PDF Note on Effective Interaction Theory and it Applications

Note on Effective Interaction Theory and it Applications

R.Okamoto

1 Introduction-motivations and perspectives-

1.1 Development of effective interaction/operator theories and unitary transformation methods

1. FST-Okubo method for deriving nucleon-nucleon interaction in meson theory 1954 [10][11]

2. Hyuga-Otsubo method for meson-exchange current operator(1977)

3. Unitary-transformation method (and cluster expansion) in nuclear physics;Providencia and Shakin)1967

4. Unitary-transformation method in solid state physics:Wegenar et al 1985 5. Folded-diagram theory (by Brandow)1960’s-

6. Q-box expansion (by Kuo et al ) 1970’-

7. Similarity transformation method for effective interaction(Suzuki,Lee) 1982 8. Unitary-model-operator approach(=UMOA) (Suzuki,Okamoto,...) 1983-

1.2 Perspectives of spin-boson system as an interesting model- quantum system

1. Possible relevance to macroscopic tunneling problem[1]

2. Possible relevance to quantum dissipation problem [2]

3. Possible relevance to understanding of chemical reaction [3]

4. Possible connection to impurity Kondo model[4]

5. Possible relevance to entanglment problem(Quantum information science) [4]

1.3 Comments on existing methods for spin-boson system

1.3.1 Flow equation method by Wegener et al

1994- also Glazaeck and K.G. Wilson 1994-(similarity renormalization method)

1.3.2 Continuous unitary transformation method by Safonov (1982,2002 [5]

1.3.3 Dynamic compensation method by T. Tsuzuki(1989) [6],[7]

2 Brief review of theory of effective interactions

2.1 Elementary Effective Hamiltonian Theory

It is common practice in various research fields to reduce the infinitely many degrees of freedom of the Hilbert space to those represented by a physically motivated subspace, the model subspace. One partitions the entire Hilbert space into two disjoint subspaces the model space (P space) and its complement (Qspace), such that the Schr¨oedinger equation becomes block matrix equations.[9] (We use hereafter the symbols P and Q for projection operators onto the respective subspaces, as well as for labeling the subspaces themselves.)

We consider Schr¨odinger equation of a system as

HΨ =EΨ (2.1)

We assume thatH is decomposed into two parts, the unperturbed Hamiltonian,H₀, and the perturbation V as

H =H₀+V. (2.2)

The eigen value equation for H₀ is given as

H₀|ϕ_i=ε_i|ϕ_i, (2.3)

ϕ_i|ϕ_j=δ_ij, (i, j = 1,2,· · ·, n), (2.4)

n

i=1|ϕ_iϕ_i|= 1. (2.5)

The projection operators defining the model and excluded subspaces are defined by P ≡ ^d

i=1

|ϕ_iϕ_i|, (2.6)

Q ≡ ⁿ

i=d+1

|ϕ_iϕ_i|. (2.7)

with d(< n) being the dimension of the model subspace. Then the projection operatorsP and Q satisfy

P +Q= 1, P² =P, Q² =Q, (2.8)

and

P Q=QP = 0. (2.9)

By definitions (2.6) and (2.7) we have

P^† = P, (2.10)

Q^† = Q. (2.11)

Then we see that

[H₀, P] = [H₀, Q] = 0. (2.12) This leads us

H₀ =P H₀P +QH₀Q. (2.13)

Using the projection operators the Schr¨odinger equation is rewritten as

(P +Q)H(P +Q)(P +Q)Ψ =E(P +Q)Ψ, (2.14)

→ (P HP +QHP)(PΨ) + (P HQ+QHQ)(QΨ)

=E(PΨ) +E(QΨ). (2.15) Multiplying P and Q from the left side of eq.(2.15) respectively, we have the coupled equations

(P HP)(PΨ) + (P HQ)(QΨ) = E(PΨ), (2.16)

(QHP)(PΨ) + (QHQ)(QΨ) = E(QΨ). (2.17)

These coupled equations can be expressed in block-matrix form

P HP P HQ

QHP QHQ

PΨ QΨ

=E

PΨ QΨ

. (2.18)

