Effective interaction in the Rayleigh–Schrödinger perturbation theory

Annals of Physics 350 (2014) 501–532

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Effective interaction in the Rayleigh–Schrödinger perturbation theory Kazuo Takayanagi Department of Physics, Sophia University, 7-1 Kioi-cho, Chiyoda-ku, Tokyo 102-8554, Japan

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Article history: Received 7 May 2014 Accepted 2 August 2014 Available online 8 August 2014 Keywords: Effective interaction Folded diagram Bracketing representation Rayleigh–Schrödinger perturbation theory

abstract We present a unified description of the effective interaction v in the Rayleigh–Schrödinger perturbation theory. First, we generalize the well-known bracketing expression for the energy shift 1E in a one-dimensional model space to express the effective interaction v in a multi-dimensional model space. Second, we show that the generalized bracketing representation has a natural graphic expression in terms of folded diagrams. The present work thus gives a unified understanding of the effective interaction (i) in one- and multi-dimensional model spaces and (ii) in algebraic (bracketing) and graphic (folded diagram) representations. © 2014 Elsevier Inc. All rights reserved.

1. Introduction In the description of quantum systems, the effective interaction has been one of the central issues [1–16]. We first divide a large Hilbert space into a model space (P-space) of a tractable size and its complement (Q -space). Then we define the effective interaction v in the P-space to describe a set of selected eigenstates of the full Hamiltonian H. In one-dimensional P-space, the problem reduces to the well-known Rayleigh–Schrödinger perturbation theory (RSPT); the energy shift 1E of the (one-dimensional) P-space state can be expressed conveniently by the so-called bracketing technique [9]. In multi-dimensional P-space, on the other hand, we do not know its counterpart. In the framework of time-independent perturbation theory, with which we work throughout this paper, the effective interaction v is usually introduced via the Bloch equation, which leads to its own algebraic and graphic (folded diagram) representations for v [7–9]. This is not, however, a generalization of the bracketing

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.aop.2014.08.002 0003-4916/© 2014 Elsevier Inc. All rights reserved.

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K. Takayanagi / Annals of Physics 350 (2014) 501–532 Table 1 Current description of the effective interaction v in time-independent perturbation theory. ‘‘RS’’ means the Rayleigh–Schrödinger perturbation theory, which only gives the bracketing expression for v = 1E in onedimensional P-space. Multi-dimensional P-space is treated on the basis of the Bloch equation [9].

One-dim. Multi-dim.

Bracketing

Folded diagram

RS Bloch

– Bloch

technique of the RSPT to multi-dimensional P-space. Also in time-dependent perturbation theory, we naturally introduce the folded diagram technique to isolate possible divergences from transition amplitudes [4]. Although this technique has been used widely, it does not have any bracketing expressions. The above uncomfortable situation, where one- and multi-dimensional P-spaces are treated in different schemes, has made it difficult to obtain a unified understanding of v in the RSPT. In this situation, which is summarized in Table 1, we generalize the bracketing technique of the RSPT in one-dimensional P-space so that it applies to multi-dimensional P-space. Then we show that the generalized bracketing scheme leads to its own definition of the ‘‘folded diagram’’ for its graphic expression. We thus enter ‘‘RS’’ in all four spaces in Table 1, establishing a unified description of the effective interaction in the RSPT. This paper is structured as follows. In Section 2, we explain the effective interaction v fixing the notation. In Section 3, we briefly review the bracketing technique of the usual RSPT in one-dimensional P-space. In Section 4, we generalize the above bracketing technique for 1E to express the effective interaction v in multi-dimensional degenerate P-space. Then we introduce the folded diagram in our convention to visualize the generalized bracketing scheme. In Section 5, we generalize the theory so that it applies to multi-dimensional nondegenerate P-space. In Section 6, we discuss several aspects of the present theory and also compare our theory with other existing theories. In Section 7, we illustrate the coherence and usefulness of the present theory using toy model calculations. Finally in Section 8, we present a brief summary. 2. Effective interaction In this section, we briefly review the concept of the effective interaction and its formal derivation using the notation of Refs. [10–12]. 2.1. Model space We describe a quantum system in a Hilbert space of dimension D with the following Hamiltonian: H = H0 + V ,

(1)

where H0 is the unperturbed Hamiltonian and V is the perturbation. By diagonalizing the full Hamiltonian H in the whole Hilbert space, we obtain D eigenstates of H. In many cases, however, the above diagonalization is beyond the current computational capacity because D is usually a huge number, and we are generally not interested in all D eigenstates. Therefore, we divide the whole Hilbert space of dimension D into a model space (P-space) of tractable dimension d and its complement (Q -space), and choose d eigenstates of H to describe: H |Ψα ⟩ = Eα |Ψα ⟩,

α = 1 , . . . , d.

(2)

The projection operators onto the P- and Q -spaces are denoted as P and Q , respectively, which satisfy P 2 = P , Q 2 = Q , and PQ = QP = 0. Here we require that the P-space be spanned by a set of d eigenstates {|i⟩, i = 1, . . . , d} of H0 that satisfy H0 |i⟩ = ϵi |i⟩,

i = 1 , . . . , d.

(3)

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Then the Q -space is spanned by {|I ⟩, I = d + 1, . . . , D} that satisfy H0 |I ⟩ = ϵI |I ⟩,

I = d + 1 , . . . , D.

(4)

Throughout this work, the P- and Q -space basis states are denoted by lowercase and uppercase letters, respectively. Note that the projection operators P and Q are given by P =

d 

|i⟩⟨i|,

Q =

i=1

D 

|I ⟩⟨I |,

(5)

I =d+1

and that they satisfy

[H0 , P ] = [H0 , Q ] = 0,

PHQ = PVQ ,

QHP = QVP .

(6)

Now we split the eigenstate |Ψα ⟩ in Eq. (2) into the P- and Q -space components as

|Ψα ⟩ = P |Ψα ⟩ + Q |Ψα ⟩ = |φα ⟩ + |Φα ⟩,

α = 1, . . . , d.

In the following, we derive an effective Hamiltonian H |φα ⟩ = P |Ψα ⟩ as its eigenstate with eigenenergy Eα .

eff

(7)

in d-dimensional P-space that describes

2.2. Energy-dependent effective Hamiltonian H BH Before introducing the effective Hamiltonian H eff , which is energy-independent by definition, it is convenient to derive the energy-dependent effective Hamiltonian of Bloch and Horowitz [8,9,17,18] for later use. Let us first write Eq. (2) in the following block form:



PHP QHP

PHQ QHQ



   |φα ⟩ |φα ⟩ = Eα , |Φα ⟩ |Φα ⟩

α = 1 , . . . , d.

(8)

Then we can easily solve Eq. (8) for |φα ⟩ to obtain H BH (Eα )|φα ⟩ = Eα |φα ⟩,

α = 1, . . . , d.

Here we have defined the Bloch–Horowitz Hamiltonian H H BH (E ) = PHP + PVQ

1 E − QHQ

(9) BH

(E ) as

QVP

= PH0 P + Qˆ (E ),

(10)

using the so-called Qˆ -box Qˆ (E ) = PVP + PVQ

1 E − QHQ

QVP .

(11)

Note that H BH (E ) is dependent on the energy parameter E, and therefore Eq. (9) requires a selfconsistent solution obtained with a special numerical technique [19]. 2.3. Energy-independent effective Hamiltonian H eff For the d chosen eigenstates {|Ψα ⟩, α = 1, . . . , d} of H, the energy-independent effective Hamiltonian H eff , which we derive here, describes their projections onto the P-space, |φα ⟩ = P |Ψα ⟩, by H eff |φα ⟩ = Eα |φα ⟩,

α = 1, . . . , d.

(12)

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 Because {|φα ⟩, α= 1, . . . , d} is not generally an orthogonal set, we define its biorthogonal set | φα ⟩, α = 1, . . . , d that satisfies ⟨ φβ |φα ⟩ = δβ,α ,

d 

P =

|φα ⟩⟨ φα |.

(13)

α=1

Then, using Eq. (9), we realize that H eff can be formally expressed as H eff =

d 

Eα |φα ⟩⟨ φα | =

d 

H BH (Eα )|φα ⟩⟨ φα |.

(14)

α=1

α=1

Finally, we define the effective interaction, v = P v P, via H eff = PH0 P + v.

(15)

In the following, we derive various expressions for v , which lead to a unified understanding of the effective interaction. 3. v in one-dimensional model space When the P-space is of one dimension, the effective interaction theory reduces to the well-known RSPT. In this section, we briefly review the bracketing technique in the RSPT using notation that is ready to be generalized to multi-dimensional P-space. Here we concentrate on the description of eigenenergies; the bracketing expression for the wave function in multi-dimensional P-space is briefly explained in Appendix A. 3.1. RSPT In one-dimensional P-space, where the P-space is spanned by a single component P |Ψ ⟩ = |φ⟩, we may eliminate the P-space indices ‘‘α ’’ and ‘‘i’’ in Eqs. (2) and (3) to obtain H |Ψ ⟩ = E |Ψ ⟩,

(16)

H0 |φ⟩ = ϵ |φ⟩.

(17)

Here we adopt the normalization ⟨φ|φ⟩ = 1, which in turn fixes the normalization of |Ψ ⟩ to ⟨φ|Ψ ⟩ = 1. Then it is easy to derive the following expressions for the energy shift 1E and wave function |Ψ ⟩:

1E = E − ϵ = ⟨φ |V |Ψ ⟩,

|Ψ ⟩ = |φ⟩ +

Q

ϵ − H0

(V − 1E )|Ψ ⟩,

(18) (19)

from which we can construct the well-known RSPT. Similarly, we see that the effective Schrödinger equation (12) reduces to H eff |φ⟩ = (H0 + v) |φ⟩ = E |φ⟩

(20)

in one-dimensional P-space. The most important point that distinguishes one-dimensional P-space from multi-dimensional P-space is that the effective interaction v in Eq. (20) is a c-number:

v = 1E , which can be seen from Eqs. (20) and (17).

(21)

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3.2. Bracketing—algebraic expression for v = 1E Here we describe the bracketing representation in the RSPT. Let us first note that an iterative solution to Eq. (19) can be written as

        −1E −1E −1E |Ψ ⟩ = 1 + RV + R RV + R R RV + · · · |φ⟩. V

V

V



−1E

(22)



Here R = ϵ−QH is the Rayleigh–Schrödinger propagator, and V means that we choose either −1E 0 or V when writing down a specific term in the expansion, and we sum up over all possibilities. Then 1E of Eq. (18) is given by

        −1 E −1E −1E 1E = V + VRV + VR RV + VR R RV + · · · , V

V

V

(23)

where we have used the single brackets ⟨· · ·⟩ to express ⟨φ| · · · |φ⟩. Noting v = 1E, we see that the above expression (23) gives

v = ⟨V ⟩ + ⟨VRV ⟩ + ⟨VRVRV ⟩ − ⟨VRv RV ⟩ + ⟨VRVRVRV ⟩ − ⟨VRVRv RV ⟩ − ⟨VRv RVRV ⟩ + ⟨VRv Rv RV ⟩ + · · · .

(24)

It is easy to see that a general term in the expansion (24) can be obtained as follows. (i) Take a bracketed product ⟨VRVRV · · · VRV ⟩, and choose any V ’s in the product except for those at the two ends. (ii) Then replace each of the chosen V ’s with −v . Now we expand v = 1E in powers of V as

v = 1E = v (1) + v (2) + v (3) + · · · ,

(25)

(n)

where v stands for the nth order term in the perturbation. By substituting the above expansion (25) into Eq. (24), we arrive at the so-called bracketing expression for v (n) = 1E (n) , of which the first few terms are as follows [9]:

v (1) = 1E (1) = ⟨V ⟩, v (2) = 1E (2) = ⟨VRV ⟩, v (3) = 1E (3) = ⟨VRVRV ⟩ − ⟨VR v (1) RV ⟩ = ⟨VRVRV ⟩ − ⟨VR⟨V ⟩RV ⟩, v (4) = 1E (4) = ⟨VRVRVRV ⟩ − ⟨VRVR v (1) RV ⟩ − ⟨VR v (1) RVRV ⟩ + ⟨VR v (1) R v (1) RV ⟩ − ⟨VR v (2) RV ⟩ = ⟨VRVRVRV ⟩ − ⟨VRVR⟨V ⟩RV ⟩ − ⟨VR⟨V ⟩RVRV ⟩ + ⟨VR⟨V ⟩R⟨V ⟩RV ⟩ − ⟨VR⟨VRV ⟩RV ⟩. (26) Now we give two examples to show how we calculate these expressions. Let us first consider ⟨VRVRV ⟩ in v (3) = 1E (3) . It is calculated by inserting the projector Q of Eq. (5) in

⟨VRVRV ⟩ =



⟨V ⟩⟩I RI I ⟨⟨V ⟩⟩J RJ J ⟨⟨V ⟩.

