effective interaction in non-degenerate model space

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Nuclear Physics A 852 (2011) 61–81 www.elsevier.com/locate/nuclphysa Effective interaction in non-degenerate model space Kazuo Takayanagi Department of Physics, Sophia University, 7-1 Kioi-cho, Chiyoda-ku, Tokyo 102, Japan Received 12 November 2010; accepted 4 January 2011 Available online 8 January 2011 Abstract The effective interaction in a model space has been calculated by the Krenciglowa–Kuo (KK) and the Lee–Suzuki (LS) iterative methods, both of which assume that the unperturbed energies in the model space are degenerate. We generalize these two methods in a natural and simple manner so that they apply also to non-degenerate model spaces. The key to the generalization is to use the effective hamiltonian instead of the effective interaction in the formulation of iterative schemes. Using test calculations in a simple model, we demonstrate that the new methods work excellently. © 2011 Elsevier B.V. All rights reserved. Keywords: Effective interaction; Effective hamiltonian; Non-degenerate model space; Krenciglowa–Kuo; Lee–Suzuki 1. Introduction In quantum many-body problems, especially in the field of nuclear physics, the concept of the effective hamiltonian has been, and will continue, playing a very important role [1–5]. Many- body systems necessarily require Hilbert spaces of huge dimensions for their description. We divide, therefore, the whole Hilbert space into a model space (P -space) of a controllable size and its complement (Q-space). Then we search for the effective hamiltonian H eff in the chosen P -space, that is designed to reproduce exact eigenenergies of the full hamiltonian H and the projections of true eigenstates onto the P -space. The problem is how to calculate such H eff . The effective hamiltonian H eff is written as a sum of the unperturbed hamiltonian H 0 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] methods. Both of these methods assume that the P -space unperturbed energies are completely degenerate. This E-mail address: [email protected]. 0375-9474/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2011.01.003

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Page 1: Effective interaction in non-degenerate model space

Nuclear Physics A 852 (2011) 61–81

www.elsevier.com/locate/nuclphysa

Effective interaction in non-degenerate model space

Kazuo Takayanagi

Department of Physics, Sophia University, 7-1 Kioi-cho, Chiyoda-ku, Tokyo 102, Japan

Received 12 November 2010; accepted 4 January 2011

Available online 8 January 2011

Abstract

The effective interaction in a model space has been calculated by the Krenciglowa–Kuo (KK) and theLee–Suzuki (LS) iterative methods, both of which assume that the unperturbed energies in the model spaceare degenerate. We generalize these two methods in a natural and simple manner so that they apply also tonon-degenerate model spaces. The key to the generalization is to use the effective hamiltonian instead ofthe effective interaction in the formulation of iterative schemes. Using test calculations in a simple model,we demonstrate that the new methods work excellently.© 2011 Elsevier B.V. All rights reserved.

Keywords: Effective interaction; Effective hamiltonian; Non-degenerate model space; Krenciglowa–Kuo; Lee–Suzuki

1. Introduction

In quantum many-body problems, especially in the field of nuclear physics, the concept of theeffective hamiltonian has been, and will continue, playing a very important role [1–5]. Many-body systems necessarily require Hilbert spaces of huge dimensions for their description. Wedivide, therefore, the whole Hilbert space into a model space (P -space) of a controllable sizeand its complement (Q-space). Then we search for the effective hamiltonian H eff in the chosenP -space, that is designed to reproduce exact eigenenergies of the full hamiltonian H and theprojections of true eigenstates onto the P -space. The problem is how to calculate such H eff.

The effective hamiltonian H eff is written as a sum of the unperturbed hamiltonian H0 and theeffective interaction V eff. Two iterative methods have been used for more than three decades tocalculate V eff, i.e., the Krenciglowa–Kuo (KK) [6] and the Lee–Suzuki (LS) [7] methods. Bothof these methods assume that the P -space unperturbed energies are completely degenerate. This

E-mail address: [email protected].

0375-9474/$ – see front matter © 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.nuclphysa.2011.01.003

Page 2: Effective interaction in non-degenerate model space

62 K. Takayanagi / Nuclear Physics A 852 (2011) 61–81

is, however, far from being reality; for example, single particle energies of 1s1/2,0d5/2,0d3/2states are not degenerate, and therefore the 1s0d shell is not a degenerate P -space for 18O. Inthis situation, it is strongly desirable to set up an effective hamiltonian in the non-degenerateP -space.

An obvious way to incorporate the non-degeneracy of the P -space is to introduce an artificialenergy shift in the unperturbed hamiltonian H0 to prepare a completely degenerate P -space [2].This is, however, quite inconvenient; interacting single particle states do not have the energiesobserved by experiments, and therefore it becomes difficult to derive simple pictures of physics.Another method is to write the P -space as a direct sum of subspaces defined by eigenstates ofH0 with possibly different eigenenergies [8]. In this scheme, at the cost of introducing projectionoperators on these subspaces separately, we arrive at an expression for V eff in the non-degenerateP -space. However, the above cost seems too high, and this scheme has not been used widely.

In this situation, we generalize the KK and the LS schemes in this work so that they applyto the non-degenerate P -space in a surprisingly natural way. The new iterative schemes are assimple as the original KK and the LS schemes, and can be implemented at the same computingcost as the original ones. We shall see that the new schemes revise H eff at each step of iteration,while the original schemes revise V eff.

The plan of this paper is the following. In Section 2, we explain the description of the modelspace. In Section 3, we introduce the energy-dependent effective hamiltonian HBH of Bloch andHorowitz. Then in Section 4, we explain the energy-independent effective hamiltonian H eff andthe decoupling equation to fix the notation. Here we present, for comparison, all of the fourequations that define the iterative schemes which we discuss in this paper; two of them lead tothe well known KK and LS schemes and the other two are their generalizations. In Section 5, webriefly review the standard KK and LS methods in the degenerate P -space, to be contrasted tothe new methods. In Sections 6 and 7, by generalizing these two methods, we propose two newiterative methods to calculate H eff. The derivation is designed to clarify their similarities anddifferences with the original KK and LS methods. The obtained expressions for H eff manifestclearly that the new methods are generalizations of the standard KK and LS methods to non-degenerate P -spaces. We also discuss the convergence conditions of these new iterative methodsand their implications. In Section 8, we present test calculations to show that the new methodswork in an excellent fashion. Finally in Section 9, we present a brief summary.

2. Model space

We describe a many-body system in a Hilbert space of dimension D with the following hamil-tonian:

H = H0 + V, (1)

where H0 is the unperturbed hamiltonian and V is the perturbation. Then the system is describedby the following Schrödinger equation:

H |Ψλ〉 = Eλ|Ψλ〉, λ = 1, . . . ,D. (2)

By diagonalizing the full hamiltonian H in the whole Hilbert space, we obtain D solutions toEq. (2). In many cases, however, the required task exceeds the current computer capacity, andmore importantly, we are not interested in all of the D eigenstates. We divide, therefore, thewhole 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,

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K. Takayanagi / Nuclear Physics A 852 (2011) 61–81 63

which satisfy P 2 = P , Q2 = Q, and PQ = QP = 0. We usually make the most convenientchoice for the P -space; we require that the P -space be spanned by a set of d eigenstates of H0.Then our choice means

[H0,P ] = [H0,Q] = 0, PHQ = PV Q, QHP = QV P. (3)

Accordingly, the eigenstate |Ψλ〉 in Eq. (2) is split into the P - and Q-space components as

|Ψλ〉 = P |Ψλ〉 + Q|Ψλ〉 = |φλ〉 + |Φλ〉, λ = 1, . . . ,D. (4)

Now our task is to derive the effective hamiltonian H eff in the d-dimensional P -space thatdescribes |φλ〉 = P |Ψλ〉 as its eigenstate with eigenenergy Eλ.