The Q-space state QΨ is easily eliminated to produce the projected Schr¨odinger equation

P HP +P HQ 1

E−QHQQHP

(PΨ) =E(PΨ). (2.19)

Since this equation essentially equivalent to the original Schr¨odinger equation,  the oper- ator [P HP +P HQ(E−QHQ)⁻¹QHP] must be formally capable of describing all of the eigenstates Ψk, k = 1,2,· · ·, n,

Let us suppose, now, that

1. the dimension of the subspaceP is finite,d < n,

2. there existed d low-lying eigenvalues {E_k, k = 1,2,· · ·, d}, which are well separated from all of the eigen values of QHQ,

3. that for thesed eigenstates QΨ|QΨ<< PΨ|PΨ. Eq.(2.21) exhibits some defects.[8]

1. this form of effective Hamiltonian (=Feshbach form) depends explicitly onE, which is unknown, and, moreover, it will generally assume different values for different members of the desired set of eigenstates.  

2. One should note that the PΨ’s are generally not mutually orthogonal, since they are merely projections (onto subspaceP) of orthogonal states vectors Ψ_k. This means that the present form of the effective Hamiltonian is, in effect, a non-Hermitian matrix operator. (Note that its eigen values remain real, this non-Hermiticity refers only to the lack of orthogonality. The operator [P HP+P HQ(E−QHQ)⁻¹QHP] is, of course, Hermitian for any fixed (and realE) ; the vareity ofEs needed for various eigenstates is what destroys the Hermiticity of its overall effect. But the empirical Hamiltonian one wishes to explain are generally Hermitian.

2.2 Effective interaction in similarity transformation

2.2.1 Effective interaction by means of similarity transformation

A general equation for constructing effective interaction can be derived by applying the similarity transformation theory. We consider a transformed Hamiltonian in terms of an operator X

H˜ ≡X⁻¹HX. (2.20)

Even for the transformed Hamiltonian we can consider the projected Schr¨odinger equation

PHP˜ +PHQ˜ 1

E−QHQ˜ QHP˜

(PΨ) =E(PΨ). (2.21)

If the operatorX is a solution to the decoupling equation

QHP˜ =Q(X⁻¹HX)P = 0, (2.22)

the P-space operator PHP˜ can be an effective Hamiltonian

PHP˜ (PΨ) =E(PΨ), (2.23)

and the relevant effective interaction V_int is defined by

V_int=PHP˜ −P H₀P =P(X⁻¹HX)P −P H₀P. (2.24) A different choice ofX leads us to a different effective interaction.

2.2.2 Non-Hermtian effective interaction We define a transformation X as

X ≡e^ω, (2.25)

where ω is an operator which satisfies the following properties

ω = QωP, (2.26)

P ω = ωQ= 0, (2.27)

ωP = ω, Qω =ω, (2.28)

ω² = ω³ =· · ·= 0, (2.29)

ω^† = P ω^†Q=−ω, (2.30)

ω^†ω = P ω^†ωP, (2.31)

ωω^† = Qωω^†Q. (2.32)

Then we have e^ω = 1 +ω.The transformation X given in eq.(2.25) is not unitary so that the effective interaction is non-Hermitian.

The decoupling equation (2.22) becomes 0 = Q(e^−ωHe^ω)P

= QHP +QHωP −QωHP −QωHωP ( = QHP +Q[H, ω]P −QωHωP)

= QV P +QHQω−ωP HP −ωP V Qω, (2.33) Here we have used eq.(2.13). We note that the decoupling equation (2.33) is non-linear with respect toω, and generally difficult to be solved, as mentioned by Okubo.[11] However, since then several methods have been developed for solving the decoupling equation.(ref.????)