(27)

I

Here we have used the double brackets, ‘‘⟨⟨’’ and ‘‘⟩⟩’’, to indicate the Q -space matrix elements, and we Q for the propagator. The next example is the final have introduced the shorthand notation RI = ϵ−ϵ I

term in v (4) = 1E (4) , which can be calculated as Eq. (27) to give

⟨VR⟨VRV ⟩RV ⟩ =

 IJ

⟨V ⟩⟩I RI ⟨V ⟩⟩J RJ J ⟨⟨V ⟩ RI I ⟨⟨V ⟩.

(28)

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Here we stress the following point. The bracketing representation for v = 1E only specifies by brackets ⟨· · ·⟩ where to take the P-space expectation values in the bracketed product ⟨VRVRV · · · VRV ⟩. This is in fact sufficient to indicate how we calculate each term, as can be easily confirmed by the above examples. Eq. (26) suggests that each term of the expansion is obtained by inserting bracket pairs at suitable positions in a bracketed product ⟨VRVR · · · RV ⟩. To transform the above observation into a precise statement, here we define valid bracketing schemes as follows. (i) Valid bracketing schemes are obtained by inserting bracket pairs around V ’s and around VR · · · RV products in the bracketed product ⟨VRVR · · · RV ⟩. (ii) They include multiple brackets such as ⟨VR⟨V ⟩R⟨V ⟩RV ⟩ and correctly nested brackets such as ⟨VR⟨VR⟨V ⟩RV ⟩RV ⟩. (iii) They exclude schemes such as ⟨VR⟨VR⟨VRV ⟩⟩RV ⟩, where an interaction V is doubly bracketed directly. Then, we can show that the perturbation expansion of v is given by the following proposition. Proposition (Bracketing for v = 1E). The nth order term of v = 1E is given by n V ′s

v

(n)

= 1E

(n)



=

   ⟨VRV · · · VRV ⟩.

(29)

bracketing

Here the summation runs over all the valid bracketing schemes, and it is assumed that each pair of brackets carries a minus sign. The above proposition can be proven easily by induction. First, we observe in Eq. (26) that the proposition is true for n = 1, 2, 3. Second, assuming that the proposition holds up to n = k, we calculate v (k+1) by collecting all the (k + 1)th order terms in Eq. (24). It is then easy to see that they contain v (1) , v (2) , . . . , v (k−1) only, which are already given by the bracketing by assumption. We can conclude, therefore, that each term in the expansion of v (k+1) is also expressed by a valid bracketing scheme. The reverse is also true; a little consideration shows that each valid bracketing scheme of the (k + 1)th order does exist in the (k + 1)th order terms of Eq. (24). This proves the above proposition. In this section, we have explained the well-known bracketing technique of the RSPT. Although it is a very convenient method, neither its multi-dimensional generalization nor its graphic expression is known to date. In the following sections, we develop the theory in these two directions. 4. v in degenerate model space In this section, we derive the effective interaction v in multi-dimensional P-space by generalizing the RSPT in Section 3. The present section is composed as follows. In Section 4.1, we derive an operator equation for v using the basis set {|φα ⟩}. Then in Section 4.2, we express its matrix elements, ⟨i|v|j⟩ = i ⟨v⟩j , in the basis set {|i⟩}. In Section 4.3, we derive a generalized bracketing representation of the perturbation expansion of i ⟨v⟩j , for which we introduce a graphic (folded diagram) representation in Section 4.4. Finally in Section 4.5, we establish a one-to-one correspondence between a valid bracketing and a valid folded diagram. Here in Section 4, the P-space is assumed to be degenerate at energy ϵ . Then Eq. (3) reduces to H0 |i⟩ = ϵ |i⟩,

i = 1, . . . , d,

(30)

which also means H0 |φα ⟩ = ϵ |φα ⟩,

α = 1, . . . , d.

Generalization to nondegenerate P-space will be made in Section 5.

(31)

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4.1. From ⟨ β|v|α⟩ to v In this section, working with the basis set {|φα ⟩}, we derive an operator equation for v [1]. In the following, we adopt the shorthand notation of α for φα and  β for  φβ . In degenerate P-space, because Eq. (31) holds, Eqs. (9) and (12) show that |φα ⟩ = |α⟩ is also an eigenstate of v and Qˆ (Eα ) as

v |α⟩ = Qˆ (Eα ) |α⟩ = 1Eα |α⟩,

α = 1 , . . . , d,

(32)

where the energy shift 1Eα is given by

1Eα = Eα − ϵ = ⟨ α |v|α⟩.

(33)

Then, Eq. (32) shows that ⟨ β|v|α⟩ can be written as

⟨ β|v|α⟩ = ⟨ β|Qˆ (Eα )|α⟩ = δβ,α 1Eα           −1Eα −1Eα −1Eα   RV + VR R RV + · · · α , = β V + VRV + VR V

V

V

(34)

where R = ϵ−QH as in one-dimensional P-space. In the above, we have expanded the Brillouin–Wigner 0 propagator in the Qˆ -box of Eq. (11) as Q Eα − H

=

Q

ϵ + 1Eα − H0 − V

 =R+R

−1Eα



V

R + ···.

(35)

Note that Eq. (34) is the multi-dimensional counterpart of Eq. (23). Now we examine a general term in the above expansion (34). Let us take ⟨ β| VR (−1Eα ) RVR (−1Eα ) RV |α⟩ as an example, which we now transform as follows:

⟨ β| VR (−1Eα ) RVR (−1Eα ) RV |α⟩ = ⟨ β|VRRVRRV (−v)(−v)|α⟩,

(36)

where we have used the fact that v is diagonal in the basis set {|α⟩}, as shown in Eq. (32), and therefore (−1Eα )(−1Eα )|α⟩ = (−v)(−v)|α⟩. The above observation for the example (36) obviously applies to all the terms that appear in the expansion (34) to give

⟨ β|v|α⟩ =



⟨ β|VR · · · RV (−v)(−v) · · · |α⟩.

(37)

term

Because Eq. (37) holds for any ⟨ β| and |α⟩, we can eliminate ⟨ β| and |α⟩ to obtain the following operator equation:

v=



VR · · · RV (−v)(−v) · · · ,

(38)

term

where we have assumed implicitly that the right hand side is always multiplied by the projector P both from the right and left. It is now easy to see that a general term on the right hand side of Eq. (38) is given as follows. (i) Take a product VRVRV · · · VRV , and eliminate any number of V ’s from the product except for those at the two ends. (ii) Then multiply the obtained expression by the same number of (−v)’s from the right. The explicit form of Eq. (38) is therefore

v = V + VRV + VRVRV − VRRV v + VRVRVRV − VRRVRV v − VRVRRV v + VRRRV vv + · · · , which is the starting point of the next section.

(39)

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K. Takayanagi / Annals of Physics 350 (2014) 501–532

4.2. From v to i ⟨v⟩j We have derived the operator equation (39) using the basis set {|α⟩}. Here we evaluate its matrix elements in the basis set {|i⟩} at hand, which leads to the bracketing and also to the folded diagram representations of i ⟨v⟩j = ⟨i|v|j⟩. Eq. (39) immediately gives the following expression for i ⟨v⟩j : i

⟨v⟩j = i ⟨V ⟩j + i ⟨VRV ⟩j + i ⟨VRVRV ⟩j − i ⟨VRRV v⟩j + i ⟨VRVRVRV ⟩j − i ⟨VRRVRV v⟩j − i ⟨VRVRRV v⟩j + i ⟨VRRRV vv⟩j + · · · .

(40)

This is not, however, the generalization of Eq. (24) to multi-dimensional P-space, which we will derive in this section. Let us observe the following difference between Eqs. (24) and (40); v = ⟨v⟩ in Eq. (24) is a c-number and is placed between two R’s, while v in Eq. (40) is a q-number and is placed at the right end. Here we explain, by introducing the concept of reducible and irreducible, how to transform Eq. (40) into a form that corresponds to Eq. (24) in one-dimensional P-space. We shall see that the final expression (46) for i ⟨v⟩j in this section can be obtained by replacing v with ⟨v⟩ in Eq. (24) and by assigning P-space indices to all the brackets ⟨· · ·⟩ in a well-defined way. In the following, we proceed using several examples. As a first example, let us take i ⟨VRVRV ⟩j , the third term on the right hand side of Eq. (40). It is calculated in the same way as Eq. (27) using Q -space intermediate states only: i

⟨V R V R V ⟩j =

 i

⟨V ⟩⟩I RI I ⟨⟨V ⟩⟩J RJ J ⟨⟨V ⟩j .

(41)

IJ

This term is irreducible in the sense that it cannot be written as a product of two or more P-space matrices. We leave such irreducible terms as they stand in Eq. (40). Comparing Eqs. (27) and (41), we note that the single brackets ‘‘⟨’’ and ‘‘⟩’’ representing the P-space matrix elements carry the P-space indices in Eq. (41), while this was not the case in Eq. (27). As a second example, let us take i ⟨VRRV v⟩j with the insertion of a single v , i.e., the fourth term in Eq. (40). By inserting the projector P of Eq. (5) in front of v , we rewrite this term in a way that features its matrix structure in the P-space: i

⟨V R R V v⟩j =

 i

⟨V R R V ⟩k k ⟨v⟩j

i

⟨V R k ⟨v⟩j R V ⟩k .

k

=



(42)

k

The first line shows that this is a product of two matrices defined in the P-space, i ⟨V R R V ⟩k and k ⟨v⟩j , showing explicitly that it is reducible. Note here that i ⟨V R R V ⟩k is an irreducible matrix and cannot be decomposed into smaller P-space matrices. In going to the second line, which is our desired expression here, we have moved k ⟨v⟩j , now being a c-number, to between the two R’s. We then realize that the resultant expression is obviously a multi-dimensional counterpart of the fourth term ⟨VRv RV ⟩ in Eq. (24). This is not, however, the only reason to go to the second line in Eq. (42). Another reason, which is more important, is that, by the transfer of k ⟨v⟩j , the expression acquires a desirable property; in the second line, the repeated index ‘‘k’’ appears as k ⟨· · · ·⟩k , i.e., ‘‘ ⟩k ’’ is placed to the right of ‘‘k ⟨’’. Because  the P-space matrix multiplication ‘‘ k · ·⟩k k ⟨··’’ requires ‘‘ ⟩k ’’ to be read first and then ‘‘k ⟨’’, we read the repeated index ‘‘k’’ from the right ‘‘ ⟩k ’’ to the left ‘‘k ⟨’’ in the second line. This is in contrast to the first line, where we read the repeated index ‘‘k’’ from the left ‘‘ ⟩k ’’ to the right ‘‘k ⟨’’. This point will soon turn out to be essential when we introduce its graphic (folded diagram) expression in Section 4.3. As a third example, let us take i ⟨VRRVRRV vv⟩i , which has two insertions of v and is therefore reducible. This is transformed as follows: i

⟨VRRVRRV vv⟩j =

 i

⟨VRRVRRV ⟩k k ⟨v⟩l l ⟨v⟩j

i

⟨V R l ⟨v⟩j RVR k ⟨v⟩l R V ⟩k .

kl

=

 kl

(43)

K. Takayanagi / Annals of Physics 350 (2014) 501–532

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The first line shows the structure in the P-space: this is a product of three P-space matrices. In going from the first line to the second line, we have moved the two ⟨v⟩’s into the two spaces between the R’s in the reverse order; first we have moved the left matrix, k ⟨v⟩l , into the right space, and then we have moved the right matrix, l ⟨v⟩j , into the left space. The reason for this ordering is as follows. In the second line, the brackets with the repeated  index ‘‘k’’ are placed as ‘‘k ⟨·· ··⟩k ’’. Then the P-space matrix multiplication, which is specified as ‘‘ k · ·⟩k k ⟨··’’ in the first line, requires that we read the repeated index ‘‘k’’ from the right ‘‘ ⟩k ’’ to the left ‘‘k ⟨’’ in the second line. We can also confirm the same for the repeated index ‘‘l’’. The second line thus guarantees that we always read repeated P-space indices from the right to the left, which is the reason for the above ordering of k ⟨v⟩l and k ⟨v⟩l in the second line. The above examples can be easily generalized; we evaluate the matrix element of the right hand side of Eq. (38) by inserting the projector P of Eq. (5) in front of each v to obtain i

⟨v⟩j =

 i

⟨VR · · · RV ⟩k k ⟨v⟩l · · · ⟨v⟩j ,

(44)

term kl···

where we have assumed that each ⟨v⟩ carries a minus sign. Each term in Eq. (44) is composed of an irreducible matrix i ⟨VR · · · RV ⟩k and several ⟨v⟩’s at the right end, all of which have been assigned appropriate P-space indices to specify the matrix multiplications. Then, in each term on the right hand side, we move ⟨v⟩’s into the irreducible matrix i ⟨VR · · · RV ⟩k in the reverse order as we did in going from the first line to the second line in Eq. (43); we first move the leftmost ⟨v⟩ into the rightmost space between the two R’s, and then move the second leftmost ⟨v⟩ into the second rightmost space between the two R’s, and so on. This rule completely determines how to place P-space indices in the resultant expression: i

⟨v⟩j =

 i

⟨VR · · · ⟨v⟩j · · · k ⟨v⟩l · · · RV ⟩k .

