Effective Hamiltonian in the extended Krenciglowa–Kuo method

Nuclear Physics A 864 (2011) 91–112 www.elsevier.com/locate/nuclphysa

Effective Hamiltonian in the extended Krenciglowa–Kuo method Kazuo Takayanagi Department of Physics, Sophia University, 7-1 Kioi-cho, Chiyoda-ku, Tokyo 102, Japan Received 22 April 2011; received in revised form 18 June 2011; accepted 21 June 2011 Available online 29 June 2011

Abstract The extended Krenciglowa–Kuo (EKK) method allows us to calculate the effective Hamiltonian in a nondegenerate model space. We show that the EKK method can be implemented numerically in two iterative schemes, which are explained in detail with emphasis on convergence conditions. Using test calculations in a simple model, we clarify how and on what conditions we can calculate the effective Hamiltonian. © 2011 Elsevier B.V. All rights reserved. Keywords: Effective Hamiltonian; Effective interaction; Non-degenerate model space; Krenciglowa–Kuo

1. Introduction In quantum many-body problems, the effective Hamiltonian has been one of the central issues especially in the field of nuclear physics [1–5]. A huge Hilbert space is indispensable normally to describe a quantum many-body system, which is then divided into a model space (P -space) of a tractable size and its complement (Q-space). Then the question to be answered is the following; how can we calculate the effective Hamiltonian H eff in the chosen P -space, that is designed to reproduce exact eigenenergies of the full Hamiltonian H and projections of the true eigenstates onto the P -space? The effective Hamiltonian H eff = H0 + V eff is composed of the unperturbed Hamiltonian H0 and the effective interaction V eff . Two iterative methods have been used for more than three decades to calculate V eff , i.e., the Krenciglowa–Kuo (KK) [6] and the Lee–Suzuki (LS) [7] E-mail address: [email protected]. 0375-9474/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2011.06.025

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methods. Both of these methods assume that the P -space unperturbed energies are completely degenerate. This is really a strong restriction; in a shell model calculation of 18 O within the P -space of a single major (1s0d) shell, for example, we cannot use the empirical single particle energies of 1s1/2 , 0d5/2 , 0d3/2 states that are not degenerate. The assumption also hinders us from incorporating two or more major shells in the P -space in a harmonic oscillator shell model calculation. Quite naturally, generalizations of the KK and the LS methods to the non-degenerate P -spaces have been proposed [8,9] using the so-called multi-energy Q-box. However, these methods are rather complicated for practical calculations, and have not been widely used. In this situation, by generalizing the KK and the LS methods, we have proposed the extended Krenciglowa–Kuo (EKK) and the extended Lee–Suzuki (ELS) methods, both of which apply quite easily to non-degenerate P -spaces [10]. The generalization is most naturally presented by setting up an iterative method to revise H eff rather than V eff at each step of iteration. This cannot be, however, the whole story. In order to make the best use of these methods in non-degenerate P -spaces, we are still in need of a clear understanding of the methods especially from a practical point of view. It is, therefore, highly desirable to have a detailed investigation of practical schemes to carry out numerical calculations in the EKK and the ELS methods. In the market of effective interaction, the KK method has been used more widely than the LS method because of its numerical feasibility. In this work, we examine, therefore, the EKK method which replaces the original KK method, and present the following results. First, we show that eff → H eff , of the EKK method can be implemented in practice in two the iterative step, Hn−1 n schemes; one iterative scheme (EKKS scheme) makes use of a series expansion of Hneff in terms eff , and the other (EKKD scheme) diagonalizes H eff , where H eff being H eff at the nth of Hn−1 n n−1 step of iteration. Second, we investigate convergence conditions of these two schemes in detail, and clarify how they work and what conditions are required in practical calculations. The plan of this paper is the following. In Section 2, we explain the effective Hamiltonian to fix the notation. In Section 3, the EKK method is reviewed briefly. Then in Section 4, we derive eff (EKKD the effective Hamiltonian H eff in the EKK method using the diagonalization of Hn−1 scheme). We also present the convergence condition of the iterative scheme. Next in Section 5, we calculate H eff in the EKK method using the Taylor series expression for Hneff (EKKS scheme). Here we discuss the convergence condition of the Taylor series. These two (EKKD and EKKS) numerical schemes are contrasted with each other in Section 6. Then in Section 7, we present test calculations to refine as well as visualize the discussions in Sections 4, 5, and 6. We shall see clearly how and on what conditions we can use these two numerical schemes to calculate H eff . Finally in Section 8, we present a brief summary. 2. Effective Hamiltonian In this section, we briefly review the concept of the effective Hamiltonian and its formal derivation using the notation of Ref. [10]. 2.1. Model space We describe a many-body system in a Hilbert space of dimension D with the following Hamiltonian: H = H0 + V ,

(1)

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where H0 is the unperturbed Hamiltonian and V is the perturbation. Then the system is described by the following Schrödinger equation: H |Ψλ  = Eλ |Ψλ ,

λ = 1, . . . , D.

(2)

Now we divide the whole Hilbert space into a model space (P -space) of tractable dimension d and its complement (Q-space). Let us denote the projection operators onto these spaces as P and Q, respectively, which satisfy P 2 = P , Q2 = Q, and P Q = QP = 0. Here we require that the P -space be spanned by a set of d eigenstates of H0 , which means [H0 , P ] = [H0 , Q] = 0,

P H Q = P V Q,

QH P = QV P .

(3)

Accordingly, we split the eigenstate |Ψλ  in Eq. (2) into the P - and Q-space components as |Ψλ  = P |Ψλ  + Q|Ψλ  = |φλ  + |Φλ ,

λ = 1, . . . , D.

(4) H eff

We can now state our task as follows; we want to derive the effective Hamiltonian in the d-dimensional P -space that describes |φλ  = P |Ψλ  as its eigenstate with eigenenergy Eλ . 2.2. Energy-dependent effective Hamiltonian H BH (E) Projecting Eq. (2) onto the P - and Q-spaces, we can easily derive the following equation for |φλ : H BH (Eλ )|φλ  = Eλ |φλ ,

λ = 1, . . . , D,

(5)

where we have defined the energy-dependent Bloch–Horowitz Hamiltonian [11,12] H BH (E) as H BH (E) = P H P + P V Q

1 QV P . E − QH Q

(6)

Eq. (5) requires that the eigenenergy Eλ and the corresponding eigenstate |φλ  be determined self-consistently, because H BH (Eλ ) itself depends on the eigenenergy Eλ . Recently, a big step has been made in this direction by Suzuki et al. [13]. They have proposed an efficient method to solve Eq. (5) both iteratively and non-iteratively, which has proven to be useful in practical use [14]. In this work, however, we take another way that we believe is very convenient; instead of attacking the eigenvalue equation (5) of the energy-dependent Hamiltonian H BH (E) directly, we introduce the energy-independent effective Hamiltonian H eff in the next section. 2.3. Energy-independent effective Hamiltonian H eff A straightforward argument [10] based on the similarity transformation [7] shows that H eff can be constructed as follows. First, we determine the (D − d) × d matrix ω by QV P + QH Qω − ωP H P − ωP V Qω = 0,

(7)

which is referred to as the decoupling equation. Second, by defining the d × d matrix H eff = P H P + P V Qω,