Several methods for solving/choosing the generator for the unitary transfor- mation

1. to solve directly/iteratvely the non-linear decoupling equation – Suzuki-Lee method,

– Okubo method to scaler meson system (= anharmonic oscillator system with linear coupling (displaced harmonic oscillator system)

2. to solve the eigenvalue equation of an entire system of interest:

- UMOA application to low-momentum NN interaction

 - two-level problem/simple spin system under transverse field

3. to solve the eigenvalue equation of a two-body subsystem of the many-body system of interest:

– UMOA application to effective interaction in finite nuclei 4. T-matrix method by Gleokcle and Eppelbaum(ref.???)

5. (*) Flow equation method(= continuous unitary transformation) by Wegnar1994, and collaborators (ref.????)

H = H_diagonal+H_{non−diagonal}, (2.34)

T = [H, H_diagonal]

= [H_{non−diagonal}, H_diagonal]. (2.35)

problem: inconsistency in dimensional analysis.

The exponentT should be non-dimensional.

6. (*) one-step unitary transformation by Stafonov1982/2002(Russia-US) critical com- ment against the Flow-equation method

The non-Hermite effective Hamiltonian and effective interaction, which are denoted re- spectively by ˜H_NH and R, are given by

H˜_NH = Pe^−ωHe^ωP

= P HP +P V Qω, (2.36)

= P H₀P +P RP, (2.37)

R ≡ P V P +P V Qω. (2.38)

The non-hermitian effective interaction R has been called thereaction matrix(G-matrix).

In order to solve eq.(2.22), we consider the deigenvalue equations

(H₀+V)|Ψ_k=E_k|Ψ_k, (k = 1,2,· · ·, d). (2.39) It is usually assumed that the states{|Ψ_k} are orthogonalized

Ψ_k|Ψ_k=δ_kk. (2.40)

We partition the eigen state |Ψ_k into the P- and Q-space components as

|Ψ_k = P|Ψ_k+Q|Ψ_k

= |φ_k+Q|Ψ_k

= |φ_k+ω|φ_k

= e^ω|φk, (2.41)

where

|φ_k ≡ P|Ψ_k, (2.42)

ω|φ_k = Q|Ψ_k. (2.43)

Here we note for the wave operator defined as Ω =P +ω

Ω|φ_k = P|φ_k+ω|φ_k=P|Ψ_k+Q|Ψ_k

= |Ψk, (2.44)

Ω|Ψ_k = P|Ψ_k+ω|Ψ_k=P|Ψ_k+Q|Ψ_k

= |Ψ_k. (2.45)

Using the eigenstate of the unperturbed HamiltonianH₀ we may write

|Ψk =

d i=1

Cik|ϕi+

n j=d+1

Cjk|ϕj, (2.46)

|φ_k =

d i=1

C_ik|ϕ_i,(ϕ_i|φ_k ≡C_ik) (2.47) Q|Ψ_k =

n j=d+1

C_jk|ϕ_j. (2.48)

It should be noted that {|Ψ_k, k = 1,2,· · ·, d} is selected so that they have the largest P-space overlapsO_k

O_k ≡ ^d

i=1

|ϕ_i|P|Ψ_k|²

=

d

i=1|C_ik|² (2.49)

among all the eigenstates.

We note that the state vectors |φ_k are, in general, not mutually orthogonal, φ_k|φ_k = δ_kk, since they are merely projections (onto P space) of the orthogonal state vectors |Φ_k. If we introduce the bi-orthogonal state, |φ˜_k, corresponding to the model-subspace state

|φ_k as

φ˜_k|φ_k=δ_kk,(k, k = 1,2,· · ·, d). (2.50) We see this relation means the matrix inversion as follows;

δ_kk = φ˜_k|φ_k=φ˜_k|P|φ_k=φ˜_k|^d

i=1

|ϕ_iϕ_i|φ_k

=

d i=1

φ˜_k|ϕ_iϕ_i|φ_k=

d i=1

D_kiC_ik

= (DC)_kk, (2.51)

where we have used the notation C_ik =ϕ_i|φ_kand D_ki ≡ φ˜_k|ϕ_i. Then we have φ˜_k| =

d i=1

φ˜_k|ϕ_iϕ_i|=

d i=1

D_kiϕ_i|, (2.52)

|φ˜k =

d i=1

(φ˜k|ϕi)^∗|ϕi=

d i=1

D_ki^∗|ϕi. (2.53) Here we obtain the transformation from |φ_kto |φ˜_k by using eqs.(2.47) and (??)