(45)

term kl···

Here we realize that the P-space matrix multiplications in Eq. (44) require that we read each of the repeated P-space indices ‘‘kl · · ·’’ in Eq. (45) from the right to the left. Brackets ‘‘k ⟨’’ and ‘‘ ⟩k ’’ with the repeated index ‘‘k’’, for example, appear in the order ‘‘k ⟨· · · ·⟩k ’’ in our final expression (45), i.e., ‘‘ ⟩k ’’ appears to the right of ‘‘k ⟨’’, which we refer to as the ‘‘bracket product rule’’ for simplicity in the following. It is now easy to see that a general term in the above expansion (45) can be given by the following rules: (i) Take a bracketed product ⟨VRVRV · · · VRV ⟩, and choose any V ’s in the product except for those at the two ends. (ii) Then replace each of the chosen V ’s with −⟨v⟩. (iii) In the obtained expression, assign P-space matrix indices to each bracket pair ⟨· · ·⟩ in accordance with the above bracket product rule. The third rule (iii), which is absent in one-dimensional P-space in Section 3.2, specifies how to place P-space indices; rule (iii) embeds the P-space matrix structure in Eq. (24) to give i ⟨v⟩j of Eq. (45). Let us consider an example. Suppose we have ⟨VR⟨v⟩RVR⟨v⟩RV ⟩ at hand following rules (i) and (ii), which also appears in one-dimensional P-space where v = ⟨v⟩. Then rule (iii) assigns P-space indices to the brackets as

 i

⟨V R l ⟨v⟩j RVR k ⟨v⟩l R V ⟩k ,

kl

to give the second line of Eq. (43), which originates from i ⟨VRRVRRV vv⟩j in the first line. Conversely, to calculate i ⟨VRRVRRV vv⟩j in multi-dimensional P-space, we simply need to embed the matrix structure by rule (iii) in the bracketing ⟨VR⟨v⟩RVR⟨v⟩RV ⟩ given by rules (i) and (ii). Let us give another example. Suppose that rules (i) and (ii) give −⟨VR⟨v⟩RVR⟨v⟩R⟨v⟩RV ⟩. Then rule (iii) allots P-space indices to the brackets to give

−

 i

⟨V R

m

⟨v⟩j RVR l ⟨v⟩m R k ⟨v⟩l R V ⟩k ,

klm

which yields −i ⟨VRRVRRRV vvv⟩j in multi-dimensional P-space.

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K. Takayanagi / Annals of Physics 350 (2014) 501–532

It is now straightforward to write down the expansion (45) explicitly as follows: i

⟨v⟩j = i ⟨V ⟩j + i ⟨VRV ⟩j + i ⟨VRVRV ⟩j − i ⟨VR k ⟨v⟩j RV ⟩k + i ⟨VRVRVRV ⟩j − i ⟨VRVR k ⟨v⟩j RV ⟩k − i ⟨VR k ⟨v⟩j RVRV ⟩k + i ⟨VR l ⟨v⟩j R k ⟨v⟩l RV ⟩k + · · · ,

(46)

where we have implicitly assumed the summation over repeated P-space indices; the fourth term reads as given in Eq. (42). To summarize, we have generalized Eq. (24) to multi-dimensional P-space, which is simply Eq. (46), which we have just derived. Our next task is to express i ⟨v⟩j in Eq. (46) in terms of V and R only. 4.3. Generalized bracketing—algebraic expression for i ⟨v⟩j In one-dimensional P-space, we have derived, starting from Eq. (24), the perturbation expansion (26) for v = 1E = ⟨v⟩ in the bracketing representation. Also in multi-dimensional P-space here, we proceed in the same fashion; starting from Eq. (46), we derive the perturbation expansion for i ⟨v⟩j , which we call the generalized bracketing representation. Let us expand i ⟨v⟩j in powers of V as i

⟨v⟩j = i ⟨v (1) ⟩j + i ⟨v (2) ⟩j + i ⟨v (3) ⟩j + · · · .

(47)

By substituting the above expression into Eq. (46), we find, as in Eq. (26), the following expressions: i

⟨v (1) ⟩j = i ⟨V ⟩j ,

i

⟨v (2) ⟩j = i ⟨VRV ⟩j ,

i

⟨v (3) ⟩j = i ⟨VRVRV ⟩j − i ⟨VR k ⟨V ⟩j RV ⟩k ,

i

⟨v (4) ⟩j = i ⟨VRVRVRV ⟩j − i ⟨VRVR k ⟨V ⟩j RV ⟩k − i ⟨VR k ⟨V ⟩j RVRV ⟩k + i ⟨VR l ⟨V ⟩j R k ⟨V ⟩l RV ⟩k − i ⟨VR k ⟨VRV ⟩j RV ⟩k ,

(48)

where ‘‘i’’ and ‘‘j’’ are fixed external indices, while ‘‘k’’ and ‘‘l’’ are dummy indices for summation. In Eq. (48) and also in what follows, we adopt the following simplified notation: when a bracketing scheme is written solely in terms of V and R, summations over repeated dummy indices are implicitly assumed. In other cases, summations are explicitly indicated. Eq. (48) clearly corresponds to Eq. (26) in one-dimensional P-space. There is, however, an important difference here; each ⟨v⟩ has been assigned P-space indices that stem from rule (iii) in Section 4.2, for which we refer to Eq. (48) as the generalized bracketing representation. In writing down the explicit forms of i ⟨v (n) ⟩j in Eq. (48), we repeatedly substitute the right hand side of Eq. (46) itself in each ⟨v⟩ on the right hand side of Eq. (46). Then, by construction, the P-space indices necessarily obey the bracket product rule in the resultant expressions in Eq. (48). The above discussion is sufficient to explain the following proposition for i ⟨v⟩j , which can be proven easily by induction in the same way as for (29) in one-dimensional P-space. Proposition (Generalized Bracketing for i ⟨v⟩j in Degenerate P-Space). The nth order term of i ⟨v⟩j is given by n V ′s i

⟨v (n) ⟩j =



   i ⟨VRV · · · VRV ⟩.

(49)

bracketing

Here the summation runs over all valid ways of bracketing in the same way as in one-dimensional P-space. On the right hand side, each pair of brackets ⟨· · ·⟩ carries a minus sign and appropriate P-space indices specified by the bracket product rule. With the above proposition, we can express all the terms in the perturbation expansion of i ⟨v⟩j in the generalized bracketing representation. Now we examine their matrix structures in P-space.

K. Takayanagi / Annals of Physics 350 (2014) 501–532

511

As an example, let us take i ⟨VR k ⟨VRV ⟩j RV ⟩k , the final term in the expression for i ⟨v (4) ⟩j in Eq. (48). We note that it is calculated as 1

i

3

4

2

⟨V R k ⟨V RV ⟩j RV ⟩k =

1

 i

2

3

4

⟨V RRV ⟩k k ⟨V RV ⟩j .

(50)

k

Here the left hand side shows the bracketing expression with suitable P-space indices that obey the bracket product rule, and the right hand side explicitly shows that this is a product of two irreducible P-space matrices. Numbers placed above the V ’s, which indicate the order of multiplication in actual calculations on the right hand side, will help to distinguish interaction operators. Now we give two more examples. 1

i

5

6

3

4

2

⟨V R l ⟨V R V ⟩j R k ⟨V R V ⟩l R V ⟩k =

1

 i

2

3

4

5

6

⟨V RRR V ⟩k k ⟨V R V ⟩l l ⟨V R V ⟩j .

(51)

kl 1

i

3

5

4

2

⟨V R k ⟨V R l ⟨V ⟩j R V ⟩l R V ⟩k =

1

 i

2

3

4

5

⟨V RR V ⟩k k ⟨V RR V ⟩l l ⟨V ⟩j .

(52)

kl

Eqs. (51) and (52) respectively represent the multiple and nested bracketing schemes. On the left hand sides, P-space indices obey the bracket product rule, indicating the P-space matrix structures given on the right hand sides. Here we note that the right hand sides of Eqs. (50), (51), and (52) respectively represent the following matrix elements of the corresponding operator products: 1

2

3

4

i

⟨V RRV P V RV ⟩j ,

i

⟨V RRR V P V R V P V R V ⟩j ,

i

⟨V RR V P V RR V P V ⟩j ,

1

1

2

2

3

3

4

4

(50a) 5

6

(51a)

5

(52a)

where the inserted projector P specifies the P-space matrix structure of each term. Note also that we have placed numbers above the V ’s in Eqs. (50), (51), and (52) so that they are in ascending order in the corresponding expressions (50a), (51a), and (52a). The above examples (50), (51), and (52) are sufficient to show that each generalized bracketing scheme in Eq. (49) is calculated as a product of irreducible matrices, which are multiplied in the order specified by the bracket product rule. Here we make a comment on the notation for the bracketing representation. As an example, let us take the generalized bracketing scheme i ⟨VR k ⟨V R V ⟩j RV ⟩k in Eq. (50). It is easily noted that we can dispense with P-space indices in this expression; to specify its P-space matrix structure, it would be sufficient simply to write ⟨VR⟨VRV ⟩RV ⟩ as in Eq. (28) in one-dimensional P-space. We can further simplify the expression by dropping all R’s; we realize that the abbreviation ⟨V ⟨VV ⟩V ⟩ can uniquely specify the expression in Eq. (50). In this work, however, we display all the propagators and P-space indices explicitly for the sake of clarity; bracketing schemes without P-space indices are used only in one-dimensional P-space. Let us summarize what we have shown in this section. In one-dimensional P-space, the bracketing scheme (26) for v = 1E indicates by brackets ⟨· · ·⟩ where to take the P-space matrix element. Here in multi-dimensional P-space, the generalized bracketing scheme (48) for i ⟨v⟩j indicates not only where but also with which states to take the P-space matrix elements, i.e., the brackets ⟨· · ·⟩ carry the P-space matrix indices that specify how we calculate i ⟨v⟩j as a product of irreducible P-space matrices. In short, the generalized bracketing scheme here is obtained by embedding a P-space matrix structure in the bracketing scheme of the RSPT discussed in Section 3. We can now replace ‘‘Bloch’’ on the lower left of Table 1 with ‘‘RS’’. 4.4. Folded diagram—diagrammatic expression for ⟨v⟩ In Section 4.3, we have derived the perturbation expansion of i ⟨v⟩j in the generalized bracketing scheme (49). Here we show that it has a natural diagrammatic expression, which is very convenient

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K. Takayanagi / Annals of Physics 350 (2014) 501–532

4

3

1

2

1

3

4

2

Fig. 1. Folded diagram representation of the bracketing expression (50) presented below the diagram. Circles stand for the interaction vertices V , and dashed lines the propagators R. Solid lines with single and double arrows represent, respectively, the P- and Q -space lines.

in actual calculations. We refer to our diagram simply as the folded diagram, because we shall see that it is closely related to the folded diagrams that have been introduced in several conventions. In the following, we explain our folded diagram rule using an example, i.e., Eq. (50), and the corresponding diagram shown in Fig. 1. A comparison with other rules will be briefly made in Section 6. We can evaluate i ⟨V R k ⟨VRV ⟩j R V ⟩k , the left hand side of Eq. (50), by inserting the projector Q of Eq. (5) in all possible places as we did in Eq. (28) to obtain 1

i

3

4

2

⟨V R k ⟨V R V ⟩j R V ⟩k =

1

 i k

3

4

2

⟨V ⟩⟩I RI k ⟨V ⟩⟩J RJ J ⟨⟨V ⟩j RI I ⟨⟨V ⟩k .