(8)

we can easily confirm H eff |φp  = Ep |φp ,

p = 1, . . . , d,

(9)

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showing that H eff defined in Eq. (8) is nothing but the energy-independent effective Hamiltonian in the d-dimensional P -space that we are looking for. Two points should be noticed on the decoupling equation (7) and its solution ω. First, ω is useful not only for calculating H eff of Eq. (8), but also for other purposes. For example, once we obtain the P -space component, |φp  = P |Ψp , of the true eigenstate |Ψp  by solving Eq. (9), we can express the corresponding Q-space component, |Φp  = Q|Ψp , as |Φp  = ω|φp , if necessary. In this work, however, we shall not be interested in the Q-space component, and will not use an explicit form of ω. Second, depending on how we solve the decoupling equation (7), being nonlinear, we obtain different solutions of ω which then lead to different results for H eff of Eq. (8). Therefore, the set of the eigenstates {|φp , p = 1, . . . , d} of H eff in Eq. (9) depends on how we solve Eq. (7). 3. EKK method In this section, we briefly explain the extended Krenciglowa–Kuo (EKK) method to solve Eq. (7) to give the effective Hamiltonian H eff of Eq. (8) in the d-dimensional P -space [10]. We also explain on what conditions the EKK iterative method converges to give H eff , and which eigenstates {|φp , p = 1, . . . , d} are described by the obtained H eff . 3.1. Definition of EKK method Solving the decoupling equation (7), which is nonlinear, necessarily requires an iterative method, that is specified by a relation between ωn and ωn−1 , where ωn means ω at the nth step. The EKK method is defined by the following set of equations: QV P + QH Qωn − ωn P H P − ωn P V Qωn−1 = 0, Hneff

= P H P + P V Qωn .

(10) (11)

Eq. (10) specifies the iterative method by fixing ωn implicitly in terms of ωn−1 , and gives a solution ω to the decoupling equation (7) in an evident way in the limit of ωn → ω, if the iteration converges. Eq. (11) defines Hneff in terms of ωn , and eventually the effective Hamiltonian H eff of Eq. (8) as H eff = lim Hneff . n→∞

(12)

In Sections 4 and 5, we give two practical schemes to solve Eqs. (10) and (11). We will see that both schemes combine Eq. (10), that defines implicitly the iterative step ωn−1 → ωn for ω, eff → H eff , for H eff . Both of the numerical with Eq. (11), to give an explicit iterative step, Hn−1 n schemes are so formulated that they can be implemented without using ωn explicitly. This is firstly because we do not need the explicit form of ω, and secondly because H eff is a d × d matrix of a tractable size while ω is usually a much bigger (D − d) × d matrix. Here we make a comment on other iterative methods to calculate H eff . Another iterative eff → H eff , is given by the extended Lee–Suzuki (ELS) method [10]. The ELS converstep, Hn−1 n gence condition is different from the EKK convergence condition that we examine in this work, and therefore the eigenstates of H eff in the ELS method are different from those of the EKK method. It is also possible to make an explicit iterative step, ωn−1 → ωn , for ω that works in non-degenerate P -spaces [15]. For these methods, interested readers are referred to the original works. Note also that our EKK method is different from the EKK method proposed in Ref. [8] which addresses Eq. (5) directly.

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3.2. Convergence condition for EKK method Eqs. (10) and (11) define the EKK method completely. Therefore, before going into practical schemes to carry out the EKK method, we had better present here the condition to guarantee that eff → H eff , converges to give H eff of Eq. (12). Here we just go quickly the EKK iterative step, Hn−1 n through the derivation of the condition, which is explained in detail in the original work [10]. Let us write the nth approximation ωn as ωn = ω + δωn , and correspondingly Hneff = H eff + δHneff . We start with writing the convergence condition for the EKK method as follows:     eff δH |φp  < δH eff |φp , p = 1, . . . , d. (13) n n−1 Noting that |δHneff |φp | = |(Hneff − H eff )|φp | is the norm of the error at the nth step in reproducing Eq. (9), we see that the inequality (13) is a natural way to express the convergence condition. Then, after some manipulations, we can convert the condition (13) into ρP =

Ψp |P |Ψp  1 > , Ψp |Ψp  2

p = 1, . . . , d,

(Condition (A))

which means that the probability ρP of the P -space components in |Ψp  — which is reproduced by H eff — is larger than one half. We call such states as the P -space dominant states in the following. Here we make several points on Condition (A). First, Condition (A) is a direct consequence of Eqs. (10) and (11), and is therefore an inherent condition of the EKK method defined by Eqs. (10) and (11). In other words, no matter how we solve Eqs. (10) and (11), Condition (A) is required. Second, Condition (A) should read as follows; if there are d (P -space dominant) eigenstates {|Ψp , p = 1, . . . , d} that satisfy Condition (A), the EKK method certainly gives H eff that reproduces these d states. Third, Condition (A) cannot always be satisfied, i.e., a system may not support d of the P -space dominant states in general. In such a case, the EKK method does not converge, and fails to give H eff . Such a case shall be shown in Section 7. Fourth, to be strict mathematically, Condition (A) is neither necessary nor sufficient to guarantee the convergence eff → H eff [10]. However, we shall see in Section 7.2 that the of the EKK iterative step Hn−1 n convergence of the EKK method is well under the control of Condition (A). 4. EKKD — EKK by diagonalization In this section, we derive a practical scheme to implement the EKK method defined by Eqs. (10) and (11). The numerical scheme is based on the diagonalization of Hneff at each step, and is referred to as the EKKD (EKK by diagonalization) scheme in what follows. 4.1. Derivation of EKKD scheme Using Eq. (11), we first write Eq. (10) as eff ωn Hn−1 − QH Qωn = QV P .

(14)

eff at hand. Then by solving the P -space Schrödinger equation (9) at the Suppose we have Hn−1 (n − 1)th step,     eff  (n−1) φp = Ep(n−1) φp(n−1) , p = 1, . . . , d, (15) Hn−1

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K. Takayanagi / Nuclear Physics A 864 (2011) 91–112 (n−1)

(n−1)

eff . Multiplying |φ we have the set of d eigenstates, {|φp , p = 1, . . . , d}, of Hn−1 p Eq. (14), we arrive immediately at     1 QV P φp(n−1) , p = 1, . . . , d. ωn φp(n−1) = (n−1) Ep − QH Q (n−1)

Then, using Eq. (11) for Hneff , we can derive the following expression for Hneff |φp       1 eff  (n−1) Hn φp = P H P + P V Q (n−1) QV P φp(n−1) Ep − QH Q  (n−1)  (n−1)  BH φ E . =H p

p

 by (16)

:

(17)

In the last line, we have used the energy-dependent Bloch–Horowitz Hamiltonian H BH (E) of Eq. (6). (n−1) , p = 1, . . . , d} of Eq. (15) is not an orthogonal set in general, we define its Because {|φp (n−1)  , p = 1, . . . , d} that satisfies biorthogonal set {|φp   p(n−1) φ (n−1) φ = δp,p , p



P=

d

 (n−1)  (n−1)  φ .  φ p

p

(18)

p=1

Using the completeness relation in Eq. (18), we can convert Eq. (17) into the following expression for Hneff : Hneff =

d

   (n−1)  . p H BH Ep(n−1) φp(n−1) φ

(19)

p=1 eff → H eff that starts from ω = 0 and H eff = Eq. (19) defines the EKKD iterative scheme Hn−1 0 n 0 P H P . When the iteration converges, we obtain the following expression for H eff :