|φ˜_k=

d k,j=1

(D^∗C⁻¹)_kj|φ_j (2.54)

We note here

P|φ_k = |φ_k, φ_k|P =φ_k|, (2.55) P|φ˜_k = |φ˜_k, φ˜_k|P =φ˜_k|. (2.56) We have

d k=1

|φkφ˜k| =

d i,j,k=1

|ϕiϕi|φkφ˜k|ϕjϕj| =

d i,j,k=1

|ϕiCikDkjϕj|

=

d i,j=1

|ϕ_i(CD)_ijϕ_j|=

d i=1

|ϕ_iϕ_i|. (2.57)

Similarly we have

d k=1

|φ˜_kφ_k| =

d i,j,k=1

|ϕ_iϕ_i|φ˜_kφ_k|ϕ_jϕ_j| =

d i,j,k=1

|ϕ_iD_ki^∗C_jk^∗ ϕ_j|

=

d i,j=1

|ϕ_i(CD)^∗_jiϕ_j|=

d i=1

|ϕ_iϕ_i|. (2.58)

Thus we have another expressions for P-space projection operator P =

d k=1

|φkφ˜k|=

d k=1

|φ˜kφk|. (2.59) Using eqs.(2.43) and (2.59) we have a formal expression for the mapping opertaor ω as

ω|φ_k = Q|Ψ_k

→ω

d k=1

|φ_kφ˜_k|P =

d k=1

Q|Ψ_kφ˜_k|P

→ω =

d k=1

Q|Ψ_kφ˜_k|P. (2.60)

2.2.3 The general solution for the decoupling equation

It is written in [12] that the formal solution is proved to satisfy the decoupling equation (2.33) expressed in terms of the mapping operator ω. Here we give an explicit derivation of the very fact. Substituting this formal solution (2.60) into eq.(2.33) we obtain

QHP +QHωP −QωHP −QωHωP

=QH

d k=1

P|φ_kφ˜_k|+QH

d k=1

Q|Ψ_kφ˜_k|P −^d

k=1

Q|Ψ_kφ˜_k|P H

d k=1

P|φ_kφ˜_k|

−^d

k=1

Q|Ψkφ˜k|P H

d k=1

Q|Ψkφ˜k|P

=Q

d k=1

H(P|φk+Q|Ψk)φ˜k|P −^d

k=1

Q|Ψkφ˜k|P

d k=1

H(P|φk+Q|Ψk)φ˜k|P

=Q

d k=1

H|Ψ_kφ˜_k|P −^d

k=1

Q|Ψ_kφ˜_k|P

d k=1

H|Ψ_kφ˜_k|P

=Q

d k=1

Ek|Ψkφ˜k|P −^d

k=1

Q|Ψkφ˜k|P

d k=1

Ek|Ψkφ˜k|P

=Q

d k=1

E_k|Ψ_kφ˜_k|P −^d

k=1

Q|Ψ_k ^d

k=1

E_kφ˜_k|P|φ_kφ˜_k|

=

d k=1

E_kQ|Ψ_kφ˜_k|P −^d

k=1

E_kQ|Ψ_kφ˜_k|

= 0. (2.61)

Here we have used eqs.(2.50), (2.56), (2.55),(2.59),(2.60). The formal relation can be used for calculating ω.

2.2.4 Hermitian effective interaction

The Hermitian Effective Interaction is obtained through unitary transformation U = e^T  and the generatorT is given in terms of the generator (the mapping operator)ω for the non unitary transformation as(ref.???)

T = arctanh(ω−ω^†)

=

∞ n=0

(−1)ⁿ 2n+ 1

ω(ω^†ω)ⁿ−(ω^†ω)ⁿω^† (2.62) It should be noted that the solutionT is not unique and determined for eq.(? to be written down ) dependently on the choice of a set of d eigen states {|Ψ_k, k = 1,2,· · ·, d }. We choose the set of the |Ψk so that they have the largest P-state overlaps among all the eigen states in eq.(?-to be write down-). This choice is the usual way in the construction of the effective interaction.