(53)

IJ

The following diagram rule gives an alternative way to obtain the right hand side of Eq. (53) starting from the bracketing scheme on the left hand side. (i) Given a generalized bracketing scheme representing i ⟨v (n) ⟩j , we first draw a circle, representing V , immediately above each V in the bracketing expression; the positions of V ’s in the bracketing expression represent their horizontal positions in the diagram. We thus draw four circles on top of the four V ’s in the bracketing scheme presented in Fig. 1. Our diagram rule assumes that we read the diagram horizontally, while usual conventions prefer to read it vertically [4,8,9]. For the sake of clarity, here we assign numbers to the interactions as in Eq. (50). (ii) Then we connect all the circles by solid lines in the order of matrix multiplications, i.e., in ascending order of the assigned numbers, 1 → 2 → 3 → · · · → n. We start from the leftmost 1

n

interaction V and arrive at V , and add a line leaving the diagram (slightly upwards) to the left from n

V [20]. We organize the diagram so that the solid lines are folded to form a zigzag as in Fig. 1. Now we follow the obtained zigzag line in the reverse order, i.e., n → (n − 1) → · · · → 2 → 1. If p+1

p

a line V →V goes (slightly downwards) to the right, it is called a P-space line and is assigned a single arrow pointing to the right.

:

P-space line.

Now we label each P-space line as indicated in the bracketing representation. Here the internal P-space line ‘‘k’’, for example, represents a line going from ‘‘k ⟨V ’’ to ‘‘V ⟩k ’’, which are placed as k ⟨V · · · V ⟩k in the bracketing by our bracket product rule. This explains why the P-space line ‘‘k’’ 3

2

necessarily points to the right. In Fig. 1, we have an internal P-space line V →V representing



k

4

and an external P-space line ‘‘j’’ entering V . q

q +1

(iii) Similarly, each line going to the left as V ← V is called a Q -space line and is assigned a dummy Q -space index and a double arrow pointing to the left.

:

Q -space line.

K. Takayanagi / Annals of Physics 350 (2014) 501–532 q +1

q

513 q +1

q

Because V is placed to the left of V in a pair of brackets as ⟨· · · V · · · V · · ·⟩, each Q -space 1

2

3

4

line necessarily points to the left. In Fig. 1, we have two Q -space lines, V ←V and V ←V , which   signify I and J , respectively. At this stage, we have fixed all the indices in Fig. 1, where each element of V on the right hand side of Eq. (53) is now visible in the diagram. Note also that the horizontal span of each Q -space line in the diagram is the span of the corresponding irreducible P-space matrix in the bracketing expression immediately below the diagram. In Fig. 1, the span 3

4

of the Q -space line ‘‘J’’ (V ←V ), for example, is the same as the span of the irreducible matrix 3

k

4

⟨V R V ⟩j .

(iv) Now we express each propagator by a perpendicular dashed line immediately above each R in the bracketing representation in such a way that it crosses the relevant Q -space line that has given its Q -space index to the propagator. For each R in the bracketing, the relevant Q -space line q +1

q

can be identified as follows. If a propagator R is placed between V and V and is bracketed in q

q +1

q

q +1

the same way as V and V , the propagator R belongs to the Q -space line V ← V and shares its Q -space index. This can be easily confirmed by recalling the process of inserting the projector Q in Eq. (53). On the right hand side of Eq. (53), the two RI ’s on both sides belong to the Q -space 1

2

3

4

line ‘‘I’’ (V ←V ), and the middle RJ belongs to the Q -space line ‘‘J’’ (V ←V ), which fixes all three propagators and their correct positions in Fig. 1. (v) The overall sign of the diagram derives from the minus sign allotted to each pair of brackets ⟨· · ·⟩ in the proposition (49). It is obviously given by (−1)nf , where nf is the number of folded (internal) P-space lines. Note here that the final P-space line ‘‘j’’ entering the diagram in Fig. 1 is not included in nf . (vi) Given a generalized bracketing, the above rules completely specify how to draw its corresponding folded diagram, in which all the elements (V ’s and R’s) are explicitly visible. By multiplying them all, and by summing over all the dummy indices, we obtain the complete expression for actual calculation. Finally, by taking into account the phase (−1)nf , we obtain the contribution of the diagram to i ⟨v⟩j . In our example, starting from the generalized bracketing i ⟨V R k ⟨VRV ⟩j R V ⟩k , the left hand side of Eq. (53), we first draw the diagram in Fig. 1 and then read off the expression on the right hand side from the diagram. Its contribution to i ⟨v⟩j is (−1) times the right hand side of Eq. (53) because nf = 1. Several points should be made regarding the above diagram rule. First, each Q -space line in the folded diagram corresponds to an irreducible P-space matrix ⟨· · ·⟩ in the bracketing. These irreducible matrices are multiplied as specified by the P-space lines in the diagram. The folded diagram in Fig. 1 visualizes the fact that the two irreducible P-space matrices ⟨VRRV ⟩ and ⟨VRV ⟩ are multiplied in the order ⟨VRRV ⟩ ⟨VRV ⟩. Second, let us recall that a bracketing expression is reducible if it contains an internal pair of brackets ⟨· · ·⟩. Our diagram rule shows that its corresponding diagram has a folded internal P-space line. We therefore refer to such a diagram as being reducible. We can easily realize that a reducible diagram can be split into two independent parts by cutting a folded internal P-space line. In Fig. 1, we can split the diagram into two parts by cutting the folded line ‘‘k’’, showing that the diagram is reducible. Third, we can now understand the role played by our bracket product rule. In our folded diagram, the P- and Q -space lines respectively point to the right and left, which obviously stems from our bracket product rule. This is convenient because we can distinguish between P- and Q -space lines by their directions in the diagram. Note that, in transforming Eq. (44) into Eq. (45), we could have chosen another ‘‘bracket product rule’’. Then the resultant expression would not have any clear diagrammatic expression. The above observation explains that our bracket product rule is crucial in deriving our convenient bracketing scheme that can be translated in a simple way into a clear diagrammatic expression. Fourth, we comment on the procedure of actual calculations. We wrote down the right hand side of Eq. (53) by inserting the projector Q in all possible places on the left hand side. The folded diagram

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K. Takayanagi / Annals of Physics 350 (2014) 501–532

Fig. 2. Bracketing and folded diagram representations for i ⟨v⟩j in degenerate P-space. The notation is the same as for Fig. 1.

technique here has established another method of writing down the right hand side of Eq. (53) starting from the left hand side. We might think here that we could dispense with the folded diagram because the former method is apparently simpler. This is not, however, the case in the nondegenerate P-space that we examine in Section 5. We shall see that the former method does not work in nondegenerate P-space, and therefore the folded diagram is indispensable for writing down the right hand side of Eq. (69), the generalization of Eq. (53) to nondegenerate P-space. Now we present several examples in Fig. 2, in which we can see, for a given bracketing scheme, how to draw the corresponding folded diagram and how to calculate it. The first diagram shows the second order irreducible bracketing i ⟨VRV ⟩j , i

⟨VRV ⟩j =

 i

⟨V ⟩⟩I RI I ⟨⟨V ⟩j ,

(54)

I

where we have explicitly written the complete form for actual calculation on the right hand side. The second diagram shows a third order irreducible bracketing i ⟨VRVRV ⟩j , which can be calculated in the same manner as Eq. (54) to give i

⟨VRVRV ⟩j =

 i

⟨V ⟩⟩I RI I ⟨⟨V ⟩⟩J RJ J ⟨⟨V ⟩j .

(55)

IJ

The third diagram shows a third order reducible bracketing i

⟨V R k ⟨V ⟩j R V ⟩k =

 i

⟨VRRV ⟩k k ⟨V ⟩j

k

=

 i k

⟨V ⟩⟩I RI k ⟨V ⟩j RI I ⟨⟨V ⟩k ,

(56)

I

where the first line shows that this is a product of two irreducible P-space matrices, and the second line displays its complete expression for actual calculation. In this example, we can realize that k ⟨V ⟩j = k ⟨v (1) ⟩j is a point-like Q -space line and that the line ‘‘k’’ is a folded internal P-space line.

K. Takayanagi / Annals of Physics 350 (2014) 501–532

515

The fourth diagram shows the following sixth order reducible term, which is an example of nested bracketing: i

⟨V R k ⟨VR l ⟨V R V ⟩j R V ⟩l R V ⟩k =

 i

⟨V RR V ⟩k k ⟨VRRV ⟩l l ⟨VRV ⟩j

kl

=

 i kl

⟨V ⟩⟩I RI k ⟨V ⟩⟩J RJ l ⟨V ⟩⟩K RK

K

⟨⟨V ⟩j RJ J ⟨⟨V ⟩l RI I ⟨⟨V ⟩k ,

(57)

IJK

where the second line shows that this is a product of three irreducible P-space matrices. Correspondingly, we note two folded internal P-space lines ‘‘k’’ and ‘‘l’’. The third line shows its complete expression for actual calculation that can be read off immediately from the folded diagram. The fifth diagram shows the following sixth order reducible term, which is an example of multiple bracketing: i

⟨V R l ⟨VRV ⟩j R V R k ⟨V ⟩l R V ⟩k =

 i

⟨V RR V RR V ⟩k k ⟨V ⟩l l ⟨VRV ⟩j

kl

=

 i kl

⟨V ⟩⟩I RI l ⟨V ⟩⟩K RK

K

⟨⟨V ⟩j RI I ⟨⟨V ⟩⟩J RJ k ⟨V ⟩l RJ J ⟨⟨V ⟩k .

(58)

IJK

To summarize this section, we have established a folded diagram representation to express the generalized bracketing scheme for i ⟨v⟩j . We can now replace ‘‘Bloch’’ on the lower right of Table 1 with ‘‘RS’’. 4.5. Correspondence between bracketing and folded diagram Here we explain, by introducing the concept of a valid folded diagram, that we can guarantee oneto-one correspondence between the bracketing and folded diagram representations of i ⟨v⟩j , i.e., valid bracketing ←→ valid folded diagram.

(59)

Let us note that the construction of the folded diagram in Section 4.4 shows that each valid bracketing scheme corresponds uniquely to its own diagrammatic representation, which we call a valid folded diagram. Then the direction −→ in (59) is obvious. We must therefore explain the opposite direction, ←−, i.e., that each valid folded diagram corresponds uniquely to a valid bracketing scheme of i ⟨v⟩j . For the above purpose, we now explain that a valid folded diagram can be easily distinguished from other diagrams by examining the bracketing expression, which in turn gives a criterion to identify valid folded diagrams.  First, let us consider two pairs of brackets that are multiplied as k ⟨V · · · V ⟩k k ⟨V · · · V ⟩ in actual calculations. Then, in a valid bracketing scheme, these two pairs of brackets should appear in either of the following relations to satisfy the bracket product rule:

(i) k ⟨V · · · V ⟩ · · · ⟨V · · · V ⟩k ,

(ii) ⟨V · · · k ⟨V · · · V ⟩ · · · V ⟩k .

(60)

Conversely, if all of the multiplied pairs of brackets meet the above requirement in a given bracketing scheme, we can easily show that the bracketing scheme is valid. Second, we translate the above conditions (i) and (ii) for valid bracketing schemes into the following conditions (i′ ) and (ii′ ) for valid folded diagrams. Consider two Q -space lines connected by a P-space line. It is then easy to see that, in a valid diagram, they satisfy either of the following two conditions. (i′ ) There is no horizontal overlap between them. (ii′ ) The horizontal span of the lower Q -space line, corresponding to ⟨V · · · V ⟩k , contains that of the upper Q -space line representing k ⟨V · · · V ⟩ in such a way that there is no interaction V on the lower Q -space line in the span of the upper Q -space line.