H eff =

d

p |, H BH (Ep )|φp φ

(20)

p=1

which proves Eq. (9) by itself because of Eq. (5). 4.2. Convergence condition for EKKD scheme In Section 4.1, we have derived the EKKD scheme (19) directly from Eqs. (10) and (11) without any assumptions, which is in contrast to the derivation based on a Taylor series expansion presented in Ref. [10]. We can conclude, therefore, that the convergence condition of the EKKD scheme is given by Condition (A) only; if the system satisfies Condition (A), the EKK method of Eqs. (10) and (11) is to give a convergent solution Hneff → H eff , which can certainly be obtained numerically by the EKKD scheme. To put it another way, the EKKD scheme surely converges to give H eff that describes the d of the P -space dominant eigenstates of H , if they exist. 5. EKKS — EKK by series summation In this section, we derive another numerical scheme to implement the EKK method of calculating H eff . The scheme is based on the Taylor series expansion of Hneff , and is referred to as the EKKS (EKK by series summation) scheme in what follows.

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5.1. Derivation of EKKS scheme Let us first write Eq. (14) as

 eff  (m − QH Q)ωn = QV P − ωn Hn−1 −m ,

(21)

where we have introduced an arbitrary parameter m — which was set to be zero in Ref. [10] — eff , Eq. (21) is a linear equation of which the meaning shall soon become clear. For a fixed Hn−1 for ωn , and can be solved formally as ωn =

∞

(−1)k k=0

 eff k 1 QV P Hn−1 −m , k+1 (m − QH Q)

(22)

which can be confirmed easily by a direct substitution in Eq. (21). Substituting Eq. (22) for ωn in Eq. (11), we arrive immediately at Hneff = H BH (m) +

∞

  k (m) H eff − m k , Q n−1

(23)

k=1

where we have used that the Bloch–Horowitz Hamiltonian of Eq. (6) satisfies H BH (m) =  and we have defined the so-called Q-box and its derivatives as P H0 P + Q(m),  Q(E) = PVP + PVQ

1 QV P , E − QH Q

(24)

k k (E) = 1 d Q(E)  Q k! dE k

1 QV P , k = 1, 2, . . . . (25) (E − QH Q)k+1 In actual calculations, we usually calculate the Q-box using a perturbation theory based on diagrammatic expansions [3]. It is pointed out that there may be qualitative differences between the exact Q-box and approximate ones [16]. In this work, however, we just assume that we can calculate the exact Q-box and its derivatives. Eq. (23) defines the EKKS scheme to calculate Hneff starting from H0eff = P H P . When the iteration converges to give Hneff → H eff , Eq. (23) reduces to the following expression for H eff : = (−1)k P V Q

H eff = H BH (m) +

∞

  k (m) H eff − m k . Q

(26)

k=1

It should be stressed here that, when the EKKS series expression (23) for Hneff is convergent, it is equivalent to the EKKD expression (19) for Hneff , and therefore that the EKKS and the EKKD eff → H eff defined by the EKK method. This can schemes represent the same iterative step Hn−1 n (n−1)

eff and its eigenstates {|φ } in Eq. (15) be easily proven as follows. Suppose we have Hn−1 p eff eff at the (n − 1)th step of the EKKD scheme. Now we define Hn in terms of the EKKD Hn−1 using the EKKS iterative step of Eq. (23), and see if it is identical with what we would obtain by the EKKD iterative step of Eq. (19). Let us multiply |φp(n−1)  by Eq. (23), where we use the eff in the right-hand side. Then, using Eq. (15), we realize that the right-hand side is a EKKD Hn−1 (n−1)

Taylor series expansion of H BH (Ep by assumption. We thus obtain

(n−1)

)|φp

(n−1)

 in powers of Ep

− m, which is convergent

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     Hneff φp(n−1) = H BH Ep(n−1) φp(n−1) ,

p = 1, . . . , d,

(27)

which coincides with Eq. (17), and therefore we immediately arrive at the EKKD expression (19) for Hneff . The above observation clearly shows that the EKKS iterative step (23) creates the same Hneff as the EKKD step (19). This is sufficient to show that these two schemes give the same sequence {Hneff , n = 0, 1, 2, . . .}, because the starting point, H0eff = P H P , is the same. Note that Eq. (23) is seemingly dependent on the parameter m, and that Eq. (19) is not. Because these two expressions (19) and (23) are proven to be identical in the above, Eq. (23) is in fact independent of m, if it converges. This can be also explained by the above observation; (n−1) m stands for the origin of the Taylor series expansion of H BH (Ep ), and the sum of the Taylor series, of course, is independent of the origin of expansion, if the series is convergent. Then, natural questions arise. First, it might seem that the EKKS scheme (23) would be redundant because it coincides with the EKKD scheme (19) when it converges. This is not, however, true. Eq. (23) is a useful expression for Hneff in its own right because of its numerical feasibility as shall be explained in Section 6. Second, what is the role of m then? The answer to this question shall be found later in several contexts. Here we just point out the following; because m is the origin of the Taylor series expansion of H BH (Ep(n−1) ), m should be, in a sense, close to the set of target eigenenergies {Ep , p = 1, . . . , d} of the EKK method to guarantee the convergence of the Taylor series. In the following, we translate the above statement into a mathematically clear condition. 5.2. Convergence condition for EKKS scheme We have seen in Section 5.1 that, if the series in Eq. (23) is convergent, the EKKS scheme eff → H eff as the EKKD scheme. This means that the congenerates the same iterative step Hn−1 n vergence of the EKKS scheme is, in addition to Condition (A), that the series in Eq. (23) be convergent. A necessary and sufficient condition for the convergence of the series in Eq. (23) is that the matrix norm, ωn , of ωn defined by Eq. (22) is finite. The matrix norm is explained in Appendix A, and the following discussion holds for any matrix norm (Euclidean or spectral or any other norm). Taking the matrix norm of Eq. (22), we see that ωn  is bounded from the above as follows (see Appendix A): k+1 ∞

eff k 1 ωn   QV P  · · Hn−1 − m . (28) m − QH Q k=0

Then a sufficient condition for ωn  being finite is that the geometric series in the right-hand side of Eq. (28) is convergent, i.e., eff 1 (29) m − QH Q · H − m < 1, where we have taken the limit of n → ∞. Let us note here that, using the inequality (A.2) in Appendix A, we have 1 1 , (30)  min |m − εq | m − QH Q max |Ep − m|  H eff − m , (31)

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where εq and Ep represent eigenenergies of QH Q and H eff , respectively. Then we realize that a necessary condition for the inequality (29) is    Ep − m    (Condition (B))  ε − m  < 1, for any p, q. q