3 Applications of effective interaction theory to sim- ple model systems

3.1 A two-level system

We consider a two-level problem of fermions which is described by the following Hamil- tonian

Hˆ = ε₁c^†₁c₁+ε₂c^†₂c₂+V(c^†₁c₂+c^†₂c₁) (3.1)

= Hˆ₀+ ˆV , (3.2)

Hˆ₀ = ε₁c^†₁c₁+ε₂c^†₂c₂, (3.3) Vˆ = V(c^†₁c₂+c^†₂c₁). (3.4) Here we assume that ε₁ < ε₂ and the relevant basis states are supposed to be orthonor- malized as

i|j=δ_ij,(i, j = 1,2). (3.5) We introduce here the vacuum state for fermion operator |0, which has the properties

c^†₁|0 ≡ |1, (3.6)

c^†₂|0 ≡ |2, (3.7)

c₁|0 ≡ c₂|0= 0, (3.8)

0|c₁ = 1|, (3.9)

0|c₂ = 2|, (3.10)

0|c^†₁ = 0|c^†₂ = 0. (3.11) We may write the Hamiltonian as

H =ε₁|11|+ε₂|22|+V(|21|+|12|) (3.12) We choose {|1} as a model subspace(= P space) . Then the projection operator of a state onto the model subspace is written as P ≡ |11|. And we define the projection operator of a state onto the complement subspace (=Q space) Q≡ |22|.

One see that P and Q satisfy

P +Q = 1, (3.13)

P² = P, (3.14)

Q² = Q, (3.15)

P Q = QP = 0. (3.16)

We may write P and Q in the matirx form if we adopt Dirac bracket notation as

|1 ≡

1 0

, (3.17)

|2 ≡

0 1

, (3.18)

P = |11|=

1 0

×(1,0), (3.19)

=

1 0 0 0

, (3.20)

Q =

0 0 0 1

. (3.21)

We obtain

[ ˆH₀, P] = [ε₁c^†₁c₁+ε₂c^†₂c₂,|11|]

= 0. (3.22)

Similarly we have

[ ˆH₀, Q] = 0. (3.23)

We consider Schr¨odinger eq.

Hˆ|Ψ=E|Ψ. (3.24)

We write an orthonormalized wave function for the lower energy solution

|Ψ = a₁|1+a₂|2, (3.25)

a²₁+a²₂ = 1,(|a₁|>|a₂|) (3.26) P-space and Q-space component (of P space-component-main state) are defined as

|φ ≡ P|Ψ=a₁|1, (3.27)

Q|Ψ = a₂|2. (3.28)

We note thate |φ is not normalized because |Ψ is supposed to be orthonormalized. So we define its bi-orthogonal state as

φ˜| ≡ 1

a₁1|, (3.29)

φ˜|φ = 1|1= 1. (3.30)

The mapping operator ω is defined as ω ≡ ^d

k=1

Q|Ψφ˜|,(d = 1), (3.31)

= a₂|21|1

a₁ (3.32)

One can easily see the properties which ω should satisfies:

ω² = a₂|21|1

a₁ ×a₂|21| 1

a₁ (3.33)

= 0. (3.34)

→ ωⁿ= 0 for n ≥2. (3.35)

We calculate the matrix elements ofω as

1|ω|1 = 1|a₂|21|1

a₁|1= 0 (3.36)

2|ω|2 = 0 (3.37)

2|ω|1 = 2|a₂|21|1

a₁|1= a₂

a₁ (3.38)

1|ω|2 = 1|a₂|21|1

a₁|2= 0. (3.39)

We expressω in a matrix form as ω=

0 0

a₂ a₁ 0

(3.40) Møller’s wave operator is given as

Ω ≡ P +ω

= |11|+

a₂

a₁ |21|. (3.41)

We can easily check the properties which Ω  should satifies.