516

K. Takayanagi / Annals of Physics 350 (2014) 501–532

Fig. 3. Invalid (upper) and valid (lower) folded diagrams. Valid diagrams correspond to valid bracketing schemes, while invalid diagrams do not. The valid diagrams represent Eqs. (51) and (52) in Section 4.3, whose simplified forms without R are presented below the diagrams.

We can easily realize that the converse is also true. We conclude, therefore, that a folded diagram is valid if and only if each pair of successive Q -space lines satisfies either of the above two conditions. Using this criterion, we can tell immediately whether a given diagram is valid or not. The above discussion shows that a valid folded diagram, being defined by the above conditions (i′ ) and (ii′ ), leads naturally to its corresponding valid bracketing scheme that is defined by conditions (i) and (ii). We can thus explain the direction ←− in (59), establishing a clear and natural one-to-one correspondence between the valid bracketing and valid folded diagram representations of i ⟨v⟩j . Before completing this section, to better understand the valid folded diagram, we show several valid and invalid diagrams in Fig. 3. On the left side, we show two sixth order diagrams. The lower diagram is a valid folded diagram that corresponds to the valid bracketing presented below the diagram; the two 3

4

5

6

Q -space lines V ←V and V ←V have no horizontal overlap satisfying condition (i′ ), to guarantee that 3

4

5

6

their corresponding brackets ⟨V · · · V ⟩ and ⟨V · · · V ⟩ satisfy condition (i) in (60). The upper diagram, 3′

4′

5′

6′

on the other hand, is invalid; the two Q -space lines V ←V and V ←V overlap partially, which does not have a counterpart in valid bracketing schemes. On the right side, we show fifth order diagrams, 1

2

3

4

where the span of the Q -space line V ←V contains that of V ←V . Here the lower diagram is valid; the relationship between these two Q -space lines is given by condition (ii′ ), and their corresponding 1

2

3

4

brackets ⟨V · · · V ⟩ and ⟨V · · · V ⟩ satisfy condition (ii) in (60). On the other hand, the upper diagram 3

4

5′

is invalid; in the span of the upper Q -space line V ←V , there is an interaction V on the lower Q -space 1

2

line V ←V that cannot be expressed by a valid bracketing scheme satisfying condition (ii) in (60). 5. v in nondegenerate model space In degenerate P-space, we now have both the (generalized) bracketing and folded diagram representations of the effective interaction v . In many realistic systems, however, we have to work in nondegenerate P-space [10,11,13]. We therefore generalize the results in Section 4 to nondegenerate P-space.

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The above generalization is most easily established by decomposing the total Hamiltonian H into ˜ 0 = P ϵ˜ P + QH0 Q and the perturbation V˜ = V + P (H0 − ϵ˜ )P, which the unperturbed Hamiltonian H is expressed as [13]

˜ 0 + V˜ = H =H



ϵ˜

0 QH0 Q

0



 ˜ + PV P QVP

PVQ QVQ



.

(61)

˜ 0 P = P ϵ˜ P and allows us to The decomposition (61) defines an artificially degenerate P-space by P H use the results obtained in Section 4. 5.1. ⟨˜v ⟩ in terms of V˜ and R˜ For the above decomposition (61) of H, the effective Schrödinger equation (12) becomes

(P ϵ˜ P + v˜ )|α⟩ = Eα |α⟩,

α = 1, . . . , d

(62)

where v˜ is the effective interaction corresponding to the decomposition (61). Note that the desired effective interaction v in Eq. (15) is given by H eff = P ϵ˜ P + v˜ = PH0 P + v, −→ v = v˜ − P (H0 − ϵ˜ )P .

(63)

With the new decomposition (61) at hand, we repeat the discussion leading to the proposition (49) to obtain the following expression for v˜ : n V˜ ′ s i

⟨˜v ⟩j =

  n

   ˜˜˜ ˜˜˜ i ⟨V RV · · · V RV ⟩,

(64)

bracketing

where the right hand side is expressed in terms of V˜ and R˜ = ϵ˜ −QH . In the following, we show how to 0 replace V˜ and R˜ in Eq. (64) with V and R, respectively. 5.2. ⟨˜v ⟩ in terms of V and R Let us first note that the difference P (H0 − ϵ˜ )P between V˜ and V appears only in the diagonal matrix elements inside P-space, i.e., i ⟨V˜ ⟩j = i ⟨V ⟩j + i ⟨H0 − ϵ˜ ⟩j = i ⟨V ⟩j + δij (ϵi − ϵ˜ ). The above observation suggests that it is convenient to perform the perturbation expansion (64) of ⟨˜v ⟩ in powers of V˜ in two steps; first, we perform the perturbation expansion in powers of V only, to end up with the following ⟨˜u⟩: n V ′s i

⟨˜u⟩j =

  n

   ˜ ˜ ⟨ V RV · · · V RV ⟩, i

(65)

bracketing

which is obtained by replacing each V˜ with V in the expansion (64). Second, we introduce V˜ − V = P (H0 − ϵ˜ )P perturbatively up to infinite order in each term in Eq. (65), which obviously transforms ⟨˜u⟩ of Eq. (65) into ⟨˜v ⟩ of Eq. (64). Now we explain how the above two-step perturbation works in practice. Suppose we have completed the first step and we have Eq. (65) at hand. As an example, let us take i ⟨V R˜ k ⟨V R˜ V ⟩j R˜ V ⟩k in the expansion (65), which reads i

⟨V R˜ k ⟨V R˜ V ⟩j R˜ V ⟩k =

 i k

⟨V ⟩⟩I R˜ I k ⟨V ⟩⟩J R˜ J J ⟨⟨V ⟩j R˜ I I ⟨⟨V ⟩k ,

(66)

IJ

and is shown in Fig. 4(a). In the second step, by introducing V˜ − V = P (H0 − ϵ˜ )P perturbatively, we change each propagator R˜ on the left hand side of Eq. (66) as R˜ −→ R˜ + R˜ ⟨H0 − ϵ˜ ⟩R˜ + R˜ ⟨H0 − ϵ˜ ⟩R˜ ⟨H0 − ϵ˜ ⟩R˜ + · · · .

(67)

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(a)

(b)

Fig. 4. (a) Folded diagram for i ⟨V R˜ k ⟨V R˜ V ⟩j R˜ V ⟩k in the expansion (65) given by the first step of the two-step perturbation. (b) Folded diagram for i ⟨V R k ⟨V R V ⟩j R V ⟩k in Eq. (69) obtained by the second step shown in Fig. 5.

Fig. 5. Diagrammatic expression for Eq. (68) that replaces R˜ I with RkI , the second step in the two-step perturbation explained

in the text. Open squares represent k ⟨H0 − ϵ˜ ⟩k = ϵk − ϵ˜ . On the left hand side, the propagator R˜ I crosses the Q -space line only, while on the right hand side the propagator RkI crosses both the Q - and P-space lines.

To be specific, let us replace the rightmost propagator R˜ on the left hand side of Eq. (66) as in the above. Then, noting that k ⟨H0 − ϵ˜ ⟩k′ = δkk′ (ϵk − ϵ˜ ), we realize that the corresponding rightmost propagator R˜ I on the right hand side of Eq. (66) changes as R˜ I =

=

1

ϵ˜ − ϵI 1

ϵk − ϵI

−→ R˜ I + R˜ I (ϵk − ϵ˜ )R˜ I + R˜ I (ϵk − ϵ˜ )R˜ I (ϵk − ϵ˜ )R˜ I + · · · = RkI ,

(68)

which is expressed diagrammatically in Fig. 5. The important point here is that R˜ I belongs to the Q -space line ‘‘I’’, and (ϵk − ϵ˜ ) attaches to the folded P-space line ‘‘k’’. Therefore, we have defined RkI in Eq. (68) that carries the P- and Q -space indices ‘‘k’’ and ‘‘I’’, respectively, in clear contrast to R˜ I , which carries the Q -space index ‘‘I’’ only. Note also that the arbitrary parameter ϵ˜ has disappeared from RkI . In the folded diagram representation, RkI is expressed by a perpendicular dashed line that crosses both the Q -space line ‘‘I’’ and the P-space line ‘‘k’’ as shown in Fig. 5. Then we can assign the energy difference, ϵk − ϵI , of the two lines bridged by the dashed line to the propagator RkI . By treating the other two propagators in the same way, the second step of the two-step perturbation transforms Eq. (66) into the right hand side of i

⟨V R k ⟨V R V ⟩j R V ⟩k =

 i k

⟨V ⟩⟩I RjI k ⟨V ⟩⟩J RjJ J ⟨⟨V ⟩j RkI I ⟨⟨V ⟩k ,

(69)

IJ

which is expressed in Fig. 4(b). In nondegenerate P-space, the crucial point is that we define the generalized bracketing i ⟨V R k ⟨V R V ⟩j R V ⟩k , the left hand side of Eq. (69), by the right hand side, or equivalently by the folded diagram in Fig. 4(b). This can be explained as follows: we can easily realize that the right hand side of Eq. (69) is not an (i, j) component of a product of operators. We cannot obtain the right hand side of Eq. (69) by inserting the projector Q in the left hand side. This is in contrast to degenerate P-space,

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519

where Eq. (53) can be expressed as i ⟨VRRV P VRV ⟩j of (50a). We therefore have to use the right hand side of Eq. (69), an explicit form of entangled multiplications, as the definition of the generalized bracketing i ⟨V R k ⟨V R V ⟩j R V ⟩k in nondegenerate P-space. Several points should be made regarding Eq. (69) and Fig. 4(b). First, the above discussion on Eq. (69) explains how we perform actual calculations in nondegenerate P-space. For a given term in the generalized bracketing, we first draw the corresponding folded diagram, from which we read off all the elements to write down its complete expression for actual calculation, as we explained in Section 4.4. Note that we cannot dispense with the folded diagram to arrive at the final expression, the right hand side of Eq. (69). j

Second, note in Fig. 4(b) that the dashed line for RJ extends upwards from the Q -space line ‘‘J’’ to the P-space line ‘‘j’’ but not downwards; to draw a valid folded diagram in the process of changing R˜ J in Fig. 4(a), we have to introduce the perturbation H0 − ϵ˜ to the P-space line ‘‘j’’ above, but not to the P-space line ‘‘k’’ below, in accordance with condition (ii′ ) in Section 4.5. Accordingly, we have to change diagram rule (iv) in Section 4.4 slightly; in nondegenerate P-space, the propagator line is drawn using the following rule. (iv′ ) A propagator line extends from its relevant Q -space line upwards to cross a P-space line. Now we understand that the second step of the two-step perturbation replaces each R˜ with R in Eq. (65). The above discussion applies to all the terms in the expansion (65) except for the first order term i ⟨˜u(1) ⟩j = i ⟨V ⟩j , which does not contain the propagator R˜ in its expression and changes as i

⟨˜u(1) ⟩j = i ⟨V ⟩j → i ⟨V ⟩j + i ⟨H0 − ϵ˜ ⟩j in the second step.

To summarize, the above two-step perturbation expansion of i ⟨˜v ⟩j gives, in place of Eq. (64), n V ′s i

⟨˜v ⟩j = i ⟨H0 − ϵ˜ ⟩j +

  n

   i ⟨VRV · · · VRV ⟩,

(70)

bracketing

where the right hand side is now written in terms of V and R. 5.3. Generalized bracketing and folded diagram in nondegenerate model space With Eq. (70) at hand, we are now ready to write down the following proposition for v =

v˜ − P (H0 − ϵ˜ )P of Eq. (63).

Proposition (Generalized Bracketing for i ⟨v⟩j in Nondegenerate P-Space). The nth order term of i ⟨v⟩j is given by n V ′s i

(n)

⟨v ⟩j =



   i ⟨VRV · · · VRV ⟩.