3. Energy-dependent effective hamiltonian

Here we solve Eq. (2) for |φλ〉 = P |Ψλ〉, to arrive at the energy-dependent effective hamilto-nian of Bloch and Horowitz [9,10].

Let us write Eq. (2) in the following block form(PHP PHQ

QHP QHQ

)( |φλ〉|Φλ〉

)= Eλ

( |φλ〉|Φλ〉

), λ = 1, . . . ,D. (5)

We can easily solve the above equation for |φλ〉 to obtain

HBH(Eλ)|φλ〉 = E|φλ〉, λ = 1, . . . ,D, (6)

where we have defined, using Eq. (3), the energy-dependent Bloch–Horowitz hamiltonianHBH(E) as

HBH(E) = PHP + PV Q1

E − QHQQV P. (7)

Eq. (6) requires that the eigenenergy Eλ and the corresponding eigenstate |φλ〉 be determinedself-consistently, because HBH(Eλ) itself depends on the eigenenergy Eλ. It should be notedhere that the d-dimensional equation (6) describes all of the D eigenstates of H .

This is not, however, our goal; we are to derive an energy-independent H eff, which we estab-lish in the following.

4. Energy-independent effective hamiltonian

Here we derive the energy-independent effective hamiltonian H eff by the similarity transfor-mation following Ref. [7]. In Section 4.1, we explain the decoupling equation that leads to H eff.In Section 4.2, we present a formal solution to the decoupling equation, and also a formal ex-pression for H eff. In Section 4.3, we enumerate all of the four iterative schemes to calculate H eff

that we discuss in this work.

4.1. Decoupling equation and H eff

The simplest way to derive H eff is to use the similarity transformation by [7]

S =(

1ω 1

), S−1 =

(1

−ω 1

), (8)

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64 K. Takayanagi / Nuclear Physics A 852 (2011) 61–81

which are expressed in the block form as in Eq. (5), and ω = QωP is a (D − d) × d matrixwhich we will specify shortly. Then the similarity transformation with the above S converts H

into H = S−1HS, and the corresponding Schrödinger equation is readily obtained as(P HP P HQ

QHP QHQ

)( |φλ〉|Φλ〉 − ω|φλ〉

)= Eλ

( |φλ〉|Φλ〉 − ω|φλ〉

). (9)

Note that the similarity transformation does not change the eigenenergy Eλ. Let us now chooseω that cancels the lower left submatrix QHP in Eq. (9), i.e., we specify ω by

QHP = QV P + QHQω − ωPHP − ωPV Qω = 0, (10)

which is referred to as the decoupling equation. Then Eq. (9) clearly shows that an eigenvalueEλ of H (and therefore of H ) is an eigenvalue of either P HP or QHQ, which we explain inorder in the following.

Let us start with d eigenenergies {Ep,p = 1, . . . , d} of P HP , for which the decoupling con-dition (10) decouples Eq. (9) to give

P HP |φp〉 = Ep|φp〉, p = 1, . . . , d, (11)

|Φp〉 = ω|φp〉, p = 1, . . . , d. (12)

Eq. (11) clearly reveals that

H eff = P HP = PHP + PV Qω, (13)

is the energy-independent effective hamiltonian H eff in the P -space that we are looking for. Werewrite, therefore, Eq. (11) as

H eff|φp〉 = Ep|φp〉, p = 1, . . . , d. (14)

Because H eff of Eq. (13) is an energy-independent d-dimensional matrix, it has d eigenstates asdenoted in Eq. (14). Here the normalization of |φp〉 is still open. In this work, we normalize |φp〉(not |Ψp〉) to unity, i.e., 〈φp|φp〉 = 1. Once the P -space component |φp〉 = P |Ψp〉 is determinedby Eq. (14), the Q-space component |Φp〉 = Q|Ψp〉 is obtained from Eq. (12). The above dis-cussion is summarized as follows; first, we solve Eq. (10) for ω to derive H eff of Eq. (13). ThenEq. (14) gives d eigenstates |φp〉 of H eff. The corresponding Q-space component |Φp〉 is thendetermined by Eq. (12), if necessary.

It is customary to define the effective interaction

V eff = PV P + PV Qω, (15)

to write Eq. (13) as

H eff = PH0P + V eff. (16)

In the following, we use both notions of H eff and V eff.Now we briefly explain the rest of the eigenenergies, i.e., the D − d eigenenergies of QHQ.

By applying the above similarity transformation to the hermitian conjugate of Eq. (5), we in-stantly arrive at the following decoupled set of equations:

〈Φq |QHQ = Eq〈Φq |, q = d + 1, . . . ,D, (17)

〈φq | = −〈Φq |ω, q = d + 1, . . . ,D, (18)

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K. Takayanagi / Nuclear Physics A 852 (2011) 61–81 65

where

QHQ = QHQ − ωPV Q, (19)

is referred to as the effective hamiltonian in the Q-space. The above equations show the follow-ing: given a solution ω to Eq. (10), we obtain QHQ of Eq. (19), which in turn determines 〈Φq |as its left eigenstate by Eq. (17). Then the corresponding P -space component 〈φq | can be cal-culated by Eq. (18). In this work, however, we will not be interested in these D − d eigenstates,and will not solve Eq. (17) in practice.

In the above discussion, we have denoted the eigenenergies of H eff = P HP as {Ep,p =1, . . . , d}, and those of QHQ as {Eq,q = d + 1, . . . ,D}. It should be pointed here that thedecoupling equation (10) is a nonlinear equation, and in general has many different solutions.Therefore the set of eigenenergies {Ep,p = 1, . . . , d} of H eff depends on the choice of ω. Inother words, different solutions of ω lead to different results for H eff that describe differentsubsets of eigenstates of H .

4.2. Formal solution

It is helpful here to note the following formal expression for ω. First, let us choose d eigen-states {|Ψp〉,p = 1, . . . , d} of H that we are going to describe. Then our H eff should describetheir projections onto the P -space, |φp〉 = P |Ψp〉, by Eq. (14). Because {|φp〉,p = 1, . . . , d} isnot an orthogonal set in general, we define its biorthogonal set {|φ̃p〉,p = 1, . . . , d} that satisfies

〈φ̃p|φp′ 〉 = δp,p′ , P =d∑

p=1

|φp〉〈φ̃p|. (20)

Second, by taking the projection of Eq. (2) for {|Ψp〉,p = 1, . . . , d} onto the Q-space, we obtain

|Φp〉 = 1

Ep − QHQQV P |φp〉 = ω(Ep)|φp〉, p = 1, . . . , d, (21)

where we have defined ω(Ep) in an obvious way. Then a comparison of Eqs. (12) and (21) leadsimmediately to the following formal expression for ω:

ω =d∑

p=1

ω(Ep)|φp〉〈φ̃p| =d∑

p=1

|Φp〉〈φ̃p|, (22)

where we have used the completeness relation in Eq. (20). Eq. (22) shows explicitly that ω isdefined by the set of eigenstates {|Ψp〉,p = 1, . . . , d} that we choose first.