This means that all of the EKK target eigenenergies {Ep , p = 1, . . . , d} should be closer to m than any of the eigenenergies {εq , q = d + 1, . . . , D} of QH Q. Two points should be noted on Condition (B). First, it might seem that Condition (B) would not be very useful because εq is not an eigenenergy of H . This is not, however, the case. In Section 7.3, we show that Condition (B) has a clear interpretation in terms of the so-called intruder states. Second, mathematically speaking, the above derivation shows that Condition (B) is neither necessary nor sufficient for the convergence of the series in Eq. (23). However, we shall see in Section 7.3 that Condition (B) controls the convergence of the series very well. At the end of this section, we emphasize that Conditions (A) and (B) are different by nature; eff → H eff defined by Condition (A) represents an inherent feature of the EKK iterative step Hn−1 n Eqs. (10) and (11), which is independent of how we solve Eqs. (10) and (11). On the other hand, Condition (B) certifies the convergence of the series expression (23) for Hneff , which is specific to the EKKS scheme of solving Eqs. (10) and (11). 6. Comparison of EKKD and EKKS schemes The EKK method applies to almost all the systems that satisfy Condition (A), meaning that the systems have d of the P -space dominant (ρP > 1/2) eigenstates. Having shown that it can be carried out by the two numerical schemes EKKD and EKKS, we compare these two schemes to clarify the sharp contrast to each other. The EKKD scheme is summarized as follows. First, in going from the (n − 1)th step eff (see to the nth step, one has to diagonalize the non-Hermitian effective Hamiltonian Hn−1 Eq. (15)), which is sometimes quite inconvenient. Second, one has to calculate the biorthogop , p = 1, . . . , d} using a matrix inversion technique (see Eq. (18)), which is also nal set {|φ (n−1) time-consuming for a P -space of large dimension d. Third, one has to calculate H BH (Ep )  p(n−1) )) at different energies for each eigenstate at each step of (and therefore the Q-box, Q(E iteration (see Eq. (19)), which is very difficult to perform with good numerical accuracy. Fourth, though the EKKD scheme has the above numerical inconvenience, it requires for its convergence Condition (A) only, i.e., it certainly converges to give H eff that describes d eigenstates {|φp , p = 1, . . . , d} that satisfy Condition (A), if they exist. Next, the EKKS scheme is summarized as follows. First, in going from the (n − 1)th step eff − m)k , to the nth step, one just needs to implement matrix multiplications to calculate (Hn−1 which is obviously very easy (see Eq. (23)). Second, the EKKS scheme incorporates an arbitrary parameter m. In Section 7.3.2, it shall be shown that, by optimizing m, one can considerably k (m), is calculated at a accelerate the convergence of the series in Eq. (23). Third, the Q-box, Q fixed value of m (see Eq. (23)) for all the steps of iteration, which is very convenient in practical calculations. Fourth, though there being the above many advantages, the EKKS scheme requires Condition (B), in addition to Condition (A), for its convergence. In Section 7.3, we examine Condition (B) in detail, and we shall see that the EKKS scheme works only on the condition that there are no intruder states. In the above, we have enumerated advantages and disadvantages of the EKKD and EKKS schemes. We have stressed, in particular, that the EKKD scheme applies to the wider class of

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Hamiltonians, and that the EKKS scheme has the better numerical feasibility. At the end of Section 7, after presenting numerical results of both schemes, we explain the criteria for deciding which scheme to use. 7. Calculations and discussions In this section, we make discussions on the two numerical schemes EKKD and EKKS using test calculations. After explaining the model Hamiltonian in Section 7.1, we study the EKKD and the EKKS schemes in Sections 7.2 and 7.3, respectively, putting special emphasis on convergence conditions. Finally in Section 7.4, we compare these two schemes, and show which scheme we should use on what conditions. 7.1. Model We adopt the model Hamiltonian studied in Ref. [10] with minor modifications. We take a four-dimensional (D = 4) Hilbert space that is spanned by the orthonormal basis vectors {|λ, λ = 1, 2, 3, 4}. Our Hamiltonian is H = H0 + V , where H0 = diag[2, 3, 7, 9] in the above basis. The two-dimensional (d = 2) P -space is spanned by {|1, |2}, and therefore the P -space is non-degenerate. The perturbation V is given by the following form with a strength parameter x: ⎛ ⎞ −5 5 −4 5 ⎜ 5 12 4 −8 ⎟ ⎟ V =x⎜ (32) ⎝ −4 4 −23 1 ⎠ . 5 −8 1 −5 By diagonalizing H = H0 + V , we obtain four eigenstates which we denote as {|Ψλ , λ = 1, 2, 3, 4} with eigenenergies E1 < E2 < E3 < E4 . Here the eigenstates {|Ψλ } are normalized to unity, i.e., Ψλ |Ψμ  = δλ,μ , and they reduce to the unperturbed states as lim |Ψλ  = |λ,

x→0

λ = 1, 2, 3, 4.

(33)

7.2. EKKD calculations Here we present numerical results of the EKKD scheme and make discussions on Condition (A). Section 7.2.1 is devoted to a system that satisfies Condition (A). We shall see how the condition specifies which eigenstates of H be reproduced by H eff . Then in Section 7.2.2, we examine a system that does not satisfy Condition (A). We shall see that the EKK method does not give H eff for such a system. 7.2.1. Study of Condition (A) — being satisfied In Fig. 1, eigenenergies Eλ for the interaction strength 0.0  x  0.2 are shown by solid curves. We see that the second and the third lowest levels (|Ψ2  and |Ψ3 ) show a level crossing around x ∼ 0.11. This can be confirmed by Fig. 2 which shows the probability amplitudes μ|Ψλ , λ, μ = 1, 2, 3, 4. It is visible that the dominant components of |Ψ2  and |Ψ3  are exchanged at x ∼ 0.11. In Fig. 2, we also show the P -space probability ρP of each state, which is defined for normalized |Ψλ  as ρP = Ψλ |P |Ψλ .

(34)

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Fig. 1. Open squares represent eigenenergies of H eff obtained by the EKKD scheme as functions of the strength parameter x. Solid curves show exact eigenenergies of the full Hamiltonian H .

Fig. 2. Probability amplitudes μ|Ψλ  for λ, μ = 1, 2, 3, 4. Lines are distinguished by index μ as denoted in each panel. Open triangles show the P -space probability ρP of Eq. (34) for each state.

Fig. 2 clearly shows that |Ψ1  and |Ψ2  are the P -space dominant (ρP > 1/2) states for x  0.11, while |Ψ1  and |Ψ3  have ρP > 1/2 for x  0.11; we have two (= d) P -space dominant eigenstates for any value of 0  x  0.2.

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Fig. 3. Left: Eigenenergies of H (solid lines), and of H eff in the EKKD scheme (open squares). EKKD scheme converges to give H eff for x  0.13. Right: P -space probabilities ρP of the six (= D) eigenstates of H . Note that ρP of |Ψ2  crosses the horizontal dashed line (ρP = 0.5) at x ∼ 0.13.