(71)

bracketing

This is formally equivalent to Eq. (49) in degenerate P-space. Here the only difference is that each propagator R carries both P- and Q -space indices, and its energy denominator is the energy difference between the P- and Q -space lines bridged by the propagator line; propagators are to be read off from the folded diagram. The above difference is easily explained with an example. We can use the bracketing expression i ⟨V R k ⟨V R V ⟩j R V ⟩k in both degenerate and nondegenerate P-spaces. In degenerate P-space, it is expressed by Fig. 1, and in nondegenerate P-space it is expressed by Fig. 4(b); these diagrams read in different ways as shown in Eqs. (53) and (69). To become familiarized with the folded diagram in nondegenerate P-space, we present several examples in Fig. 6, which are to be considered in contrast to Fig. 2. It is now easy to understand that these examples read as

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Fig. 6. Bracketing and folded diagram representations for i ⟨v⟩j in nondegenerate P-space. These diagrams should be compared with those in Fig. 2.

i

⟨VRV ⟩j =

 i

⟨V ⟩⟩I RjI I ⟨⟨V ⟩j ,

(72)

I i

⟨VRVRV ⟩j =

 i

⟨V ⟩⟩I RjI I ⟨⟨V ⟩⟩J RjJ J ⟨⟨V ⟩j ,

(73)

IJ i

⟨V R k ⟨V ⟩j R V ⟩k =

 i k

i

⟨V R k ⟨VR l ⟨V R V ⟩j R V ⟩l R V ⟩k  j j j = i ⟨V ⟩⟩I RI k ⟨V ⟩⟩J RJ l ⟨V ⟩⟩K RK kl

i

⟨V ⟩⟩I RjI k ⟨V ⟩j RkI I ⟨⟨V ⟩k ,

⟨⟨V ⟩j RlJ J ⟨⟨V ⟩l RkI I ⟨⟨V ⟩k ,

(75)

⟨⟨V ⟩j RlI I ⟨⟨V ⟩⟩J RlJ k ⟨V ⟩l RkJ J ⟨⟨V ⟩k .

(76)

K

IJK

⟨V R l ⟨VRV ⟩j R V R k ⟨V ⟩l R V ⟩k  j j = i ⟨V ⟩⟩I RI l ⟨V ⟩⟩K RK kl

(74)

I

K

IJK

5.4. Summary of perturbation expansion of v Because we have developed our theory in a heuristic manner, it is appropriate here to summarize our method of calculating the effective interaction v . In the following, we present the procedure of calculation in multi-dimensional nondegenerate P-space that we have developed in this work. (i) We first give the perturbation expansion of i ⟨v⟩j in the generalized bracketing scheme of Section 4.3, which is defined via the valid bracketing schemes in Section 3.2 and the P-space indices given by the bracket product rule in Section 4.2. (ii) For each generalized bracketing scheme in the expansion, we draw its corresponding valid folded diagram and enter all the external and dummy indices in it, as explained by rules (i)–(v) in Section 4.4 and rule (iv′ ) in Section 5.2.

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(iii) Finally, we read off all the matrix elements of V and R from the diagram and multiply them all to obtain the contribution of the diagram to i ⟨v⟩j , as explained by rule (vi) in Section 4.4. 6. Discussion Here we discuss our results from several viewpoints. In Section 6.1, we compare our results in degenerate and nondegenerate P-spaces, and explain why we needed to start with degenerate P-space. In Section 6.2, we clarify the relationship between the results in one- and multi-dimensional P-spaces in the folded diagram representations. In Section 6.3, we compare our theory of folded diagrams with other theories that lead to their own folded diagram representations. We will show that our approach alone has a clear correspondence to the well-known bracketing technique in the RSPT. 6.1. Degenerate versus nondegenerate In Section 5, we have shown that the difference in practical calculations between degenerate and nondegenerate P-spaces can be seen only in the propagators. Here we discuss the difference from the viewpoint of reducible bracketing schemes. A reducible bracketing is a product of two or more independent P-space matrices. In degenerate P-space, such a bracketing contains at least one internal pair of brackets ⟨· · ·⟩, and therefore its diagrammatic expression has at least one folded (internal) P-space line. For example, the reducible bracketing i ⟨VR k ⟨VRV ⟩j RV ⟩k in Eq. (50) is a product of two independent P-space matrices i ⟨VRRV ⟩k =   I i ⟨V ⟩⟩I RI RI I ⟨⟨V ⟩k and k ⟨VRV ⟩j = J k ⟨V ⟩⟩J RJ J ⟨⟨V ⟩j . This is also obvious in the diagram; the corresponding diagram in Fig. 1 can be split into two independent P-space factors by cutting the folded line ‘‘k’’. In nondegenerate P-space, on the other hand, we cannot apply the same reasoning. The corresponding expression (69) cannot be decomposed into two independent factors; the expression  j k I i ⟨V ⟩⟩I RI RI I ⟨⟨V ⟩k depends on the index ‘‘j’’, and cannot be expressed as i ⟨VRRV ⟩k . Accordingly, the diagram in Fig. 4(b) cannot be split into two parts by cutting the folded P-space line ‘‘k’’. This shows that i ⟨VR k ⟨VRV ⟩j RV ⟩k is irreducible by definition in the nondegenerate P-space. In fact, a little consideration shows that every bracketing scheme is irreducible in the nondegenerate P-space. The above observation clarifies the reason why we needed to start our discussion of v in degenerate P-space, i.e., why we could not start with nondegenerate P-space from the beginning. The derivation of the generalized bracketing representation (48) requires that repeated substitutions of the right hand side of Eq. (46) itself in each ⟨v⟩ on the right hand side result in a reducible form. Such an expression is obviously incompatible with the bracketing scheme in nondegenerate P-space, which is necessarily irreducible. This clearly explains that, in nondegenerate P-space, we cannot use the arguments in Section 4 without modification. 6.2. One dimension versus multiple dimensions The well-known bracketing scheme in one-dimensional P-space in Section 3 is obviously a special case of the generalized bracketing scheme in multi-dimensional P-space developed in Sections 4 and 5. Here we examine the above point in connection with the folded diagram. For the sake of simplicity, we use a degenerate multi-dimensional P-space in the following discussion. In multi-dimensional P-space, let us consider the bracketing i ⟨VR k ⟨VRV ⟩j RV ⟩k in the expansion of

(4) ⟩j . This is a matrix product of two independent P-space matrices i ⟨VRRV ⟩k and k ⟨VRV ⟩j , as shown i ⟨v in Eq. (50). It is therefore necessary to assign the external P-space indices ‘‘i’’ and ‘‘j’’ and the dummy index ‘‘k’’ to the brackets to specify how to construct the matrix product. In the folded diagram, this in turn means that the P-space line ‘‘k’’ is indispensable for specifying the matrix product, as shown in Fig. 7(a). Also in one-dimensional P-space, we can naturally express the corresponding bracketing ⟨VR ⟨VRV ⟩RV ⟩ of Eq. (28) in the same graphic way as in multi-dimensional P-space. However, it is interesting here to note the special point in one-dimensional P-space, i.e., the P-space index is unique

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K. Takayanagi / Annals of Physics 350 (2014) 501–532

(a) (b)

Fig. 7. (a) Folded diagram for i ⟨v⟩j in degenerate multi-dimensional P-space. (b) Simplified folded diagram for v = 1E in one-dimensional P-space.

and does not need to be displayed explicitly. Correspondingly, the bracketing ⟨VR⟨VRV ⟩RV ⟩ is a product of two c-numbers ⟨VRRV ⟩ and ⟨VRV ⟩. In the folded diagram, therefore, the P-space lines and indices are redundant; the product can be fully expressed by Fig. 7(b), which is obtained by removing all the P-space lines and indices from Fig. 7(a). The above observation can be easily generalized; to express 1E in one-dimensional P-space, each folded diagram for v can slough off all the P-space lines and indices to simplify itself. It is now clear that, in addition to the well-known algebraic (bracketing) expansion of 1E, we now have a graphic expansion in terms of the valid folded diagrams (without P-space lines if desired) explained in Section 4.5. To summarize, we can now treat one- and multi-dimensional P-spaces in a unified fashion in both algebraic and graphic representations. We can thus enter ‘‘RS’’ in the empty space in Table 1. 6.3. Comparison with other methods In this work, we have generalized the bracketing technique in the RSPT to multi-dimensional P-space and developed its folded diagram representation. Here we compare our results with other existing schemes to express v in perturbation theory. We will show that our theory alone can fill all four spaces in Table 1 with ‘‘RS’’. Nonperturbative methods of calculating v are briefly explained in Appendix B. 6.3.1. Bloch theory In the Bloch approach [8,9], we first introduce the wave operator Ω = Ω P that generates the true eigenstate |Ψα ⟩ of Eq. (2) from its projection |φα ⟩ onto the P-space as

Ω |φα ⟩ = |Ψα ⟩,

α = 1, . . . , d,

(77)

which in turn means that P Ω = P. From Eqs. (2) and (77), it is easy to see that PH Ω |φα ⟩ = Eα |φα ⟩,

α = 1, . . . , d,

(78)

showing that H eff = PH Ω = PH0 P + PV Ω = PH0 P + v

(79)

is obviously the effective Hamiltonian H eff in this approach. The perturbation expansion of v = PV Ω therefore reduces to that of the wave operator Ω . Using the fact that |Ψα ⟩ is an eigenstate of H, we now derive an equation for Ω . First, by combining Eqs. (2) and (77) we have H0 Ω |φα ⟩ + V Ω |φα ⟩ = Eα |Ψα ⟩,

α = 1, . . . , d.

(80)

Second, by multiplying Eq. (2) by Ω P, we obtain

Ω H0 |φα ⟩ + Ω PV Ω |φα ⟩ = Eα |Ψα ⟩,

α = 1, . . . , d.

(81)

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Then by taking the difference between Eqs. (80) and (81), we obtain

[Ω , H0 ] = V Ω − Ω PV Ω ,

(82)

which is referred to as the (generalized) Bloch equation [8,9]. Now we expand Ω in powers of V as

Ω = Ω (0) + Ω (1) + Ω (2) + · · · ,

(83)

(n)

where Ω stands for the nth order term in the perturbation and Ω for Ω (n) can be easily derived from Eqs. (82) and (83):

[Ω (n) , H0 ] = V Ω (n−1) −

n −1 

Ω (j) PV Ω (n−j−1) ,

(0)

= P. Then an iterative formula

n = 1, 2, . . . .

(84)

j=0

Note here that Ω (n) = Q Ω (n) P for n = 1, 2, . . . . Now we introduce the following notation [7] to express the matrix element of an arbitrary operator A between Q - and P-space states ‘‘I’’ and ‘‘i’’, respectively, with the corresponding energy denominator: I

(A)j =

⟨⟨I |A|j⟩ = RjI I ⟨⟨A⟩j . ϵj − ϵI

(85)

Note that the left and right parentheses ‘‘(’’ and ‘‘)’’ respectively carry the Q - and P-space indices and also stand for the energy denominator. Then we find that the matrix elements of Eq. (84) can be expressed conveniently as I

⟨⟨Ω (1) ⟩j = I (V )j ,

I

⟨⟨Ω (2) ⟩j = I (V (V ))j − I ((V )V )j ,

I

⟨⟨Ω (3) ⟩j = I (V (V (V )))j − I ((V )V (V ))j − I (V (V )V )j − I ((V (V ))V )j + I (((V )V )V )j .

(86)

Noting that Eq. (79) gives v (n) = PV Ω (n−1) , we can now write down i ⟨v (n) ⟩j = i ⟨V Ω (n−1) ⟩j explicitly;

for example, i ⟨v (3) ⟩j is given by i

⟨v (3) ⟩j =

 i

⟨V ⟩⟩I I (V (V ))j −

 i

⟨V ⟩⟩I I ((V )V )j .

(87)

I

I

Here we realize that the Bloch theory can accommodate nondegenerate P-space from the beginning; it is not necessary to start from degenerate P-space. Although this is a major merit, the Bloch approach does not correspond to the well-known bracketing representation of the RSPT, as we show below. Let us consider the following term in i ⟨v (4) ⟩j that is derived from the second term in Ω (3) in Eq. (86): 1

 i I

2

3

4

⟨V ⟩⟩I I ((V ) V (V ))j =

1

 i k

2

3

4

⟨V ⟩⟩I RjI RkI I ⟨⟨V ⟩k k ⟨V ⟩⟩J RjJ J ⟨⟨V ⟩j .