The above formal expression for ω is extremely useful. As its application, let us derive aformal expression for H eff. By substituting Eq. (22) for ω in Eq. (13), we arrive at

H eff =d∑

p=1

HBH(Ep)|φp〉〈φ̃p|, (23)

which is expressed in terms of HBH defined in Eq. (7). The above expression for H eff is explicitlycompatible with Eq. (6), and shall be used repeatedly in the derivation of H eff. Note, however,that Eq. (23) is not a very useful expression in actual calculations; one has to solve Eq. (6) before

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66 K. Takayanagi / Nuclear Physics A 852 (2011) 61–81

writing down H eff of Eq. (23). In Section 4.3, we introduce iterative schemes to obtain H eff

without explicitly solving Eq. (6).As another application of Eqs. (21) and (22), we derive the norm of |Ψp〉 for later use. Suppose

that we solve Eq. (14) for |φp〉 = P |Ψp〉, requiring that |φp〉 be normalized to unity. Then Eq. (21)shows that 〈Ψp|Q|Ψp〉 = 〈Φp|Φp〉 can be obtained from the normalized |φp〉 as

〈Ψp|Q|Ψp〉 = ⟨φp|ω†(Ep)ω(Ep)|φp

⟩ = ⟨φp

∣∣∣∣PV Q1

(Ep − QHQ)2QV P

∣∣∣∣φp

⟩= −⟨

φp

∣∣Q̂1(Ep)∣∣φp

⟩, p = 1, . . . , d, (24)

where Q̂1(Ep) in the last line will be explained shortly in Eq. (39). With the above knowledge of〈Ψp|Q|Ψp〉, we can easily obtain the norm of |Ψp〉 as

√〈Ψp|Ψp〉 = (1 + 〈Ψp|Q|Ψp〉)1/2. ThenEq. (24) allows us to calculate, for example, the Q-space probability of |Ψp〉 as

ρQ = 〈Ψp|Q|Ψp〉〈Ψp|Ψp〉 = 〈Ψp|Q|Ψp〉

1 + 〈Ψp|Q|Ψp〉 , p = 1, . . . , d. (25)

4.3. Iterative schemes

Now our task is to solve Eq. (10) to obtain ω, and to calculate the effective hamiltonian H eff

of Eq. (13). Because Eq. (10) is a complicated nonlinear equation, we have to resort to iterativemethods to obtain the solution. An iterative scheme is specified by a relation between ωn andωn−1, where ωn means ω at the nth step. In this work, we discuss the following four iterativeschemes:

QV P + QHQωn − ωnε0 − ωnPV P − ωnPV Qωn−1 = 0, (26)

QV P + QHQωn − ωnε0 − ωn−1PV P − ωn−1PV Qωn = 0, (27)

QV P + QHQωn − ωnPH0P − ωnPV P − ωnPV Qωn−1 = 0, (28)

QV P + QHQωn − ωn−1PH0P − ωn−1PV P − ωn−1PV Qωn = 0. (29)

Each of the above iterative schemes is designed to give a solution ω to the decoupling equa-tion (10) in an evident way in the limit of ωn → ω, if the iteration converges. Now we explainthe above relations in order.

Eqs. (26) and (27) define the KK [6] and the LS [7] schemes, respectively. In these twoschemes, the degenerate P -space is assumed, and the following relation is used

PH0P = ε0P, (30)

where ε0 is the degenerate P -space energy. Eqs. (28) and (29) do not assume the degenerateP -space condition (30), and shall be shown to lead to iterative schemes in the non-degenerateP -space, the main results in this work. We recognize that the assumption (30) reduces Eq. (28)to Eq. (26), and therefore that Eq. (28) is a generalization of the KK scheme to non-degenerateP -spaces. This will be shown in Section 6. The LS scheme (27) is obtained from Eq. (29) byreplacing ωn−1PH0P with ωnPH0P and by assuming the degenerate P -space. In Section 7, wewill show that Eq. (29) generalizes the LS scheme to non-degenerate P -spaces.

In actual calculations, we combine one of these equations (26)–(29) with either of the follow-ing equations:

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K. Takayanagi / Nuclear Physics A 852 (2011) 61–81 67

V effn = PV P + PV Qωn, (31)

H effn = PH0P + V eff

n = PHP + PV Qωn, (32)

that define V effn and H eff

n in terms of ωn by Eqs. (15) and (16), respectively. Then the iterativescheme gives ωn, and consequently V eff

n and H effn , and eventually the effective interaction

V eff = limn→∞V eff

n , (33)

and the effective hamiltonian

H eff = PH0P + V eff = limn→∞H eff

n , (34)

when the iteration converges. In the following, we derive iterative schemes that revise V effn or

H effn rather than ωn, because V eff

n and H effn are expressed by d × d matrices of a tractable size,

while ωn is usually a much bigger (D − d) × d matrix.Before leaving this section, one point should be noted; iterative schemes generally do not

allow us to specify d eigenstates of H eff as we like. We have to examine, therefore, which d

eigenstates of H would be described by H eff obtained by the above iterative schemes.

5. V eff in degenerate model space

Here we briefly review the derivations of the effective interaction V eff in the degenerate P -space, that have been known to date. We summarize the KK method [6] in Section 5.1, and theLS method [7] in Section 5.2.

5.1. Krenciglowa–Kuo (KK) method

Here we derive V eff by solving Eqs. (26) and (31). Using Eq. (31), we first write Eq. (26) as

(ε0 − QHQ)ωn = QV P − ωnVeffn−1. (35)

For a fixed V effn−1, Eq. (35) is a linear equation for ωn, and obviously has the following formal

solution:

ωn =∞∑

k=0

(−1)k1

(ε0 − QHQ)k+1QV P

(V eff

n−1

)k. (36)

Substituting Eq. (36) for ωn in Eq. (31), we obtain

V effn = Q̂(ε0) +

∞∑k=1

Q̂k(ε0)(V eff

n−1

)k. (37)

Here we have defined the so-called Q-box and its derivatives as

Q̂(E) = PV P + PV Q1

E − QHQQV P, (38)

Q̂k(E) = 1

k!dk

dEkQ̂(E)

= (−1)kPV Q1

k+1QV P, k = 1,2, . . . . (39)

(E − QHQ)

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68 K. Takayanagi / Nuclear Physics A 852 (2011) 61–81

In actual calculations, we have to calculate the Q-box using a perturbation theory which is nor-mally defined in terms of the diagrammatic expansion [3]. In this work, we just assume that wecan calculate the Q-box and its derivatives.