Let us recall Condition (A) in Section 3.2. Then the above observation indicates that our system satisfies Condition (A) in the whole range 0.0  x  0.2 in the figure. This in turn means that H eff of the EKK method certainly exists for 0.0  x  0.2 and can be obtained by the EKKD numerical scheme. Condition (A) also tells that the obtained H eff describes {|Ψ1 , |Ψ2 } for x  0.11, and {|Ψ1 , |Ψ3 } for x  0.11. This is confirmed numerically in Fig. 1, where eigenenergies of H eff in the EKKD scheme are plotted by open squares. In general, the EKKD scheme surely converges to give H eff whenever the system satisfies Condition (A), as explained in Section 4.2. Because Condition (A) is expressed in terms of ρP , the P -space probability of eigenstate, we can derive different forms of H eff by choosing different P -spaces. Suppose we define our P space as {|1, |3}. Then the EKKD scheme leads to H eff that describes {|Ψ1 , |Ψ3 } for x  0.11, and {|Ψ1 , |Ψ2 } for x  0.11, which can be easily confirmed by numerical calculations. 7.2.2. Study of Condition (A) — not being satisfied Here we consider a system that does not satisfy Condition (A). We can prepare such a system most easily as follows. Suppose that we have a small P -space in a very large Hilbert space (d D), and that the coupling between the P - and Q-spaces is not weak. Then we naively expect that the P -space components would spread over D ( d) eigenstates of H , and consequently that Condition (A) cannot be satisfied by any set of d eigenstates of H . For such a system, the EKK method (and therefore the EKKD numerical scheme) would not converge. In Fig. 3, we present an example of such a system. We have made the Hamiltonian of the system (D = 6, d = 2) by adding two Q-space states {|5, |6} to our model with slight modifications to the perturbation V , which is not explicitly presented here. We show the eigenenergies {Eλ , λ = 1, . . . , 6} of H and the EKKD results in the left panel, and the P -space probabilities ρP of the eigenstates, {|Ψλ , λ = 1, . . . , 6}, in the right panel. For x  0.13, we see in the right panel that two (= d) eigenstates {|Ψ1 , |Ψ2 } have ρP > 1/2, and therefore the system satisfies Condition (A). Consequently, the EKK method (and the EKKD numerical scheme) should converge for x  0.13 to give H eff that reproduces their eigenenergies {E1 , E2 } and their P -space projections {|φ1 , |φ2 } correctly. This is indeed the case as can be confirmed in the left panel. For x  0.13, on the other hand, only a single state |Ψ1  has ρP > 1/2, meaning that Condition (A) is not met by the system, and therefore that the EKK method (and the EKKD scheme) fails to converge, as can be confirmed in the figure.

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We may express the essence of Sections 7.2.1 and 7.2.2 as follows; try the EKKD numerical scheme, and see whether the calculation converges or not. Convergence means that the system satisfies Condition (A) to give H eff that describes d of the P -space dominant eigenstates, while non-convergence means that the system does not satisfy Condition (A) and that the EKK method cannot yield H eff . 7.3. EKKS calculations Let us turn to the EKKS scheme which requires Condition (B), in addition to Condition (A), to assure the convergence of the series expression (23) for Hneff . First in Section 7.3.1, we show how Condition (B) is realized in numerical calculations. Then in Section 7.3.2, we clarify the role played by the arbitrary parameter m in Condition (B), and show how to determine an optimal value of m. In Section 7.3.3, we present a simple interpretation of Condition (B) in terms of an intruder state. We also show that Condition (B) can be replaced conveniently with a much simpler criterion. Finally in Section 7.3.4, we investigate several examples to achieve a clear understanding of the convergence conditions. 7.3.1. Study of Condition (B) Here we examine Condition (B) — that depends on the arbitrarily chosen parameter m — using the model Hamiltonian in Section 7.1. In Fig. 4, we plot eigenenergies {εq , q = 3, 4} of QH Q together with {Eλ , λ = 1, 2, 3, 4} of H . Here Condition (A) specifies that the target eigenenergies {Ep } of the EKK method are {E1 , E2 } for x  0.11, and {E1 , E3 } for x  0.11, respectively, as explained in Section 7.2.1. They are obtained by the EKKD scheme, and are presented by open squares. Because ε4 is far away from the target eigenenergies, ε3 only is relevant in Condition (B), which reads as      E2 − m   E1 − m    < 1,   (35)  ε − m  < 1, for x  0.11,  ε −m  3 3      E1 − m   E3 − m      (36)  ε − m  < 1,  ε − m  < 1, for x  0.11. 3 3 Let us look at the first inequality in (35) for x  0.11. For a fixed value of x, with E1 and ε3 being fixed, the inequality specifies the range of m, i.e., m should be closer to E1 than to ε3 for x  0.11. This condition requires m < (E1 + ε3 )/2, which restricts m to the area below the dashed line, m = (E1 + ε3 )/2, which extends from (0.0, 4.5) to (0.11, 2.8) with a downward arrow. In the same way, the second inequality in (35) means m < (E2 + ε3 )/2, which is the area below the dashed curve extending from (0.0, 5.0) to (0.11, 4.3) with an attached downward arrow. Therefore, for x  0.11, Condition (B) is satisfied by m in the intersection of the above two areas, which is the region with slashes in Fig. 4. Next, let us turn to the inequalities in (36) for x  0.11. We can proceed in the same way as in the above, and find the following; the first inequality means m < (E1 + ε3 )/2, requiring m to be in the region below the dashed line extending from (0.12, 2.7) to (0.2, 1.2) with a downward arrow. The second inequality, which reads m > (E3 + ε3 )/2, restricts m to the region above the dashed line extending from (0.12, 4.5) to (0.2, 3.8) with an arrow directed upwards. The above observation shows that the two inequalities are satisfied in the two separate areas that have no intersection. This means that the condition (36) cannot be satisfied by any value of m, and that the EKKS scheme cannot give H eff for x  0.11.

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Fig. 4. Explanation of Condition (B) in the EKKS scheme. Solid curves represent eigenenergies {Eλ , λ = 1, 2, 3, 4} of the full Hamiltonian H , and circles indicate eigenenergies {εq , q = 3, 4} of QH Q as functions of the strength parameter x. Open squares show the target eigenenergies of H eff in the EKK method, and are obtained by the EKKD scheme. Four dashed lines with attached arrows denote areas of m that are specified by the four inequalities in (35) and (36). The slashed area shows the region of m that satisfies all of the four inequalities. For details, see the text.

The above investigation of Condition (B), represented by (35) and (36), is summarized as follows; in order to satisfy Condition (B) for a given value of x, m has to be in the slashed area in Fig. 4. Now we present numerical results of the EKKS calculation for m = 0.0, 2.0, 3.0, 4.0 in Fig. 5. We plot eigenenergies obtained by H eff in the EKKS scheme by solid squares. We have also shown the slashed area obtained in Fig. 4 and the horizontal dashed line representing m. Let us first look at m = 0.0 case shown in the upper left panel. We see that the EKKS scheme gives H eff that reproduces E1 and E2 for x  0.11, while it fails to give H eff for x  0.11. Now we increase m from m = 0.0. The above situation remains the same when we reach m = 2.0, the upper right panel. However, as we increase m further (beyond m ∼ 2.8) the convergence region of x decreases gradually as shown in the lower panels; at m = 3.0 (lower left panel) the convergence is achieved for x  0.09, and at m = 4.0 (lower right panel) only for x  0.03. For m  4.5, convergence cannot be attained for any value of x > 0. The previous argument with Fig. 4 clearly explains what we have observed in the above; in each panel of Fig. 5, the intersection of the horizontal line of m and the slashed area specifies the convergence region of x. For example, in case of m = 2.0 in the upper right panel, the line m = 2.0 is in the slashed area for x  0.11, which is indeed the area where the EKKS scheme converges to give H eff . Let us take another example, m = 4.0, in the lower right panel. The intersection of the line m = 4.0 and the slashed area is x  0.03, which agrees very well with the convergence area of x. The above observation shows clearly that Condition (B) specifies the convergence area almost exactly for any value of m. At the end, let us look into the above discussion from a practical point of view. Suppose that the system satisfies Condition (A), and that {Ep , p = 1, . . . , d} are to be reproduced by H eff in the EKK method. Here we want to know whether the EKKS series expression (23) for Hneff converges or not, and what value of m we may use. The argument in this section may be summarized by the following prescription; we compare, for each value of x, eigenenergies {Ep , p = 1, . . . , d} of H and {εq , q = d + 1, . . . , D} of QH Q, and determine the range of m that satisfies Condition (B). This is, however, obviously very inconvenient because εq is an