(88)

IJ

We immediately realize that this is identical to Eq. (69) in our generalized bracketing scheme. In the graphic representation, we express Eq. (88) by Fig. 8 in the usual convention [9]; we connect interaction vertices by a zigzag line in the order of matrix multiplications designated by the numbers assigned to the V ’s in such a way that the horizontal propagator lines cross the relevant P- and Q -space lines. Note that Fig. 8 looks similar to Fig. 4(b) after a rotation of π /2. Strictly speaking, however, they 2

are not exactly the same; the Bloch prescription here does not specify the relative positions of V and 4

V in the diagram, i.e., either of the two can be displayed above the other in Fig. 8. Regarding the algebraic representation, the present and the Bloch theories are in sharp contrast. Eqs. (88) and (69) express the same contribution to i ⟨v (4) ⟩j in a completely different fashion. The algebraic expression in the Bloch theory, the left hand side of Eq. (88), is designed to indicate where

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K. Takayanagi / Annals of Physics 350 (2014) 501–532

1

3

2 4 Fig. 8. Folded diagram for Eq. (88) in the Bloch approach.

to evaluate the energy denominators between the P- and Q -space states. On the other hand, our bracketing representation highlights where to take the P-space matrix elements. The difference is also clear in the order in which the V ’s are displayed in the algebraic representations. Let us examine the numbers assigned to the V ’s in Eq. (88) and Fig. 8. Eq. (88) exhibits V ’s in the connected order 1

2

3

4

in Fig. 8, which may be symbolically expressed as V ((V ) V (V )). On the other hand, in Eq. (53) of the present theory (degenerate version of Eq. (69)) which is displayed in Fig. 1, they are arranged as 1

3 4

2

⟨V ⟨V V ⟩ V ⟩, which is obviously different from Eq. (88). To summarize, the Bloch approach provides its own algebraic and folded diagram expressions for

v . It is not, however, a multi-dimensional generalization of the well-known bracketing technique of the RSPT in one-dimensional P-space. In other words, in the Bloch theory there is no way to overwrite ‘‘Bloch’’ with ‘‘RS’’ in Table 1.

6.3.2. Time-dependent perturbation theory In the field of nuclear physics, the folded diagram has been introduced in the context of manybody problems using time-dependent perturbation theory (TDPT) [4]. Here we compare the present approach with TDPT. In TDPT, we first construct Ω of Eq. (77) adiabatically and then remove infinite factors from PV Ω to extract the effective interaction v . Here we consider a one-particle system in the framework of manybody perturbation theory, because complexities inherent in the many-body problem, especially the linked diagram theorem, become irrelevant [3]. Then the effective interaction v for the one-particle system can be expressed easily by the folded diagram of TDPT with the time axis, which can be directly compared with our folded diagrams. Let us consider the example shown in Fig. 9. Although the energy denominator in TDPT is normally given by the difference between the initial and intermediate energies, here we ‘‘bend’’ the initial line ‘‘j’’ so that we can express the energy denominator as the energy difference on the corresponding horizontal line as in our diagrams. On the left hand side in Fig. 9, we have two diagrams that are only different in their time orderings t2 > t4 or t2 < t4 . Excluding the common matrix elements of V , their propagators sum up to 1

1

1

ϵj − ϵI ϵj + ϵk − ϵI − ϵJ ϵj − ϵJ =

1

1

1

ϵj − ϵI ϵj − ϵJ ϵk − ϵI

+

1

1

1

ϵj − ϵI ϵj + ϵk − ϵI − ϵJ ϵk − ϵI

= RjI RjJ RkI ,

(89)

which is assigned to the right hand side of Fig. 9. Note here that on the right hand side of Fig. 9, all propagator lines cross only two solid lines, while on the left hand side the middle propagators cross four solid lines.

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525

Fig. 9. Folded diagrams in time-dependent perturbation theory (TDPT). The two diagrams on the left hand side represent two possible time orderings t2 > t4 and t2 < t4 . The sum of these two contributions can be expressed by the single diagram on the right hand side.

We observe that the right hand side of Eq. (89) gives the same propagators as in Eqs. (69) and (88). In TDPT, therefore, the right hand side of Fig. 9 reads exactly in the same way as Fig. 4(b) and Fig. 8. However, the right hand side of Fig. 9 represents the sum of the two time orderings t2 > t4 and t2 < t4 , which cannot be translated into the language of the present approach or the Bloch theory. Furthermore, TDPT does not have any algebraic expressions that reduce to the bracketing of the RSPT in one-dimensional P-space. To summarize, TDPT gives its own folded diagram expression for v , which is different from ours. Although TDPT has the advantage that it can be applied to many-body systems in a straightforward manner, it does not have a bracketing expression. In other words, TDPT does not give any insight into the left column of the ‘‘bracketing’’ in Table 1. 7. Test calculations We have now a simple and powerful scheme for calculating the effective interaction v in perturbation theory, which has clear expressions in the bracketing and folded diagram representations. Here we present simple test calculations to visualize and to better understand the present theory. In Section 7.1, we first present a practical method of calculating ⟨v (n) ⟩ in an efficient way. Then in Section 7.2, we briefly review the well-established results [21] for the convergence property of the perturbation expansion of the effective interaction. In Section 7.3, using toy model calculations, we demonstrate that our approach is coherent and useful. 7.1. Practical method of calculation Here we explain a practical method of calculating ⟨v (n) ⟩ using ⟨v (1) ⟩ = ⟨V ⟩, ⟨v (2) ⟩, ⟨v (3) ⟩, . . . ,

⟨v (n−2) ⟩ as its building blocks.

A little consideration with some diagrams is sufficient to note that any folded diagram for i ⟨v⟩j can be expressed uniquely in the form shown in Fig. 10, which in fact is a folded diagram expression for Eq. (45). The diagram is composed of a Q -space line with an arbitrary number of V ’s on it and a folded P-space line with an arbitrary number of ⟨v⟩’s on it. Each propagator line crosses both the Q and P-space lines as shown in the figure. The important point here is that each ⟨v⟩ on the P-space line can be calculated independently of other parts of the diagram. Therefore, we can calculate each term in the expansion of ⟨v⟩ using lower order ⟨v⟩’s as its building blocks. Now we explain the above situation using examples. First, let us consider the diagram in Fig. 4(b), which represents Eq. (69). Using i ⟨v (2) ⟩j = i ⟨V R V ⟩j given in Eq. (72), which is shown by the first diagram (on the upper left) in Fig. 6, we can easily see that Eq. (69) can be expressed as i

⟨V R k ⟨V R V ⟩j R V ⟩k =

 i

⟨V R k ⟨v (2) ⟩j R V ⟩k ,

k

using ⟨v (2) ⟩ as a building block, as shown in Fig. 11(a).

(90)

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K. Takayanagi / Annals of Physics 350 (2014) 501–532

Fig. 10. Diagrammatic expression for a general term that expresses i ⟨v⟩j .

(b) (a)

Fig. 11. (a) Diagrammatic expression for the right hand side of Eq. (90), which uses ⟨v⟩ as a building block. (b) The same for Eq. (91).

Second, we can see in the same way that the last diagram (on the lower right) in Fig. 6, representing Eq. (76), can be calculated as shown in Fig. 11(b), which reads as i

⟨V R l ⟨VRV ⟩j R V R k ⟨V ⟩l R V ⟩k =

 i

⟨V R l ⟨v (2) ⟩j R V R k ⟨v (1) ⟩l R V ⟩k .

(91)

kl

The above two examples confirm the validity of the calculation scheme shown in Fig. 10. In the test calculations in Section 7.3, we have used the present method to calculate i ⟨v (n) ⟩j . 7.2. Convergence property Because we have studied the perturbation expansion of v , it is appropriate here to briefly explain its convergence property [21]. Here we introduce a complex parameter x in the Hamiltonian in Eq. (1) and write H = H0 + xV .

(92)

Then the eigenenergies are functions of x and are displayed on a complex plane. Suppose we are to describe d eigenstates with eigenenergies {Eα (x), α = 1, . . . , d} by our effective interaction v , discarding other states with {Eβ (x), β = d + 1, . . . , D}. It may be the case that, for some value of x, one of the Eα (x)’s and one of the Eβ (x)’s coincide on the complex plane. Such an x is referred to as a biexceptional point. Now we consider the order-by-order perturbation expansion of v in powers of x. We can show that its convergence radius is given by the modulus of the biexceptional point closest to the origin [21].

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527

Fig. 12. Eigenenergies of Hamiltonian (93) as a function of x. The solid lines E1 and E2 represent exact eigenenergies of Eq. (94). The perturbation expansion (95) of E1 is displayed by dotted lines with numbers representing the highest order of the calculation.

We usually rephrase the above result simply as follows; the perturbation series converges only when the perturbation V is weak enough so that there are no intruder states in the spectrum that H eff = PH0 P + v is to reproduce [21]. 7.3. Toy model calculations Here we perform toy model calculations to illustrate the coherent structure of the present theory. 7.3.1. One-dimensional P-space Let us take a two-dimensional (D = 2) Hilbert space that is spanned by the orthonormal basis vectors {|1⟩, |2⟩} of Eq. (3). Our Hamiltonian in this basis set is

 

0 0

H = H0 + xV =



0 2

 + x

1 1 2

1



2 ,  −1

(93)

where ϵ1 = 0 and ϵ2 = 2 in Eq. (3), and the perturbation is proportional to a strength parameter x. The above Hamiltonian is easily diagonalized to give eigenenergies

 E1 = 1 −

1 − 2x +

5 4

 x2

,

E2 = 1 +

1 − 2x +

5 4

x2 ,

(94)

which are shown by solid lines in Fig. 12. Now we define our one-dimensional P-space (d = 1) as being spanned by {|1⟩}, and we calculate E1 = 1E1 = E1 − ϵ1 in the perturbation theory. Because the P-space is one-dimensional, we can make use of the results in Eq. (26) to write down E1 as E1 = x −

1 8

x2 −

1 8

x3 −

15 128

x4 −

13 128

x5 −

81 1024

x6 − · · · .

(95)

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K. Takayanagi / Annals of Physics 350 (2014) 501–532

For example, the third order term is calculated as (3)

E1

= ⟨VRVRV ⟩ − ⟨VR⟨V ⟩RV ⟩     1 x x 1 1 x 1 x 1 (−x) − x = − x3 . = 2 −2 −2 2 2 −2 −2 2 8

(96)

Note that, in our simple model, we could have obtained the expression (95) directly by expanding E1 of Eq. (94) in powers of x. The above result (95) is plotted by dotted lines in Fig. 12 with numbers that denote the highest order of the corresponding perturbation expansion. The figure shows clearly that the perturbation series tends to diverge for x & 1 as the order of the perturbation increases. This is in accordance with the simple criterion in terms of the intruder state given at the end of Section 7.2; the spectrum in Fig. 12 shows that the level E1 is an intruder state above the level crossing at x ∼ 0.9, and therefore indicates that the convergence radius of the perturbation series would be given by x ∼ 0.9. The above observation has a strict explanation based on the discussion in Section 7.2. The biexceptional points of our Hamiltonian (93) are x = 4±52i on the complex x plane, where E1 and E2 coincide. Then the convergence radius of the series (95) is given by  4±52i  = √2 ∼ 0.9, as can be





5

confirmed in the figure. 7.3.2. Multi-dimensional P-space Let us consider the following Hamiltonian in a three-dimensional (D = 3) Hilbert space that is spanned by the orthonormal basis vectors {|1⟩, |2⟩, |3⟩}.