The iterative scheme (37) for V effn can be implemented as follows. Suppose we have V eff

n−1at hand. Then we solve the P -space Schrödinger equation (14) at the (n − 1)th step, which iswritten as(

PH0P + V effn−1

)∣∣φ(n−1)p

⟩ = E(n−1)p

∣∣φ(n−1)p

⟩, p = 1, . . . , d. (40)

By noting that the P -space is degenerate, and therefore that H0|φ(n−1)p 〉 = ε0|φ(n−1)

p 〉, we canwrite Eq. (40) as

V effn−1

∣∣φ(n−1)p

⟩ = (E(n−1)

p − ε0)∣∣φ(n−1)

p

⟩, p = 1, . . . , d. (41)

Multiplying |φ(n−1)p 〉 by Eq. (37) and using Eq. (41), we find that the right hand side is a Taylor

series of Q̂(E(n−1)p )|φ(n−1)

p 〉 around ε0, to arrive at

V effn

∣∣φ(n−1)p

⟩ = Q̂(E(n−1)

p

)∣∣φ(n−1)p

⟩, p = 1, . . . , d. (42)

Let us define {|φ̃(n−1)p 〉,p = 1, . . . , d} that is biorthogonal to the set of d eigenstates in Eq. (42),

{|φ(n−1)p 〉,p = 1, . . . , d}. Then Eq. (42) immediately leads to the following expression for V eff

n :

V effn =

d∑p=1

Q̂(E(n−1)

p

)∣∣φ(n−1)p

⟩⟨φ̃(n−1)

p

∣∣. (43)

The above procedure shows how to calculate V effn of Eq. (37) with the knowledge of V eff

n−1. Start-ing from ω0 = 0 and therefore V eff

0 = PV P , we can generate V effn , n = 1,2, . . . , successively,

eventually to obtain V eff of Eq. (33) if the iteration converges. The above method to calculateV eff is referred to as the KK (resummation) method.

5.2. Lee–Suzuki (LS) method

Here we derive V eff from Eqs. (27) and (31). Let us first arrange them into the following setof equations:

V effn = 1

1 + PV Q 1ε0−QHQ

ωn−1Q̂(ε0),

ωn = 1

ε0 − QHQQV P − 1

ε0 − QHQωn−1V

effn . (44)

These two equations can be solved iteratively starting from ω0 = 0, to give ωn and V effn , n =

1,2, . . . , successively. Explicit forms of the first few terms of V effn are

V eff1 = Q̂(ε0),

V eff2 = 1

1 − Q̂1(ε0)Q̂(ε0),

V eff3 = 1

1 − Q̂ (ε ) − Q̂ (ε )V effQ̂(ε0). (45)

1 0 2 0 2

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K. Takayanagi / Nuclear Physics A 852 (2011) 61–81 69

We can show easily by induction that the general form of V effn (n � 3) is the following:

V effn = 1

1 − Q̂1 − Q̂2Veffn−1 − Q̂3V

effn−2V

effn−1 − · · · − Q̂n−1V

eff2 V eff

3 · · ·V effn−1

Q̂, (46)

where the Q-boxes are evaluated at the degenerate P -space energy ε0 as in Eq. (45), i.e., Q̂k =Q̂k(ε0). The above iterative procedure gives V eff of Eq. (33) in case of convergence, and isknown as the LS method.

6. Extended Krenciglowa–Kuo (EKK) method

In this section, we develop a new iterative scheme that does not need the assumption of thedegenerate P -space. We first derive the iterative scheme in Section 6.1, and then discuss the con-vergence condition in Section 6.2. The obtained formula shows that it is clearly a generalizationof the KK scheme in Section 5.1. We refer, therefore, to this method as the extended KK (EKK)method in what follows.

6.1. Derivation of EKK scheme

Here we start with Eqs. (28) and (32). Using Eq. (32), we first write Eq. (28) as

−QHQωn = QV P − ωnHeffn−1. (47)

For a fixed H effn−1, this is a linear equation for ωn. Then, in the same way as for Eq. (35), we can

immediately write down the following formal solution for ωn:

ωn =∞∑

k=0

(−1)k1

(−QHQ)k+1QV P

(H eff

n−1

)k. (48)

Substituting Eq. (48) for ωn in Eq. (32), we obtain

H effn = HBH(0) +

∞∑k=1

Q̂k(0)(H eff

n−1

)k, (49)

where we have used that the Bloch–Horowitz hamiltonian of Eq. (7) satisfies HBH(0) = PH0P +Q̂(0).

The above iterative scheme (49) can be carried out in the same way as the KK resummationmethod in Section 5.1. Suppose we have H eff

n−1 at hand. Then by solving the P -space Schrödingerequation (14) at the (n − 1)th step,

H effn−1

∣∣φ(n−1)p

⟩ = E(n−1)p

∣∣φ(n−1)p

⟩, p = 1, . . . , d, (50)

we obtain the set of d eigenstates, {|φ(n−1)p 〉,p = 1, . . . , d}. Multiplying |φ(n−1)

p 〉 by Eq. (49), werealize

H effn

∣∣φ(n−1)p

⟩ = HBH(E(n−1)

p

)∣∣φ(n−1)p

⟩, p = 1, . . . , d, (51)

which in turn shows that H effn can be written as

H effn =

d∑HBH(

E(n−1)p

)∣∣φ(n−1)p

⟩⟨φ̃(n−1)

p

∣∣. (52)

p=1
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70 K. Takayanagi / Nuclear Physics A 852 (2011) 61–81

The above iterative scheme, which we refer to as the EKK scheme, shows that we can calcu-late H eff

n starting from H eff0 = HBH(0), and eventually the effective hamiltonian H eff and the

corresponding effective interaction V eff in Eqs. (34) and (33).We make several points on the above derivation. First, the EKK expression (52) for H eff

n

manifests in an obvious fashion that H effn approaches the formal solution (23) in the limit of

n → ∞. Second, one might have thought, from Eq. (49), that the above iterative scheme wouldbe strongly dependent on the origin of the energy, E = 0. This is not, however, the case; anychange of the origin is absorbed by E

(n−1)p in solving Eq. (50), to become unobservable after a

few steps of iteration.By comparing the above EKK scheme with the KK scheme in Section 5.1, we realize the

following points. First, and most importantly, the EKK scheme does not require the degenerateP -space, in contrast to the KK scheme of which the derivation depends evidently on the degener-ate P -space. Note that we can easily derive the KK formula (43) from the EKK formula (52) byassuming the degenerate P -space. Second, the EKK scheme revises H eff

n , while the KK schemerevises V eff

n at each step of iteration. This difference originates from the disparity between theirstarting equations (35) and (47), and is evidently the key to generalize the KK scheme (43) toobtain the EKK scheme (52). Third, the EKK formula is as simple as the original KK formula,and gives H eff at the same computing cost as the KK formula. The above observations show thatthe EKK method is a natural and ideal generalization of the KK method.