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Fig. 5. Realization of Condition (B) in the EKKS scheme. Eigenenergies of H eff in the EKKS scheme are plotted by solid squares for m = 0.0, 2.0, 3.0, 4.0. Horizontal dashed lines represent m in the corresponding panels. Other notation is the same as for Fig. 4. In each panel, the convergence region of x in the EKKS scheme is specified by the intersection of the slashed area and the horizontal dashed line of m. For the explanation, see the text.

eigenenergy of QH Q, but not of H . In Section 7.3.3, therefore, we replace Condition (B) with a much simpler condition. 7.3.2. Role of m in Condition (B) Here we clarify the role played by m in Condition (B), and explain what would be the optimal value of m for a given Hamiltonian. Fig. 5 suggests that any value of m  2.8, including negative values, would give H eff for x  0.11, the largest possible area of x in the EKKS scheme. This is indeed the case. Then natural questions arise; what do we get by tuning m? What is the role played by m in Condition (B)? We now answer these questions using Fig. 6. In the left panel of Fig. 6, we show the number of terms in the right-hand side of Eq. (26) that are necessary to achieve convergence of the series with relative accuracy of 10−6 (in the sense of the Euclidean matrix norm). The calculation is performed for several values of m that are denoted in the figure. For example, we see that 32 terms are necessary for m = 0.0 to assure the convergence at x = 0.08, while for m = 2.5 only 15 terms are necessary to achieve the convergence at the same point x = 0.08. This means that, by increasing m from 0.0 to 2.5, we can double the speed of convergence of the series in Eq. (26) at x = 0.08. However, as we increase m further from 2.5, the convergence of the series becomes slower, and the convergence region of x shrinks as explained in Figs. 4 and 5. The numerical results in the left panel show

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Fig. 6. Convergence properties of the EKKS scheme for several values of m denoted in the figure. Left: number of terms in the right-hand side of Eq. (26) that are necessary to guarantee the convergence with 10−6 accuracy. Right: number n of iterative steps for which Hneff converges to H eff with 10−6 accuracy. All the lines with different values of m fall on the EKKD results (open squares). Arrows indicate the largest value of x for which the EKKS calculation converges at the corresponding value of m. Note that the EKKS scheme converges in the range 0.0  x  0.11 for m = 0.0, 2.0, 2.5, while it converges in the reduced range 0.0  x  0.09 for m = 3.0, for example, as seen in both panels.

that the optimal value of m is given by m ∼ 2.5 for the whole range 0.0  x  0.11, in the sense that the convergence of the series in Eq. (26) is fastest for m ∼ 2.5. It is visible in the left panel that every line rises rapidly as it approaches its edge point, where one of the fractions, |Ep − m|/|εq − m|, in Condition (B) approaches unity. This in turn means that the geometric ratio (left-hand side of Eq. (29)) of the series in Eq. (28) might be close to unity, which explains that the convergence of the series in Eq. (26) becomes very slow. The above observation also clarifies the meaning of the optimal value of m, i.e., it minimizes the fraction, |Ep −m|/|εq −m|, in Condition (B) on the average to give the fastest convergence of the series in Eq. (26). Clearly, the most promising candidate for the optimal m is m = (max Ep + min Ep )/2, i.e., the middle of the spectrum {Ep , p = 1, . . . , d} of the P -space dominant states. We thus arrive at the following condition to determine m for a given Hamiltonian; optimal m in the middle of the spectrum of {Ep }.

(Condition (B ))

In practice, there is a finite range of m that satisfies Condition (B). Therefore, we had better search for the optimal value of m by trial and error starting from the middle of the spectrum {Ep , p = 1, . . . , d}. The numerical results in the left panel confirm the above consideration; the optimal value suggested in the left panel, m ∼ 2.5, coincides with the middle, (E1 + E2 )/2, of the spectrum {Ep = E1 , E2 } for 0.0  x  0.11 as can be confirmed in Fig. 4. We could have thus expected, only from the knowledge of the spectrum {Ep = E1 , E2 }, that the EKKS scheme would require least terms in Eq. (23) for m ∼ (E1 + E2 )/2 ∼ 2.5. Let us turn to the right panel of Fig. 6, where lines show the number n of iterative steps for which Hneff of Eq. (23) converges to H eff with relative accuracy of 10−6 . All the lines with different values of m overlap, and the arrows indicate the largest value of x for which the EKKS scheme converges at the corresponding value of m. For example, we see that the EKKS scheme with m = 3.0 converges for 0.0  x  0.09, in accordance with the lower left panel of Fig. 5 and the left panel of Fig. 6. We also plot the results of the EKKD scheme by open squares. Let us take m = 3.0 and x = 0.08 as an example. The EKKD squares show that the expression (19) eff approximates H eff with 10−6 accuracy. In the same way, the EKKS line for m = 3.0 for Hn=9

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eff approximates H eff with the same accuracy at x = 0.08. shows that the expression (23) for Hn=9 As the above example shows, the interesting point in the right panel is that all the EKKS lines fall on the EKKD results in the whole range of x, irrespective of the value of m. This is a direct consequence of the fact that both of the EKKD and EKKS numerical schemes generate the same eff → H eff defined by Eqs. (10) and (11), as stressed in Section 5.1. EKK iterative step Hn−1 n eff → H eff , of the The right panel also shows that the convergence of the iterative step, Hn−1 n EKK method (and consequently of the EKKD and EKKS schemes) is slowest at the level crossing point (x ∼ 0.11) between |Ψ2  and |Ψ3 . This is because the P -space probabilities, ρP , of these two states are competitive in Condition (A) at the level crossing point; the numerical scheme requires many iterative steps to determine which eigenstate to describe. In this section, we have clarified the role of m in Condition (B); we can accelerate the convergence of the series in Eq. (23) by using Condition (B ), i.e., by choosing m properly in the middle of the target spectrum {Ep , p = 1, . . . , d} specified by Condition (A). However, the number of eff → H eff , necessary to achieve the required accuracy is already the EKKS iterative steps, Hn−1 n specified by the EKK method of Eqs. (10) and (11), and is independent of m. In short, m controls the speed of convergence of the series in the right-hand side of Eq. (23), but not of the iterative eff → H eff defined by Eq. (23). step Hn−1 n At the end, let us note that the above argument to optimize m by Condition (B ) has assumed the existence of m that satisfies Condition (B), which is difficult to verify as it stands. Therefore, our next job is to find a condition that works conveniently in place of Condition (B), which we establish in the next section.