Ω|Ψ =

|11|+

a₂

a₁ |21|[a₁|1+a₂|2]

= a₁|1+a₂|2

= |Ψ. (3.42)

→Ω|φ =

|11|+

a₂

a₁ |21|a₁|1

= a₁|1+a₂|2

= |Ψ. (3.43)

Ω ≡ P +ω (3.44)

=

1 0

a₂ a₁ 0

. (3.45)

P H = |11|{ε₁c^†₁c₁+ε₂c^†₂c₂+V(c^†₁c₂+c^†₂c₁)}

(3.46)

= ε₁|11|+V|12|. (3.47)

P HΩ = {ε₁|11|+V|12|}{|11|+a₂

a₁|21|}

(3.48)

= ε₁|11|+V

a₂

a₁ |11|, (3.49)

= [ε₁+V(a₂

a₁)]P. (3.50)

Here note

a₂

a₁ = V

E−ε₂. (3.51)

This relation is derived as follows: First, we consider the eigenvalue equation for the two-

level hamiltonian:

ε₁ V V ε₂

x y

=E

x y

. (3.52)

The eigen valueE is given as the solution of the secular equation

E²−(ε₁+ε₂)E+ε₁ε₂−V² = 0, (3.53) or E =ε₁+ V²

E−ε₂. (3.54)

The eigenvalue and components of the eigen vector with the lower energy solution are given as

E₋ = ε₁+ε₂−(ε₁−ε₂)²+ 4V²

2 , (3.55)

x₋ = V

(E₋−ε₁)²+V²

, (3.56)

y₋ = E₋−ε₁

(E₋−ε₁)²+V²

. (3.57)

From these relation we have y₋

x₋ = E₋−ε₁

V = V

E₋−ε₂ (3.58)

Noting that E =E₋, a₁ =x₋, a₂ =y₋ we can derive the relation.

The operator form for ω is given as ω = a₂

a₁|21|,

= V

E−ε₂|21|. (3.59)

It’s hermitian conjugate is written as ω^† = a₂

a₁|12|,

= V

E −ε₂|12|. (3.60)

We obtain here the energy-dependent (P-space) effective interaction in its operator form Hˆ_{ef f}(E) ≡ P HΩ(E),

= P H(P +ω),

= (ε₁+ V²

E−ε₂)|11|,

= (ε₁+ V²

E−ε₂)P (3.61)

We express Schr¨odinger equation within P space as

Hˆ_{ef f}(E)·(PΨ) = Hˆ_{ef f}(E)(a₁|1)

= (ε₁+ V²

E−ε₂)(PΨ)

= E(PΨ). (3.62)

Finaly we try to calculate the operator form for the generator of unitary transformation.

In doing so we have to calculate the P-space operator   ω^†ω = V

E−ε₂|12| V

E−ε₂|21|,

= ( V

E−ε₂)²P, (3.63)

(ω^†ω)ⁿ = ( V

E−ε₂)²ⁿP. (3.64)

We obtain

T = arctanh(ω−ω^†)

=

∞ n=0

(−1)ⁿ

2n+ 1[ω(ω^†ω)ⁿ−h.c.]

=

∞ n=0

(−1)ⁿ 2n+ 1( V

E−ε₂)²ⁿ⁺¹(|21| − |12|),

= 1 i

∞ n=0

1

2n+ 1(i V

E−ε₂)²ⁿ⁺¹(|21| − |12|),

= arctan( V

E−ε₂)·(|21| − |12|). (3.65) It is seen that the anti-hermiticity, T^† = −T, is satisfied. We can confirm that e^−THe^T becomes diagonal.

3.2 A simple spin system under transverse field

The Hamiltonian of a simple spin system under transverse field inx direction is given as

H = H₀+W, (3.66)

H₀ ≡ −¯hΩs_z =−¯hΩ 2

1 0 0 −1

, (3.67)

W ≡ ¯hKs_x = hK¯ 2

0 1 1 0

. (3.68)

(We note here that Ω is not an operator but a mere energy splitting.) We may write the Hamiltonian as

H = −¯hΩ

2 c^†₁c₁+¯hΩ

2 c^†₂c₂+¯hK

2 (c^†₁c₂+c^†₂c₁), (3.69)

= ε₁c^†₁c₁+ε₂c^†₂c₂+V(c^†₁c₂+c^†₂c₁), (3.70) where we have put ε₁ =−¯hΩ/2, ε₂ = ¯hΩ/2 and V = ¯hK/2.