6.2. Condition for convergence of EKK scheme

Here we study in detail the convergence condition for the above EKK scheme by reexaminingthe discussion on the KK scheme in Ref. [7]. Suppose ω is the exact solution to the decouplingequation (10). We write the nth approximation ωn as ωn = ω+ δωn, and correspondingly H eff

n =H eff + δH eff

n . Then Eq. (47) gives, up to the first order in δω,

QHQδωn − δωnHeff − ωδH eff

n−1 = 0. (53)

In order to make a discussion on the basis of Eq. (53), we state the convergence condition forthe EKK scheme as follows:∣∣δH eff

n |φp〉∣∣ <∣∣δH eff

n−1|φp〉∣∣, p = 1, . . . , d. (54)

Note that |δH effn |φp〉| represents the norm of the error made by H eff

n |φp〉 in expressing H eff|φp〉,the left hand side of the P -space Schrödinger equation (14) that we are to reproduce. This obser-vation shows that the inequality (54) guarantees the convergence of the EKK iterative scheme,and vice versa.

Now we translate the convergence condition (54) into a form that is convenient for practicaluse. Multiplying |φp〉 of Eq. (14) by Eq. (53), we obtain

δωn|φp〉 = − 1

(Ep − QHQ)ωδH eff

n−1|φp〉, p = 1, . . . , d. (55)

Then, by multiplying both sides of Eq. (55) by PV Q, we end up with

δH effn |φp〉 = −ω†(Ep)ωδH eff

n−1|φp〉, p = 1, . . . , d, (56)

where we have used δH effn = PV Qδωn and the expression for ω(Ep) defined in Eq. (21). By

taking the norm of both sides of Eq. (56), we arrive at

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K. Takayanagi / Nuclear Physics A 852 (2011) 61–81 71

∣∣δH effn |φp〉∣∣ �

∥∥ω†(Ep)ω∥∥

S

∣∣δH effn−1|φp〉∣∣, p = 1, . . . , d, (57)

where ‖ · ‖S is the spectral norm of matrix [11–13] that is briefly explained in Appendix A. Theabove inequality (57) clearly shows that a sufficient condition for the convergence condition (54)is ∥∥ω†(Ep)ω

∥∥S

< 1, p = 1, . . . , d. (58)

Next, noting that {|φp〉} is a normalized set, i.e., 〈φp|φp〉 = 1, we can derive the following nec-essary condition for the inequality (58):∣∣〈φp|ω†(Ep)ω(Ep)|φp〉∣∣ < 1, p = 1, . . . , d, (59)

where we have used the expression (22) for ω. Finally, using Eqs. (24), and (25), we can trans-form the condition (59) into the following convenient form:

ρQ = 〈Ψp|Q|Ψp〉〈Ψp|Ψp〉 <

1

2, p = 1, . . . , d, (60)

which means that the probability ρQ of the Q-space components in |Ψp〉 is less than one half.The condition (60) is the same as the KK convergence condition given in Ref. [7] for the de-

generate P -space. Note that this should be the case, because the condition (60) does not containthe P -space energy. The condition (60) suggests that the EKK scheme would give d eigenstatesof H with the largest P -space overlaps. This convergence condition is widely accepted in themarket for the KK scheme. However, the above derivation shows, in a strict sense, that the con-dition (60) is neither necessary nor sufficient for the convergence condition (54) of the EKKscheme. In other words, the inequality (60) is a rough criterion, but not a strict condition, forconvergence. In Section 8.2, we give an example where the states with the largest overlaps arenot described by H eff obtained by the EKK method.

7. Extended Lee–Suzuki (ELS) method

Here we develop another iterative scheme that does not need the assumption of the degener-ate P -space. In Section 7.1, we derive the iterative scheme, and in Section 7.2, we discuss itsconvergence condition. The obtained iterative scheme is obviously a generalization of the LSscheme explained in Section 5.2, and will be referred to as the extended LS (ELS) scheme inwhat follows.

7.1. Derivation of ELS scheme

Our starting point here is Eqs. (29) and (32). First we cast them into the following set ofequations:

H effn = 1

1 + PV Q1

−QHQωn−1

HBH(0), (61)

ωn = 1

−QHQQV P − 1

−QHQωn−1H

effn . (62)

These two equations can be solved easily starting from ω0 = 0 in the same way as in Section 5.2,to give

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72 K. Takayanagi / Nuclear Physics A 852 (2011) 61–81

H eff1 = HBH(0),

H eff2 = 1

1 − Q̂1(0)HBH(0),

H eff3 = 1

1 − Q̂1(0) − Q̂2(0)H eff2

HBH(0). (63)

In the same way as for the LS scheme, it is straightforward to see that the general expression forH eff

n (n � 3) can be written as

H effn = 1

1 − Q̂1 − Q̂2Heffn−1 − Q̂3H

effn−2H

effn−1 − · · · − Q̂n−1H

eff2 H eff

3 · · ·H effn−1

HBH(0).

(64)

The above iterative procedure constitutes the ELS scheme to give H eff in Eq. (34), when theiteration converges. Note that the Q-box and its derivatives are evaluated at E = 0, i.e., Q̂k =Q̂k(0), in Eq. (64). In contrast to the EKK scheme, the origin of energy, E = 0, plays an importantrole in the ELS scheme. We will see shortly in Section 7.2 that we can provide the ELS schemewith wider applicability by shifting the origin properly.

Let us compare the above ELS and the original LS schemes. First, and most importantly,there is no assumption of the degenerate P -space in the ELS scheme. Second, the ELS scheme(64) revises H eff

n in the same way as the LS scheme (46) revises V effn at each step of iteration.

Third, we notice that the ELS iterative scheme (64) can be implemented at the same computingcost as the LS scheme (46). The above observations, along with the discussion in Section 7.2,convince us that the ELS scheme is a natural and optimal generalization of the LS scheme tonon-degenerate P -spaces.

7.2. Condition for convergence of ELS scheme

Here we examine the convergence condition for the ELS scheme by revisiting the discussionin Ref. [7]. Suppose that ω is the exact solution to the decoupling equation (10), and that we haveωn = ω + δωn at the nth step. Then Eq. (29) gives, up to the first order in δω,

QHQδωn = δωn−1Heff, (65)

where QHQ is the Q-space effective hamiltonian defined in Eq. (19).The convergence condition for the ELS scheme is limn→∞ δωn = 0, or equivalently

limn→∞ ‖δωn‖ = 0, in any matrix norm ‖ · ‖ (see Appendix A). In order to make a discussionon the basis of Eq. (65), we first translate the above condition into

‖δωn‖E < ‖δωn−1‖E, (66)

where ‖ · ‖E stands for the euclidean (or Frobenius) norm of matrix [11–13] that is briefly ex-plained in Appendix A.

By taking the matrix element of the relation (65) with 〈Φq | of Eq. (17) and |φp〉 of Eq. (11),we obtain

Eq〈Φq |δωn|φp〉 = Ep〈Φq |δωn−1|φp〉, for any p,q. (67)

This shows that a sufficient condition for the inequality (66) is

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K. Takayanagi / Nuclear Physics A 852 (2011) 61–81 73

|Ep| < |Eq |, for any p,q. (68)

Note that, among D eigenenergies of H , there is always one and only set of d eigenenergies{Ep,p = 1, . . . , d} that satisfy the above sufficient condition (68). This means that, if the iterationconverges in actual calculations, the obtained H eff certainly describes d eigenstates with leastabsolute eigenenergies. This is a rigorous conclusion that is in contrast to the rough criterion(59) obtained for the EKK scheme.