 

2

  3 + x 2 5 

0



2

H = H0 + xV =

1

3 2 0

−

3

1



  − . 2  3 − 3

(97)

2 2 By diagonalizing H = H0 + xV , we obtain three eigenenergies E1 < E2 < E3 , which are shown by solid lines in Fig. 13. Now we take a two-dimensional (d = 2) nondegenerate P-space spanned by {|1⟩, |2⟩} and construct the effective interaction v so that H eff = PH0 P + v describes E1 and E2 in the small-x region. We may calculate such an effective interaction v using the bracketing representation (71) directly for each order of perturbation. In actual calculations, however, we have used the efficient method explained in Section 7.1 and obtained the following expression for v = v (1) + v (2) + · · ·.

v (1) = x



2 1.5

v (4) = x4



1.5 , 0



−0.033 −0.495

  −0.2 0.5 v (2) = x2 , 0.3 −0.75   −0.19 −0.06650 , v (5) = x5 0.285 −0.09975

v (3) = x3



0.01 −0.015

0.15 , −0.225



 −0.20567 . 0.30850

For example, the third order term 1 ⟨v (3) ⟩1 = 0.01 × x3 is calculated as 1

⟨v (3) ⟩1 = 1 ⟨VRVRV ⟩1 −

2  1

⟨VR k ⟨V ⟩1 RV ⟩k

k=1

 =

x

1 −3x 1

−5 2 −5

 x

  1 1 1 3x 1 −3x 1 3 − x 2x x+x = x . −5 −5 −5 2 −3 2 100

In Fig. 13, we plot the eigenenergies of H eff = PH0 P + v by dotted lines in the same way as in Fig. 12. Note that H eff here gives two eigenenergies for each value of x. Let us examine the exact eigenenergies E1 , E2 , E3 in Fig. 13. A sharp level crossing is visible between E2 and E3 at x ∼ 1.4, which derives from the biexceptional points x = 1.38 ± 0.20i. One might then

K. Takayanagi / Annals of Physics 350 (2014) 501–532

529

Fig. 13. Eigenenergies of Hamiltonian (97) as a function of x. The notation is the same as for Fig. 12.

conclude that the intruder state enters the target spectrum {E1 , E2 } at x ∼ 1.4, and therefore the convergence radius for E1 and E2 would be ∼1.4. The figure shows, however, that perturbation theory cannot reproduce E1 for x & 1, indicating that the convergence radius of the perturbation series is less than unity, which is not in accordance with the above naive estimation. This is explained as follows: in this model, the biexceptional point with the smallest modulus is given by x = 0.205 ± 0.907i (|x| = 0.93), which determines the convergence radius. Because the above biexceptional point is far from the real axis, it can hardly be recognized from the spectrum in Fig. 13; the usual criterion used to estimate the convergence radius appears to be too naive in this case. We can thus confirm that our effective interaction v reproduces the target eigenenergies E1 and E2 inside the convergence radius, as it should. To summarize, we have demonstrated that the present theory in fact gives the perturbation expansion of an effective interaction v in a convenient and coherent way. 8. Summary In a one-dimensional model space, it is well known that the usual Rayleigh–Schrödinger perturbation theory (RSPT) gives the energy shift 1E via a convenient bracketing technique. On the other hand, in a multi-dimensional model space, where 1E is replaced by the effective interaction v , we do not have a bracketing representation for v . Therefore, the above two cases have been treated in different manners to date, as shown in Table 1, which is cumbersome and makes it difficult to obtain a unified understanding of the effective interaction. In this situation, we have presented the following results in this work. First, we have generalized the bracketing technique to a multi-dimensional model space. Our generalized bracketing expression for v reduces to the well-known result for 1E in a one-dimensional model space. Conversely, by embedding the matrix structure in the bracketing expression for 1E, we can obtain the generalized bracketing expression for v . Second, we have introduced our own folded diagram rule to visualize the generalized bracketing scheme, establishing a clear one-to-one correspondence between the algebraic (generalized bracketing) and graphic (folded diagram) representations of the effective interaction v . Third, we have compared our results with other approaches, the Bloch theory and time-dependent perturbation theory, which give different expressions for the effective interaction. We have shown

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that our approach alone gives a unified description of the effective interaction (i) in one- and multidimensional model spaces and (ii) in algebraic and graphic representations, in such a manner that the description is compatible with the well-known bracketing technique of the RSPT in a one-dimensional model space. In other words, the present work allows ‘‘RS’’ to be entered in all four spaces in Table 1. Having presented a unified description of the effective interaction in the RSPT, we believe that the concept of the effective interaction has become clearer than ever and that it will enjoy greater use in applications. Appendix A. Bracketing and folded diagram for wave operator ω In this appendix, we briefly explain the bracketing and folded diagram techniques for a wave function in multi-dimensional P-space. We define the ‘‘wave operator’’ ω = Q ωP by writing the wave operator Ω of Eq. (77) as

Ω = P + ω, and we examine its expressions. A.1. Degenerate model space Let us first assume a degenerate P-space as in Section 4. Then it is easy to show that

ω|α⟩ = |Φα ⟩ =

Q Eα − H

 =

 RV + R

V |α⟩

−1Eα V



 RV + R

−1Eα V

   −1Eα R

V

 RV + · · · |α⟩.

(A.1)

Now we expand the second line of the above expression and treat each term of the expansion in the same manner as in Eq. (36). As an example, let us consider RVR(−1Eα )RVR(−1Eα )RV |α⟩, which we transform as follows: RVR(−1Eα )RVR(−1Eα )RV |α⟩ = RVRRVRRV |α⟩⟨ α |(−v)|α⟩⟨ α |(−v)|α⟩

= RVRRVRRV (−v)(−v)|α⟩.

(A.2)

The above argument applies to each term in the expansion (A.1) to give

ω|α⟩ =



RVR · · · RV (−v)(−v) · · · |α⟩.

(A.3)

term

Because Eq. (A.3) holds for any |α⟩, we can eliminate |α⟩ to obtain

ω=



RVR · · · RV (−v)(−v) · · · ,

(A.4)

term

where we have implicitly assumed, in contrast to Eq. (38), that the right hand side is multiplied by the projector P from the right and by Q from the left. In Eq. (A.4), a general term on the right hand side is given as follows. (i) Take a product RVRVRV · · · VRV , and eliminate any number of V ’s from the product except for the rightmost V . (ii) Then multiply the obtained expression by the same number of (−v)’s from the right. The explicit form of Eq. (A.4) is therefore

ω = RV + RVRV + RVRVRV − RRV v − RRVRV v − RVRRV v + RRRV vv + · · · . Now we take the matrix element I ⟨⟨ω⟩j = ⟨I |ω|j⟩ of the above ω = Q ωP. Repeating the same discussion as in Sections 4.2 and 4.3, we can express each term in the perturbation expansion I

⟨⟨ω⟩j = I ⟨⟨ω(1) ⟩j + I ⟨⟨ω(2) ⟩j + I ⟨⟨ω(3) ⟩j + · · ·

(A.5)

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531

Fig. A.1. Folded diagram for I ⟨⟨ω(4) ⟩j of Eq. (A.8). The notation is the same as for Fig. 6.

using the generalized bracketing scheme; explicit forms of the first few terms are given by I

⟨⟨ω(1) ⟩j = I ⟨⟨RV ⟩j ,

I

⟨⟨ω(2) ⟩j = I ⟨⟨RVRV ⟩j − I ⟨⟨R k ⟨V ⟩j RV ⟩k ,

I

⟨⟨ω(3) ⟩j = I ⟨⟨RVRVRV ⟩j − I ⟨⟨R k ⟨V ⟩j RVRV ⟩k − I ⟨⟨RVR k ⟨V ⟩j RV ⟩k + I ⟨⟨R l ⟨V ⟩j R k ⟨V ⟩l RV ⟩k − I ⟨⟨R k ⟨VRV ⟩j RV ⟩k .

(A.6)

It is also straightforward to write down the generalized bracketing scheme for the nth order term ⟨⟨ω(n) ⟩j :

I

n V ′s I

(n)

⟨⟨ω ⟩j =



 I

  ⟨⟨RVRV · · · RVRV ⟩.

(A.7)

bracketing

The valid bracketing here means the same as for ⟨v⟩, except that the leftmost V is not excluded from the bracketing here, as can be confirmed in Eq. (A.6). A.2. Nondegenerate model space Using the arguments in Section 5, it is easy to generalize the above discussion to nondegenerate P-space. In the same way as for the effective interaction i ⟨v⟩j , we can show that the above bracketing expressions (A.6) and (A.7) for I ⟨⟨ω⟩j are also valid in nondegenerate P-space; we simply need to use the recipe for calculating the propagator in nondegenerate P-space explained in Section 5.2. Finally, we give the example of I ⟨⟨R V R k ⟨VRV ⟩j R V ⟩k , which contributes to I ⟨⟨ω(4) ⟩j . We can easily understand that it is expressed by the diagram in Fig. A.1 and is calculated as I

⟨⟨R V R k ⟨VRV ⟩j R V ⟩k =

 k

j

j

RI I ⟨⟨V ⟩⟩J RJ

k

⟨V ⟩⟩K RjK

K

⟨⟨V ⟩j RkJ J ⟨⟨V ⟩k .

(A.8)

JK

Appendix B. Nonperturbative expression for v In this work, we have examined the perturbation expansion for v in a unified fashion. In actual calculations of v , however, nonperturbative methods [5,6,10,11,13–16] are often used. In this appendix, therefore, we briefly derive a nonperturbative method to calculate v following Refs. [10,11]. B.1. Degenerate model space Let us start with degenerate P-space. We first write Eq. (32) as follows:

v|α⟩ = Qˆ (ϵ + 1Eα ) |α⟩ =

∞  k=0

Qˆ k (ϵ)(1Eα )k |α⟩ =

∞  k=0

Qˆ k (ϵ) v k |α⟩,

(B.1)

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K. Takayanagi / Annals of Physics 350 (2014) 501–532

where Qˆ k (ϵ) is defined by Qˆ k (ϵ) =

1 dk Qˆ (ϵ) k!

dϵ k

.

(B.2)

In Eq. (B.1), the second equality shows the Taylor series expansion of Qˆ (ϵ + 1Eα ) around ϵ , and the third equality originates from Eq. (32), i.e., v|α⟩ = 1Eα |α⟩. Because Eq. (B.1) holds for any |α⟩, we obtain the following operator equation:

v=

∞ 

Qˆ k (ϵ) v k .

(B.3)

k=0

The usefulness of Eq. (B.3) lies in the fact that we can transform it into the following recursive formula:

v n +1 =

∞ 

Qˆ k (ϵ) vnk ,

v0 = V ,

(B.4)

k=0

where vn is the effective interaction at the nth step of the iteration. The above formula (B.4) gives the effective interaction as v = limn→∞ vn , which is called the Krenciglowa–Kuo method [14]. Note that, to use the above nonperturbative method, we have to calculate the Qˆ -box and its derivatives in advance. B.2. Nondegenerate model space Now we turn to nondegenerate P-space. Starting from Eq. (62), it is easy to see that

v˜ =

∞ 

Qˆ k (˜ϵ ) v˜ k .

(B.5)

k=0

Then, noting that H eff = ϵ˜ + v˜ = H0 + v , we can immediately arrive at H eff − ϵ˜ = PH0 P − ϵ˜ +

∞ 

Qˆ k (˜ϵ )(H eff − ϵ˜ )k ,

(B.6)

k=0

from which we can easily derive a recursive formula to calculate H eff in the same way as Eq. (B.4), which is referred to as the extended Krenciglowa–Kuo method [11]. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

B.H. Brandow, Rev. Modern Phys. 39 (1967) 771. B.H. Brandow, Adv. Quantum Chem. 10 (1977) 187. P.J. Ellis, E. Osnes, Rev. Modern 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, Prog. Part. Nucl. Phys. 53 (2004) 419. I. Lindgren, J. Phys. B: At. Mol. Phys. 7 (1974) 2441. I. Lindgren, J. Morrison, Atomic Many-Body Theory, second ed., Springer-Verlag, 1986. I. Shavitt, R.J. Bartlett, Many-Body Theory in Chemistry and Physics, Cambridge, 2009. K. Takayanagi, Nuclear Phys. A 852 (2011) 61. K. Takayanagi, Nuclear Phys. A 864 (2011) 91. K. Takayanagi, Nuclear Phys. A 899 (2013) 107. N. Tsunoda, K. Takayanagi, M. Hjorth-Jensen, T. Otsuka, Phys. Rev. C 89 (2014) 024313. E.M. Krenciglowa, T.T.S. Kuo, Nuclear Phys. A 235 (1974) 171. K. Suzuki, S.Y. Lee, Progr. Theoret. Phys. 64 (1980) 2091. F. Andreozzi, Phys. Rev. C 54 (1996) 684. C. Bloch, Nuclear Phys. 6 (1958) 329. C. Bloch, J. Horowitz, Nuclear Phys. 8 (1958) 91. K. Suzuki, R. Okamoto, H. Kumagai, S. Fujii, Phys. Rev. C 83 (2011) 024304. By rotating the usual folded diagrams used in the field of nuclear physics by π/2, we obtain a diagram in which the last P-space line ‘‘j’’ leaves the diagram rightwards. Here we adopt the convention that the line ‘‘j’’ leaves leftwards, which allows us to express every energy denominator as the energy difference on a single horizontal line [9]. [21] T. Schucan, H.A. Weidenmüler, Ann. Phys. 73 (1972) 108; 76 (1973) 483.