments in Sec. 3 to be applicable even after a certain projection, it is desired to respect the UL-duality, as well as the LR-duality. Therefore the UL-dual subspace is also considered, The projector relevant to Σ The subspace Σ xW * ν and its relevant projector are defined as well. Depending on ω ν , some of W ν ,W ν , Σ x W * ν and Σ xW * ν could be identical. We shall use a collective index [ν] to stand for their direct sum, apart from their overlap. The projector Λ [ν] on W [ν] is defined by sum of the projectors. Though in restricted cases, a similar projection is considered in Ref. [10]. It is straightforward to show the following properties of Λ [ν] , Of next interest is how the projection affects the RPA equation and the dualities.
Proof. From Eq. (26), Concerning and therefore The last equality follows because the quantity summed over ν does not depend on [ν ′ ].
Lemma 7. S [ν] inherits the symmetry properties of Eq. (7), Proof. The hermiticity of S [ν] is obvious from its definition. The second equation is proven as where (det S of degeneracy, particularly of Jordan blocks, has not been examined well, except several specific NG modes. Although degeneracy occurs even in physical solutions under presence of certain symmetry (e.g., degeneracy with respect to magnetic quantum numbers under the rotational symmetry), it does not give rise to Jordan blocks. This is obvious when the conservation law allows us to separate the RPA equation into the equations according to the quantum numbers. However, it is not trivial whether the same holds for a variety of extensive applications of the RPA. For instance, energy levels are highly degenerate in continuum, as in the continuum RPA [11]. Consideration of the degeneracy could be relevant to how we can take the continuous limit from arguments on discrete levels. For the NG mode, Thouless restricted himself to the case of two-dimensional Jordan blocks. While higher-dimensional blocks are not very likely to emerge in physical situations, it will be meaningful to distinguish physical situations from facts with rigorous mathematical proof.

Solutions for real eigenvalues
Let us first consider Classes (1) and (2) of Prop. 2.

Proposition 5. If the stability matrix S is positive-definite, the RPA equation is fully solvable. If the RPA equation is fully solvable, S is positive-definite.
Although the first part of this proposition was already proven in Ref. [8], I prove it again in combination with the second part.

13
Therefore the arguments on physical solutions by Thouless are applicable even under the presence of degeneracy as in the continuum.
For solutions of Class (2), a positive eigenvalue ω ν is accompanied by an eigenvector In the former case, its UL-dual partner is normalizable but corresponds to the eigenvalue −ω ν (< 0). The submatrix S [ν] of this solution is negativedefinite, as exemplified in Appendix C.1. Therefore the stability matrix S has two negative eigenvalues at least. In the x † ν N x ν = 0 case, x ν forms a Jordan block [12], whose UL-dual partner associated with −ω ν (< 0) belongs to another Jordan block. Probably for this reason, Thouless ignored this class of solutions, having focused on his arguments near the stability.
It is noted here that Re(ω ν ) = 0 covers the solutions of Class (5) as well as (3) in Prop. 2.
If Σ x x * ν = e −iφ x ν is assumed for the Re(ω ν ) = 0 case, the lower D-dimensional components of the RPA equation (5) is only a repetition of the upper D-dimensional components.
We can choose the phase e iφ arbitrarily, because it is controllable via a transformation Let us now take c This lemma states that solutions of Classes (3) and (5) can be self UL-dual, together with basis vectors generated from them.
Proof. In the case that Re(ω ν ) = 0, Eqs. (11) and (12) come From Lemma 8 and the argument above, we can assume Σ x ξ k , the first equation of (39) indicates that the second equation has a solution fulfilling Σ x ξ k+1 . The lemma is then proven inductively.
Compatibility of this lemma with Prop. 3 is confirmed in Appendix B.1. Under the above convention for the Re(ω ν ) = 0 case, the normalization condition of ξ although ξ Unlike the self LR-duality for real eigenvalues, the self UL-duality does not forbid Jordan blocks, although most pure-imaginary eigenvalues are expected not to form Jordan blocks.
An example of Jordan blocks is presented in Appendix C.4.
Let us turn to solutions of Class (4). Quartet solutions are a manifestation of the two types of dualities. The possibility of quartet solutions was first pointed out in Ref. [13] for S = S * cases, and mentioned in Ref. [6] in more general context. A minimal model for quartet solutions is constructed by taking D = 2, and is analyzed in Appendix C.3. For quartet solutions ν, d [ν] is a multiple of four. Hence, by denoting the solutions ±α ± iβ (α, β ∈ R), det S [ν] = det(N [ν] S [ν] ) = (α 2 + β 2 ) d [ν] /2 > 0. As S cannot be positive-definite on account of the latter part of Prop. 5, Eq. (34) indicates that S has at least two negative eigenvalues for quartet solutions to come out.