Because the convergence condition (68) is expressed in terms of eigenenergies, we can derivean effective hamiltonian H eff which reproduces d eigenstates that are nearest to an arbitrarilychosen energy E = m. Let us define a modified hamiltonian H ′ = H − m that has the sameeigenstates as H but at shifted eigenenergies:

H ′|Ψλ〉 = (Eλ − m)|Ψλ〉, λ = 1, . . . ,D. (69)

Following the derivation in Section 7.1 with the modified hamiltonian H ′, we obtain, instead ofEq. (64),

H ′ effn = 1

1 − Q̂′1 − Q̂′

2H′ effn−1 − Q̂′

3H′ effn−2H

′ effn−1 − · · · − Q̂′

n−1H′ eff2 H ′ eff

3 · ·H ′ effn−1

× H ′BH(0). (70)

Here H ′BH(0) and Q̂′k = Q̂′

k(0) are obtained using H ′ in place of H in the expressions forHBH(0) and Q̂k(0). Let us note that Q̂′

k(0) = Q̂k(m) and H ′BH(0) = HBH(m)−m, and thereforethat Eq. (70) can be transformed into the following form:

H ′ effn = 1

1 − Q̂1 − Q̂2H′ effn−1 − Q̂3H

′ effn−2H

′ effn−1 − · · · − Q̂n−1H

′ eff2 H ′ eff

3 · ·H ′ effn−1

× (HBH(m) − m

), (71)

where Q̂k stands for Q̂k(m). Then we can calculate H ′ effn successively using the above expression

starting from H ′ eff1 = H ′BH(0) = HBH(m) − m. Now Eq. (67) for H is replaced by

(Eq − m)⟨Φq

∣∣δω′n

∣∣φp

⟩ = (Ep − m)⟨Φq

∣∣δω′n−1

∣∣φp

⟩, (72)

where δω′n is defined for H ′ in the same way as δωn for H . Then the convergence condition for

the iterative process in Eq. (71) is, instead of (68),

|Ep − m| < |Eq − m|, for any p,q, (73)

which shows that H ′ eff = limn→∞ H ′ effn describes d eigenstates of H ′ = H − m with least ab-

solute eigenenergies. This in turn means that

H eff = H ′ eff + m, (74)

describes d eigenstates of H whose eigenenergies are nearest to the arbitrarily chosen energy m.Another advantage of having the free parameter m at hand is the possibility of accelerating

the convergence of iteration by tuning m. It should be clear from Eq. (72) that one would beable to accelerate the convergence by minimizing the ratios (Ep − m)/(Eq − m) by tuning m

around the center of the spectra {Ep,p = 1, . . . , d} that one would like to reproduce. This willbe demonstrated in Section 8. Here, tuning m in the ELS scheme should not be confused withadjusting the degenerate P -space energy ε0 in the LS scheme [7]; m stands for a shift of thewhole spectra of H , while adjusting ε0 changes the unperturbed energy of the (degenerate) P -space only.

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74 K. Takayanagi / Nuclear Physics A 852 (2011) 61–81

Fig. 1. Eigenenergies of the effective hamiltonians obtained by the four iterative methods (KK, LS, EKK, ELS) in thedegenerate P -space (ε = 0). Solid curves show the exact eigenenergies of the full hamiltonian as functions of the strengthparameter x.

8. Test calculations and discussions

Here we discuss the iterative schemes presented in the above using test calculations. We adoptthe model hamiltonian studied in Ref. [8], and add a term that lifts the P -space degeneracy. Wetake a four-dimensional Hilbert space that is spanned by the basis vectors {|λ〉, λ = 1,2,3,4}.Our hamiltonian is H = H0 + V , where H0 = diag[1,1 + ε,3,9] in the above basis. The two-dimensional P -space is spanned by {|1〉, |2〉}, and a nonzero value of ε removes the degeneracyof the P -space. The perturbation V is given by the following form with a strength parameter x:

V =⎛⎜⎝

0 5x −5x 5x

5x 25x 5x −8x

−5x 5x −5x x

5x −8x x −5x

⎞⎟⎠ . (75)

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. In this section, we normalize |Ψλ〉 to unity,i.e., 〈Ψλ|Ψμ〉 = δλ,μ, as in usual textbook quantum mechanics. Note that

limx→0

|Ψλ〉 = |λ〉, λ = 1,2,3,4. (76)

In Sections 8.1 and 8.2, we study the degenerate case (ε = 0) and the non-degenerate case(ε �= 0), respectively.

8.1. Degenerate P -space

We first diagonalize H = H0 + V with ε = 0. In Fig. 1, eigenenergies Eλ for the interactionstrength 0 < x < 0.2 are shown by solid curves. We see that, with increasing value of x, thesecond and the third lowest levels (|Ψ2〉 and |Ψ3〉) show a level crossing around x ∼ 0.06. Thiscan be confirmed by Fig. 2 which shows the probability amplitudes 〈μ|Ψλ〉, λ,μ = 1,2,3,4. It isvisible that characters of |Ψ2〉 and |Ψ3〉 are exchanged at x ∼ 0.06, i.e., the dominant componentof |Ψ2〉 (|Ψ3〉) is |2〉 (|3〉) for x � 0.06, while for x � 0.06 the main component of |Ψ2〉 (|Ψ3〉) is

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K. Takayanagi / Nuclear Physics A 852 (2011) 61–81 75

Fig. 2. Probability amplitudes 〈μ|Ψλ〉 in the degenerate case (ε = 0). Lines are distinguished by the index μ, as denotedin the figure. Open triangles show the P -space probability ρP of Eq. (77) for each state. Note that |Ψλ〉 is normalized tounity.

|3〉 (|2〉). In Fig. 2, we also show the P -space probability ρP of each state, which is defined fornormalized |Ψλ〉 as

ρP = 〈Ψλ|P |Ψλ〉 = 1 − ρQ. (77)

We see that |Ψ1〉 and |Ψ2〉 are described predominantly in the P -space for x � 0.06, while |Ψ1〉and |Ψ3〉 have the largest ρP for x � 0.06. This means that our model hamiltonian is in the weak(strong) coupling regime in x � 0.06 (x � 0.06).

Now we examine the eigenenergies of the effective hamiltonian in the four methods (KK, LS,EKK, and ELS), which are plotted by circles and squares in Fig. 1. First, we see immediately thatthe effective hamiltonians of the LS and the ELS methods reproduce the lowest two levels, aswas proven in Section 7.2. Second, noting the level crossing at x ∼ 0.06, we realize that the KKand the EKK methods give the two states with the largest P -space overlaps, as was explained inSection 6.2. The above observations convince us that, in the degenerate P -space, the EKK andthe ELS schemes work as satisfactorily as the KK and the LS schemes.