3.2.1 Method of solving the decoupling equation for ω Here we put ¯h= 1. The decoupling equation forω is given

0 = Qe^−ωHe^ωP

= QHP +QHQω−ωP HP −ωP HQω

= Q(−Ωs_z +Ks_y)P +Q(−Ωs_z+Ks_y)Qω−ωP(−Ωs_z+Ks_y)P

−ωP(−Ωs_z+Ks_y)Qω

= Q(Ksy)P +Q(−Ωsz)Qω−ωP(−Ωsz)P −ωP(Ksy)Qω. (3.71) Noting ω =QωP we may write with c-number x

ω=

0 0 x 0

, ω^†=

0 x 0 0

, ω^†ω=

x² 0 0 0

. (3.72)

Then we have

0 = K

2

0 0 1 0

+ Ω 2

0 0 x 0

+Ω 2

0 0 x 0

− K 2

0 0 x² 0

. (3.73) We obtain the quadratic equation for ω and its two solutions

Kx²−2Ωx−K = 0 (3.74)

→ x= Ω±√

Ω²+K²

K = −K

Ω∓√

Ω² +K². (3.75)

We note that the solution with minus sign leads us indefinite solution, so it is not appro- priate for a physical solution.

The exponent T is given as T =

∞ n=0

(−1)ⁿ 2n+ 1{

0 0 x 0

x² 0 0 0

n

−

x² 0 0 0

n

0 x 0 0

}

=

∞ n=0

(−1)ⁿ

2n+ 1x²ⁿ⁺¹{

0 0 1 0

−

0 1 0 0

} (3.76)

Here we have

∞ n=0

(−1)ⁿ

2n+ 1x²ⁿ⁺¹ = 1 i

∞ n=0

i²ⁿ⁺¹

2n+ 1x²ⁿ⁺¹ = 1

iarctanh(ix)

= arctan(x) = 1

2arctan( 2x 1−x²)

= 1

2arctan(K

Ω). (3.77)

Finaly we have

T =i arctan(K

Ω)s_y. (3.78)

Here it would be instructive to notice that the ratio between the energy splitting of the un- perturbed Hamiltonian and interaction strength,K/Ω,means an effective coupling strength of the system under consideration.

Let’s check the decoupling property of the transformed Hamiltonian. In doing this, for convenience, we define arctan(K/Ω)≡λ (tanλ=K/Ω). ThenT is written asT =iλ×sy. We calculate the transformation of s_z, s_x in terms of U = e^T.

˜

s_z = e^−Ts_ze^T

= s_z + [s_z, T] + 1

2![[s_z, T], T] +· · ·

= s_zcosλ+s_xsinλ, (3.79)

˜

s_x = e^−Ts_xe^T

= s_x+ [s_x, T] + 1

2![[s_x, T], T] +· · ·

= s_xcosλ−s_zsinλ, (3.80)

H˜ = e^−T(H₀+W)e^T

= e^−T(−¯hΩs_z + ¯hKs_x)e^T

= −¯h(Ω cosλ+Ksinλ)s_z+ ¯h(−Ω sinλ+Kcosλ)s_x

= −¯h(Ω cosλ+Ksinλ)s_z. (3.81)

We see that the decoupling property of the transformed Hamiltonian is satisfied asQHP˜ = PHQ˜ = 0.