7.3.3. Interpretation of Condition (B) Here we revisit the argument in Section 7.3.1, and derive a convenient criterion for convergence that replaces Condition (B). Let us describe the spectrum in Fig. 1 in the following way; for x  0.11 the first and the second lowest levels {|Ψ1 , |Ψ2 } are the P -space dominant (ρP > 1/2) states, while for x  0.11 the first and the third levels {|Ψ1 , |Ψ3 } are P -space dominant. This means that, for x  0.11, there is an intruder state |Ψ2  inside the spectrum of the P -space dominant EKK target states {|Ψ1 , |Ψ3 }, i.e., E1 (target) < E2 (intruder) < E3 (target),

for x  0.11.

(37)

In order to study Condition (B), let us compare ε3 with E2 and E3 in Fig. 4. A notable point is that ε3 is a monotonous function of x, while E2 and E3 repel each other at the level crossing point x ∼ 0.11 because of the interaction between the P - and Q-spaces. Consequently, we have ε3 ∼ E3 for x  0.11, and ε3 ∼ E2 for x  0.11. Note that this is a general feature at the level crossing where two eigenstates exchange their wave function components. The above observation shows that the inequality (37) for x  0.11 can be replaced with E1 (target) < ε3 < E3 (target),

for x  0.11.

(38)

Now let us recall the discussion on the allowed range of m for x  0.11 in Fig. 4. In the situation of the inequality (38), Condition (B) leads to two separate regions of m — one region of m bounded from the above, m < (E1 (target) + ε3 )/2, and another from the below, m > (E3 (target) + ε3 )/2 — which have no intersection, meaning that the EKKS scheme does not converge for any value of m. We can summarize the above argument as follows; the presence of an intruder in Fig. 1 is expressed by the inequality (37), which is then replaced with the inequality (38) using the general

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Fig. 7. Eigenenergies of H eff in the EKKS scheme (solid squares), and of H (solid lines). Left: EKKS calculation with m = 7.0 for the P -space of {|3, |4}. Right: EKKS calculation with m = 5.0 for the P -space of {|2, |3}. Other notation is the same as for Fig. 4. See the text for explanations.

structure of level crossing. Because Condition (B) is not satisfied in the situation of (38), we can conclude that the EKKS scheme does not work in the presence of an intruder. The above discussion can be generalized easily; the presence of an intruder means a failure of Condition (B), and vice versa. The absence of an intruder, therefore, proves that none of {εq } is inside the spectrum of {Ep }, showing that there is certainly a range of m that satisfies Condition (B). We thus obtain the following condition to guarantee the convergence of the series in Eq. (23): no intruders inside the spectrum of {Ep }.

(Condition (B ))

If and only if Condition (B ) is satisfied, we can find m that satisfies Condition (B), and therefore we can obtain H eff in the EKKS scheme. In short, Condition (B ) replaces Condition (B). At the end, we stress that Condition (B ) is written in terms of eigenenergies of H only, and is much more convenient than Condition (B) that requires eigenenergies {εq } of QH Q. Let us look into Fig. 1, for example. We can easily identify an intruder for x  0.11 by the spectrum {Eλ , λ = 1, 2, 3, 4} of H alone as in (37), without the knowledge of the spectrum {εq } of QH Q. Therefore, we could have expected from the spectrum of H alone that the EKKS scheme would converge to give H eff only for x  0.11 even with an optimal choice of m, as can be confirmed in Fig. 5. 7.3.4. Study of Condition (B ) In order to have a clear understanding of Condition (B ), we study the Hamiltonian in Section 7.1 with several choices of the P -space. In the following, we explain the detailed study using Fig. 7. In the left panel, we show the EKKS calculation with m = 7.0 for the P -space defined as {|3, |4}. We see that the EKKS calculation converges to give {Ep = E3 , E4 } only for 0.0  x  0.1. This can be explained, and could have been expected, by Condition (B ); from the spectrum of {Eλ , λ = 1, 2, 3, 4} and the definition of our P -space, we see that the EKK target spectrum is {Ep = E3 , E4 } for 0.0  x  0.1, and {Ep = E2 , E4 } for x  0.1. We thus realize that, for x  0.1, E3 is an intruder level inside the target spectrum {Ep = E2 , E4 }. Then Condition (B ) immediately explains that H eff in the EKKS scheme can be obtained, with an optimal choice of m, for 0.0  x  0.1 where there are no intruders.

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Now we explain the optimal choice of m. In the left panel, we have determined the optimal value of m using Condition (B ) at x = 0.08, i.e., we have taken (E3 (x = 0.08) + E4 (x = 0.08))/2 ∼ 7.0 for m. Strictly speaking, we should have determined the optimal value of m using Condition (B ) at each x. However, we have used the fixed value of m in the whole range of x to simplify the figure, because we can easily verify that the EKKS calculation with m = 7.0 converges not only for x = 0.08, but also in the largest possible range 0.0  x  0.1, as shown in Fig. 7. This can be verified most easily using Condition (B); the left panel clearly shows that m = 7.0 is closer to {Ep = E3 , E4 } than to {εq = ε1 , ε2 } for 0.0  x  0.1, and therefore Condition (B) guarantees that the EKKS calculation with m = 7.0 converges for 0.0  x  0.1. Let us turn to the right panel which presents the EKKS calculation for the P -space defined as {|2, |3}. We have fixed m = 5.0 using Condition (B ) at x = 0.0, which is used in the whole range of x because of the same reason as for the left panel. We can see that the EKKS calculation converges to reproduce {Ep = E2 , E3 } in the whole range 0.0  x  0.2. This can be explained by observing that the system satisfies Condition (B ) for 0.0  x  0.2, i.e., there are no intruders inside the target spectrum {Ep = E2 , E3 } of the P -space dominant states. Note that the level crossing at x ∼ 0.11 between |Ψ2  and |Ψ3  does not create an intruder inside the spectrum {Ep = E2 , E3 }, and therefore that Condition (B ) is satisfied for x  0.11 as well as for x  0.11. In order to compare Conditions (B) and (B ), let us examine the right panel on the basis of Condition (B). It is clear that we could have used Condition (B) to expect the numerical results; we can see that m = 5.0 is closer to {Ep = E2 , E3 } than to {εq = ε1 , ε4 } for 0.0  x  0.2, showing that the system satisfies Condition (B). Let us vary m in the right panel. If we choose m = 4.0, we find that the convergence is achieved only for 0.04  x  0.2. This is understandable because m = 4.0 is closer to ε1 than to E3 for 0.0  x  0.04, indicating that the system does not meet Condition (B) for 0.0  x  0.04. In the same way, for m = 6.0, the EKKS scheme converges only for 0.0  x  0.13, which is explained easily with Condition (B), i.e., m = 6.0 is closer to ε4 than to E2 for x  0.13. Though both Conditions (B) and (B ) can explain the numerical results, the contrast between them is remarkable; Condition (B ) tells the convergence area of x using eigenenergies {Eλ } of H only, while Condition (B) requires eigenenergies {εq } of QH Q which are not available in usual systems. Let us conclude this section by the following statements. We have shown a practical way of utilizing Condition (B) and choosing the optimal value of m for a given Hamiltonian; first require Condition (B ), and then determine m by Condition (B ). Combined Conditions (B ) and (B ) conveniently replace Condition (B) which is not very practicable as it stands. 7.4. EKKD or EKKS? In Section 6, we have explained (i) that the EKKS scheme is much more feasible numerically than the EKKD scheme, and (ii) that the EKKS scheme needs Condition (B) in addition to Condition (A) for its convergence, while the EKKD scheme requires Condition (A) only. In Section 7.3, we have understood that Condition (B) can be replaced with combined Conditions (B ) and (B ). We can now answer the question: which scheme should we use on what conditions? The proper use of the EKKD and the EKKS schemes is the following. First of all, we have to assure that the system satisfies Condition (A) so that the EKK method works. This can be