Let us make a point on the P -space probability ρP in Fig. 2. In our test calculations, we havethe exact eigenstates |Ψλ〉, λ = 1,2,3,4, and therefore can calculate ρP easily for all of the fourstates. In the effective interaction theory, however, we are supposed to have the knowledge only

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76 K. Takayanagi / Nuclear Physics A 852 (2011) 61–81

Fig. 3. �n of Eq. (78) that shows convergence of iterative methods (KK, LS, EKK, and ELS) for ε = 0, left: x = 0.05,and right: x = 0.15. ELS results are shown for m = 0.0, 1.5 for x = 0.05, and m = 0.0, 1.4 for x = 0.15.

of the two normalized eigenstates |φp〉 = P |Ψp〉 of H eff. Here the index p specifies the statesthat are reproduced by H eff; in the EKK scheme, e.g., p = 1,2 for x � 0.06, and p = 1,3 forx � 0.06. In this case, we can utilize Eqs. (24) and (25) to calculate ρP of Eq. (77); they allowus to calculate ρQ with the knowledge of the normalized |φp〉 and the Q-box, which is easilyconfirmed in our model calculations.

Next, we investigate the convergence properties of the four iterative schemes. As in Ref. [8],we define the following measure of the deviation of the calculated eigenenergies, E

(n)p , at the nth

step from the exact values, Ep:

�n =[∑

p

(E(n)

p − Ep

)2]1/2

, (78)

where the index p runs over the two states that H eff reproduces in each method.In Fig. 3, we show �n for x = 0.05 and 0.15 in each method. Following points are visible

in Fig. 3. First, we see that the KK scheme converges much faster than the LS scheme, whichseems to be a general feature [5,8]. Note that, in our degenerate case, the EKK formula (52) isessentially the same as the KK one (43), and so is the convergence rate. Second, we recognizethat the level crossing point is very close to x = 0.05, and therefore the convergence at x = 0.05is slower than at x = 0.15 which is far from the level crossing. Third, let us look into the resultsof the ELS scheme, of which the convergence can be accelerated by tuning m, as explained inSection 7.2. In order to reproduce the lowest two eigenstates, |Ψ1〉 and |Ψ2〉, we have found thatthe optimal value is m ∼ 1.5 at x = 0.05, and m ∼ 1.4 at x = 0.15. The convergence rates withthese optimal values of m are plotted together with the m = 0.0 results. We realize that a sizablechange in the convergence rate can be made by tuning m; with the optimal value of m, the ELSscheme converges faster than the original LS scheme.

8.2. Non-degenerate P -space

In Section 8.1, we have confirmed that both the EKK and the ELS methods work perfectlywell in the same way as the KK and the LS methods in the degenerate P -space (ε = 0). Here weturn to the non-degenerate P -space (ε �= 0), where we can use the EKK and the ELS methodsonly. We shall see that these two methods work excellently as in the degenerate case.

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K. Takayanagi / Nuclear Physics A 852 (2011) 61–81 77

Fig. 4. Eigenenergies in the non-degenerate P -space (ε = 1). Notation is the same as for Fig. 1.

Fig. 5. Probability amplitudes 〈μ|Ψλ〉 in the non-degenerate case (ε = 1). Notation is the same as for Fig. 2.

We take the hamiltonian with ε = 1 that removes the P -space degeneracy. In Fig. 4, we showthe exact eigenenergies of H with solid curves for 0 < x < 0.2. At x = 0, we see that the eigenen-ergies are 1,2,3,9, and that the P -space is obviously non-degenerate. The level crossing takesplace at x ∼ 0.04 for ε = 1, in contrast to x ∼ 0.06 for ε = 0. This can be confirmed by Fig. 5

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78 K. Takayanagi / Nuclear Physics A 852 (2011) 61–81

Fig. 6. �n of Eq. (78) that shows convergence of iterative methods (KK, LS, EKK, and ELS) for ε = 1, left: x = 0.05,and right: x = 0.15. ELS results are shown for m = 0.0, 1.8 for x = 0.05, and m = 0.0, 1.5 for x = 0.15.

which shows the probability amplitudes 〈μ|Ψλ〉, λ,μ = 1,2,3,4, in the same way as in Fig. 2for ε = 0.

In Fig. 4, eigenenergies of the effective hamiltonians in the EKK and the ELS methods areshown by squares and circles, respectively. We see clearly that the ELS method reproduces thelowest two levels as was explained in Section 7.2, and that the EKK method gives the two lev-els with the largest P -space overlaps as discussed in Section 6.2. Fig. 4 convinces us that theEKK and the ELS schemes work in the non-degenerate P -space as nicely as the KK and the LSschemes in the degenerate P -space.

The convergence rates of the EKK and the ELS schemes are shown in Fig. 6 in the same wayas in Fig. 3 for ε = 0. We can make the following observations; first, the EKK scheme convergesfaster than the ELS scheme as in the ε = 0 case. Second, it is visible that the convergence ofthe ELS scheme at x = 0.05 is much slower in Fig. 6 than in Fig. 3, while the EKK schemeconverges almost at the same rate in Figs. 6 and 3. This can be explained as follows; let us notethat the level crossing, which is close to x = 0.05, is much sharper for ε = 1 than for ε = 0. Thisis realized by Figs. 2 and 5; the exchange of the wave function components between |Ψ2〉 and|Ψ3〉 takes place more rapidly as a function of x for ε = 1 than for ε = 0. Therefore, at x = 0.05,the ELS scheme requires more iterative steps for ε = 1 than for ε = 0 to follow the exchange ofthe wavefunction components. On the other hand, the EKK scheme describes the state which iscontinuous as a function of x, and therefore its convergence property is not worsened by the levelcrossing. Third, it can be seen that the convergence rate in the ELS scheme can be improved bytuning m, in the same way as in the degenerate P -space. The optimal value of m to reproduce thelowest two eigenstates is given by m ∼ 1.8 at x = 0.05, and m ∼ 1.5 at x = 0.15. Fig. 6 clearlyshows that we can accelerate the convergence to a large extent by optimizing m.

As a final example, we adopt the hamiltonian with ε = 0.5, and show possible limitations ofthe iterative schemes discussed in this work. In Fig. 7, we show eigenenergies of the effectivehamiltonians (marks) together with the exact eigenenergies (solid curves) for ε = 0.5. We alsopresent the P -space probability ρP of each state in a separate panel. The ELS calculations arecarried out with m = 0.0 and m = 6.0 as denoted in the figure.

First, let us look into the EKK result. A notable point in Fig. 7 is that the EKK method, whenit converges, does not necessarily reproduce the states with the largest P -space overlaps, in con-tradiction to the criterion (60). At x = 0.05, the left panel in Fig. 7 shows that the EKK scheme

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K. Takayanagi / Nuclear Physics A 852 (2011) 61–81 79

Fig. 7. Left: eigenenergies for the non-degenerate P -space (ε = 0.5). ELS results are show for m = 0.0 and m = 6.0.Other notation is the same as for Fig. 1. Right: P -space probability ρP of Eq. (77) of exact eigenstates.

reproduces |Ψ1〉 and |Ψ3〉. On the other hand, the right panel shows that ρP (Ψ3) = 0.473 issmaller than ρP (Ψ2) = 0.548, suggesting that the EKK scheme would reproduce |Ψ1〉 and |Ψ2〉.This reveals clearly that the criterion (60) is not a strict condition for the eigenstates obtained bythe EKK scheme, as discussed in Section 6.2.