NG-mode solutions
The simplest example of the NG-mode solution is given in Appendix C.1, by the 2 × 2 stability matrix. It illustrates that the null eigenvalue is often associated with a twodimensional Jordan block, as indicated by Thouless [8]. The NG modes that generate twodimensional Jordan blocks have well been investigated [1,8,9]. However, in the example of Appendix C.1, there is a trivial case of S = 0 in which two d ν = 1 eigenvectors are present for the null eigenvalues. Moreover, an example of 4-dimensional Jordan block is seen in Appendix C.5. Likely or not, it is difficult to exclude the possibilities other than the twodimensional Jordan block for the null eigenvalue only from mathematical viewpoints. Proof. Because of Prop. 1, the number of non-zero eigenvalues must be even, up to their degeneracies. Moreover, Lemma 2 ensures that sum of dimensions of Jordan blocks for nonzero eigenvalues is even. The total dimension of N S is 2D, which concludes the degeneracy of the null eigenvalue must be even. Also proven from Lemma 6.
When SSB occurs, there must be NG-mode solutions corresponding to the broken symmetry; e.g., the linear momentum in the SSB with respect to the translation and the angular momentum in the SSB with respect to the rotation in deformed nuclei. For specific NG-mode solutions with such physical interpretations, their properties can further be explored, though I do not pursue this direction in this article.
The null eigenvalues may lie at the intersection of the self LR-and the self UL-dualities. Although there is no single eigenvector having both of the self dualities as indicated by Lemma 6, there could be an even-dimensional Jordan block in which the LR-duality closes by its basis vectors, while keeping the self UL-duality of Lemma 9. In such cases the Jordan block, instead of the basis vectors, may be said self LR-dual. Proposition 6. For even-dimensional Jordan blocks associated with a null eigenvalue, it is possible to produce basis vectors η k ; k = 1, · · · , d ν having double self duality, This proposition is proven in Appendix B.2. An example of the transformation shown in Appendix B.2 is given by the NG mode of the angular momentum, under SSB with respect to the rotation. Even though the Jordan blocks corresponding to J ± are the LR-dual of each other, their linear combinations provide those corresponding to J x and J y , each of which could be self LR-dual.
If d ν = odd, there should be two Jordan blocks, which are the partner of the LR-(or UL-) duality of each other. To separate them, one may apply the projector Λ [ν] introduced in Sec. 4.

Summary
Properties of solutions of the RPA equation is reanalyzed in terms of the linear algebra. As well as eigensolutions, cases in which the matrix N S (and S N) forms Jordan blocks are examined. Two types of dualities of eigenvectors and basis vectors, which are called LRand UL-dualities in this article, are pointed out and explored. These dualities are useful to clarify properties of the RPA solutions. Projection respecting the dualities is developed.
Eigenvalues given by the RPA equation are classified into five classes, in Prop. 2. As pointed out by Thouless, all solutions are physical ones if the stability matrix is positivedefinite. Its opposite is also true (in absence of NG modes), being useful to judge stability of a MF solution from numerical calculations in the RPA. These solutions are singled out, not constituting Jordan blocks, and have the self LR-duality while are paired by the UL-duality. Eigenvectors and basis vectors for pure-imaginary eigenvalues can be made self UL-dual, and paired by the LR-duality. With no self dualities, quartet solutions manifest two types of the dualities. NG-mode solutions, which are associated with the null eigenvalue and often related to the spontaneous symmetry breaking, lie at intersection of the two self dualities. However, a single vector cannot be both self LR-dual and self UL-dual. Only even-dimensional Jordan blocks can have double self dualities. The well-known prescription of separating out the NG modes could be applicable to such cases.

A.2 Proof of Lemma 4
Proof. Suppose ω * ν = ω ν ′ . In this case Eq. (A1) yields Corresponding to the k = 0 case in Eq. (A4), c k ′ can be non-zero only when ξ (ν ′ ) k ′ +1 does not exist. Namely, x ν = ξ  k with k ≥ 3 is also transformed, accordingly. The first part of the lemma is proven by repeating this argument.
The additional part of the lemma concerns the case in which plural Jordan blocks have an equal eigenvalue ω ν . If d ν > d ν ′ , it is concluded by continuing the above argument until can be non-zero, but ξ k ′ = 0 for any k and k ′ . The same holds for the d ν < d ν ′ case.

A.3 Proof of Prop. 3
Proof. Since the basis vectors of S N spans a complete set, any non-vanishing vector overlaps with at least one of them. Consider the case that some basis vectors of N S have overlap with plural basis vectors of S N. It is sufficient to consider that one of the basis vectors of a single Jordan block (or an eigenvector as a special case) ξ (ν) k ; k = 1, · · · , d ν have non-vanishing overlaps with members of two Jordan blocks (or two eigenvectors) of S N, N ξ (ν ′ ) k ′ ; k ′ = 1, · · · , d ν ′ and N ξ (ν ′′ ) k ′ ; k ′ = 1, · · · , d ν ′′ . From Lemmas 3 and 4, ω * ν = ω ν ′ = ω ν ′′ and d ν = d ν ′ = d ν ′′ . Consider a linear combination α k ′ ξ (ν ′ )

B =
a id id f . The matrix N S has eigenvalues ±id with the associating eigenvectors . This provides an example of Jordan blocks with dimension higher than two. There is only a single eigenvector,