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verified numerically by the EKKD scheme as explained at the end of Section 7.2.2, if necessary. Then two cases should be considered separately. If Condition (B ) is satisfied, i.e., if there is no intruder inside the target spectrum {Ep , p = 1, . . . , d} of the P -space dominant states, the EKKS scheme is recommendable with m determined by Condition (B ). On the other hand, if Condition (B ) is not satisfied, we have to use the EKKD scheme. At the end, we might ask a question; what can we do if the system does not satisfy Condition (A)? Even for such a system, the extended Lee–Suzuki (ELS) method [10] works to give H eff , though the EKK method is not available irrespective of the numerical scheme we use. 8. Summary The extended Krenciglowa–Kuo (EKK) method is devised to give an effective Hamiltonian H eff that describes projections of the true eigenstates of a full Hamiltonian H onto the model eff → H eff , space (P -space) of tractable size d. The method is featured by its iterative step Hn−1 n where Hneff being H eff at the nth step of iteration. The EKK method works in non-degenerate as well as degenerate P -spaces, in contrast to the original Krenciglowa–Kuo (KK) method that works in degenerate P -spaces only. In this work, we have shown that the EKK iterative method can be implemented in practice eff to express H eff , and is referred in two schemes; one is based on the diagonalization of Hn−1 n to as the EKKD (EKK by diagonalization) scheme. The other is based on the series expansion eff − m, where m is an arbitrary parameter, and is called as the of Hneff − m in powers of Hn−1 EKKS (EKK by series summation) scheme. By examining convergence conditions of these two numerical schemes, we have clarified the following points, which are then visualized by test calculations at the end. First, H eff in the EKK method describes P -space projections of the true eigenstates of H whose P -space probabilities are more than one half (P -space dominant states). The EKK iteraeff → H eff certainly converges to give H eff whenever the system does support d of tive step Hn−1 n the P -space dominant eigenstates. eff → H eff without any Second, the EKKD scheme can carry out the EKK iterative step Hn−1 n assumption. Therefore, if there are d of the P -space dominant eigenstates of H , the EKKD scheme certainly converges to give H eff that describes these d states; the EKKD scheme applies to a very wide class of Hamiltonians. However, it requires diagonalization of non-Hermitian eff and very accurate evaluation of Q-box, and may not be very practicable for a P matrix Hn−1 space of very large dimension. Third, the EKKS scheme is numerically much more feasible than the EKKD scheme. Further, by making an optimal choice of the parameter m, one can considerably accelerate the convergence of the series expression for Hneff . However, the EKKS scheme requires that there be no intruder states inside the spectrum of the P -space dominant states that the EKK method is to describe. Fourth, we have given the clear answer to the following question: which scheme should we use on what conditions? In the absence of intruders, the EKKS scheme is recommendable. In the presence of intruders, use the EKKD scheme. Having derived two numerical schemes to calculate H eff in the EKK method, and having clarified how and on what conditions we should use them, we believe that the present work has made the concept of the effective Hamiltonian ready for immediate use in non-degenerate general model spaces.

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Appendix A. Matrix norm Here we explain the matrix norm briefly. Comprehensive explanations can be found in Refs. [17,18] for square matrices, which can be generalized to rectangular matrices easily. A brief explanation on the norm of rectangular matrices is found in Ref. [19]. Let A and B be m × n and n × q complex matrices. Then a matrix norm  ·  is defined by the following axioms. (1)

A  0,

A = 0 if and only if A = 0.

(2)

αA = |α|A,

(3)

A + B  A + B.

(4)

AB  AB.

α is any complex number. (A.1)

Note that the axiom (1) means that, in any matrix norm, A → 0 means A → 0, and vice versa. There are many different matrix norms in the market that satisfy all of the above four axioms. However, the following inequality for the square (n × n) matrix A holds for any matrix norm: A  μ(A),

(A.2)

where μ(A) = max |λi |, λi being an eigenvalue of A, is referred to as the spectral radius of A. The proof goes as follows. Let λ be an eigenvalue of A such that μ(A) = |λ|, and x be the corresponding eigenvector (column vector) satisfying Ax = λx. Defining the n × n matrix Ax = [x 0 0 . . . 0],

(A.3)

we observe that AAx = λAx .

(A.4)

By taking the matrix norm of both sides, using the axioms (2) and (4), we arrive at A · Ax   |λ| · Ax .

(A.5)

Since Ax = 0 and therefore Ax  = 0 because of the axiom (1), we obtain the desired result A  |λ| = μ(A).

(A.6)

Now we explain some of the equations in the text. First, the axioms (3) and (4) are sufficient to explain the inequality in (28). Second, the inequalities in (30) and (31) follow immediately from the inequality (A.2) in the above. References [1] [2] [3] [4] [5] [6] [7] [8] [9]

B.H. Brandow, Rev. Mod. Phys. 39 (1967) 771. P.J. Ellis, E. Osnes, Rev. Mod. Phys. 49 (1977) 777. T.T.S. Kuo, E. Osnes, Springer Lecture Notes in Physics, vol. 364, Springer-Verlag, 1990. M. Hjorth-Jensen, T.T.S. Kuo, E. Osnes, Phys. Rep. 261 (1995) 125. D.J. Dean, T. Engeland, M. Hjorth-Jensen, M.P. Kartamyshev, E. Osnes, Progress in Particle and Nuclear Physics 53 (2004) 419. E.M. Krenciglowa, T.T.S. Kuo, Nucl. Phys. A 235 (1974) 171. K. Suzuki, S.Y. Lee, Prog. Theor. Phys. 64 (1980) 2091. K. Suzuki, R. Okamoto, P.J. Ellis, T.T.S. Kuo, Nucl. Phys. A 567 (1994) 576. T.T.S. Kuo, K. Suzuki, F. Krmpoti´c, R. Okamoto, Nucl. Phys. A 582 (1995) 205.

112

[10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

K. Takayanagi / Nuclear Physics A 864 (2011) 91–112

K. Takayanagi, Nucl. Phys. A 852 (2011) 61. C. Bloch, Nucl. Phys. 6 (1958) 329. C. Bloch, J. Horowitz, Nucl. Phys. 8 (1958) 91. K. Suzuki, R. Okamoto, H. Kumagai, S. Fujii, Phys. Rev. C 83 (2011) 024304. H. Dong, T.T.S. Kuo, J.W. Holt, arXiv:1105.4169 [nucl-th]. F. Andreozzi, Phys. Rev. C 54 (1996) 684. P.J. Ellis, T. Engeland, M. Hjorth-Jensen, A. Holt, E. Osnes, Nucl. Phys. A 573 (1994) 216. P. Lancaster, M. Tismenetsky, The Theory of Matrices, second edition, Academic Press, 1985. R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, 1985. G.H. Golub, C.F. van Loan, Matrix Computations, third edition, The Johns Hopkins University Press, 1996.