Second, let us turn to the ELS results. We see that the ELS scheme with m = 0.0 convergesin the whole range of x in the figure, and reproduces the lowest two eigenstates, in accordancewith the convergence condition (68). For m = 6.0, the condition (73) tells that the effectivehamiltonian, if it converges, should give the highest two eigenstates, |Ψ3〉 and |Ψ4〉. This is indeedthe case as can be seen in the left panel. It is remarkable that the ELS method successfully givesH eff that describes |Ψ3〉 and |Ψ4〉 in the weak coupling (small x) region where ρP of thesestates are very small; at x = 0.01, for example, ρP (Ψ3) = 0.00224 and ρP (Ψ4) = 0.00016. Forx > 0.16, however, the calculation with m = 6.0 does not stabilize, and we cannot obtain theeigenstates.

Now we summarize what we have learned from test calculations with a variety of parametersets. Most importantly, we have confirmed that the EKK and the ELS schemes work very well inthe non-degenerate P -spaces in the same way as the KK and the LS schemes do in degenerateP -spaces. Looking into the details, we have made the following observations. First, the EKKscheme almost always converges to give the eigenstates with the largest P -space overlaps. Thereare, however, exceptions as we saw in the above. Second, the ELS scheme always convergesto describe the target eigenstates, if we are concerned with the lowest eigenstates. On the otherhand, if we are interested in reproducing, e.g., the highest two eigenstates — which cannot begenerated perturbatively from the P -space — in the ELS scheme by choosing m properly asexplained in Section 7.2, we sometimes find that the calculation does not stabilize. Third, we canaccelerate considerably the convergence of the ELS scheme by optimizing m.

9. Summary

The eigenvalue problem of a hamiltonian H in a full Hilbert space can be transformed into theeigenvalue problem of an effective hamiltonian H eff in a model space (P -space) of a tractablesize. The effective interaction V eff, and therefore the effective hamiltonian H eff, have been cal-culated by two iterative methods, i.e., the Krenciglowa–Kuo (KK) and the Lee–Suzuki (LS)

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80 K. Takayanagi / Nuclear Physics A 852 (2011) 61–81

methods that were proposed more than three decades ago. These two methods require that theP -space be degenerate, i.e., that the unperturbed energies of all the P -space states be the same.This is really a strong requirement that is in many cases far from reality. In this situation, we haveremoved the longstanding requirement of the degenerate P -space from both the KK and the LSmethods, and have proposed two new methods in this work, i.e., the extended KK (EKK) and theextended LS (ELS) methods. The main points we have clarified are the followings.

First, the EKK and the ELS methods revise H eff at each step of iteration in the same wayas the original KK and the LS methods revise V eff, and are applicable to the non-degenerateP -space as well as to the degenerate P -space.

Second, the extended iterative schemes (EKK and ELS) are as simple as the original schemes(KK and LS), and can be implemented at the same computing cost.

Third, using test calculations, we have shown that the extended schemes (EKK and ELS) workexcellently in non-degenerate P -spaces in the same way as the original schemes (KK and LS) doin degenerate P -spaces.

Finally, from the above points, we conclude that the EKK and the ELS schemes are naturaland ideal generalizations of the KK and the LS schemes to non-degenerate P -spaces.

Having removed the strong requirement of the degenerate P -space from the standard KK andLS frameworks, we believe that the present work has given a much wider market of activity thanever to the concept of effective hamiltonian in the field of quantum many-body problems.

Appendix A. Matrix norm

Here we explain the matrix norm briefly. Comprehensive explanations can be found inRefs. [11,12] for square matrices, which can be generalized to rectangular matrices easily. A briefexplanation on the norm of rectangular matrices is found in Ref. [13].

Let A and B be m × n and n × q complex matrices. Then a matrix norm ‖ · ‖ is defined bythe following axioms.

(1) ‖A‖ � 0, ‖A‖ = 0 if and only if A = 0.

(2) ‖αA‖ = |α|‖A‖, α is any complex number.

(3) ‖A + B‖ � ‖A‖ + ‖B‖.(4) ‖AB‖ � ‖A‖‖B‖. (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. Among them, especially the spectral

norm ‖ · ‖S and the euclidean (or Frobenius) norm ‖ · ‖E are most widely used, which we explainbriefly in order.

Let us start with the spectral norm. Suppose the matrix A is an operator on a vector spacewhere the vector norm is the usual euclidean (vector) norm, viz. for x = (x1, . . . , xn)

|x| =[

n∑i=1

|xi |2]1/2

. (A.2)

Then the spectral norm of the matrix A is defined as

‖A‖S = max |Ax|. (A.3)

|x|=1
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K. Takayanagi / Nuclear Physics A 852 (2011) 61–81 81

It is straightforward to show that the above norm satisfies all the four axioms of matrix norm inEq. (A.1). By noting

‖A‖2S = max

|x|=1x†A†Ax, (A.4)

we can easily understand that ‖A‖S = λA, where λ2A is the largest eigenvalue of A†A. In other

words, ‖A‖S is the largest singular value of A [11,12]. Note that the definition (A.3) of thespectral norm immediately leads to the following inequality:

|Ax| � ‖A‖S |x|, (A.5)

which holds for any x. Then we see at once that the inequality (A.5) explains the relation (57).Using the definition (A.3), it is also easy to see that the condition (58) leads to the inequality(59), and therefore that (59) is a necessary condition for (58).

Next we turn to the euclidean norm. For an m × n matrix A = [aij ], it is defined as

‖A‖E =(

m∑i=1

n∑j=1

|aij |2)1/2

. (A.6)

It is easy to verify that the above norm satisfies all the four axioms of matrix norm in Eq. (A.1).The definition (A.6) of the euclidean matrix norm is regarded as the euclidean vector norm inan mn-dimensional vector space, and explains right away that the inequality (68) is a sufficientcondition for (66).

References

[1] B.H. Brandow, Rev. Modern Phys. 39 (1967) 771.[2] P.J. Ellis, E. Osnes, Rev. Modern Phys. 49 (1977) 777.[3] T.T.S. Kuo, E. Osnes, Springer Lecture Notes in Physics, vol. 364, Springer-Verlag, 1990.[4] M. Hjorth-Jensen, T.T.S. Kuo, E. Osnes, Phys. Rep. 261 (1995) 125.[5] D.J. Dean, T. Engeland, M. Hjorth-Jensen, M.P. Kartamyshev, E. Osnes, Prog. Part. Nucl. Phys. 53 (2004) 419.[6] E.M. Krenciglowa, T.T.S. Kuo, Nucl. Phys. A 235 (1974) 171.[7] K. Suzuki, S.Y. Lee, Progr. Theoret. Phys. 64 (1980) 2091.[8] K. Suzuki, R. Okamoto, P.J. Ellis, T.T.S. Kuo, Nucl. Phys. A 567 (1994) 576.[9] C. Bloch, Nucl. Phys. 6 (1958) 329.

[10] C. Bloch, J. Horowitz, Nucl. Phys. 8 (1958) 91.[11] P. Lancaster, M. Tismenetsky, The Theory of Matrices, second ed., Academic Press, 1985.[12] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, 1985.[13] G.H. Golub, C.F. van Loan, Matrix Computations, third ed., The Johns Hopkins University Press, 1996.