3.2.2 Method of solving the formal solution for ω

In order to determine the exponent T, first, we need to calculate E−ε₂ = ε₁−ε₂−(ε₁−ε₂)²+ 4V²

2

= −hΩ¯ −(¯hΩ)²+ (¯hK)²

2 . (3.82)

Furthermore we note

|21| − |12| =

0 1

[1,0]−

1 0

[0,1]

=

0 0 1 0

−

0 1 0 0

=

0 −1 1 0

= −2is_y. (3.83)

T =

∞ n=0

(−1)ⁿ

2n+ 1( ¯hK

¯

hΩ +(¯hΩ)²+ (¯hK)²)²ⁿ⁺¹×2isy

=

∞ n=0

(i)²ⁿ⁺¹

2n+ 1( K

Ω +

(Ω)²+ (K)²

)²ⁿ⁺¹×2s_y

=

∞ n=0

1

2n+ 1( iK

Ω +

(Ω)²+ (K)²

)²ⁿ⁺¹×2s_y

= arctanh( iK

Ω +(Ω)²+ (K)²

)×2s_y. (3.84)

Introducing a constant

y≡ K

Ω +

(Ω)² + (K)²

, (3.85)

we have

T = arctanh(iy)×2s_y

= i arctan(y)×2s_y

= i arctan( 2y

1−y²)×s_y. (3.86)

Here note  

2y

1−y² = K

Ω. (3.87)

Inserting eq.(3.87) into eq.(3.86) we obtain T =i arctan(K

Ω)×s_y. (3.88)

Here it would be instructive to notice that the ratio between the energy splitting of the un- perturbed Hamiltonian and interaction strength,K/Ω,means an effective coupling strength of the system under consideration.

Let’s check the decoupling property of the transformed Hamiltonian. In doing this, for convenience, we define arctan(K/Ω)≡λ (tanλ=K/Ω). ThenT is written asT =iλ×s_y.

We calculate the transformation of s_z, s_x in terms of U = e^T.

˜

s_z = e^−Ts_ze^T

= s_z + [s_z, T] + 1

2![[s_z, T], T] +· · ·

= s_zcosλ+s_xsinλ, (3.89)

˜

s_x = e^−Ts_xe^T

= sx+ [sx, T] + 1

2![[sx, T], T] +· · ·

= s_xcosλ−s_zsinλ, (3.90)

H˜ = e^−T(H₀+W)e^T

= e^−T(−¯hΩsz + ¯hKsx)e^T

= −¯h(Ω cosλ+Ksinλ)s_z+ ¯h(−Ω sinλ+Kcosλ)s_x

= −¯h(Ω cosλ+Ksinλ)s_z. (3.91)

We see that the decoupling property of the transformed Hamiltonian is satisfied asQHP˜ = PHQ˜ = 0.

3.3 Anharmonic boson system in linear order ( = displaced bo- son system = neutral meson system)

The anharmonic boson system in linear order is well known and useful model system, and also called as the displaced boson system. Historically it was investigated as neutral meson system.[11]

The Hamiltonian of the anharmonic boson system in linear order and its relevant or- thonormalized states are given in its most simple form as

H = b^†b+g(b^†+b),(g : coupling strenth) (3.92) [b, b^†] = 1, [b, b] = 0, [b^†, b^†] = 0, (3.93)

b|0 = 0, 0|b^†= 0, (3.94)

|n ≡ (b^†)ⁿ

√n!|0, (3.95)

n|n = δ_nn, (3.96)

The matrix elements of the Hamiltonian are calculated n|H|n = n|b^†b|n+gn|(b^†+b)|n

= n δ_n,n +g(√

n δ_n,n+1+√

n+ 1 δ_n,n−1). (3.97)

The matrix form of the Hamiltonian is expressed in a tri-diagonal form shown as

H =

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎣

0 g 0 0 0 0 · · ·

g 1 g√

2 0 0 · · · 0 g√

2 2 g√

3 0 · · ·

0 0 g√

3 3 2g 0 · · ·

0 0 0 2g 4 g√

5 0

0 0 0 0 g√

5 5

· · ·

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎦

. (3.98)

In the next subsubsection we shall diagonalize the Hamiltonian.

3.3.1 Usual solution for the anharmonic boson system

It is known that the unitary transformation which diagonalize the anharmonic boson Hamiltonian

U = exp[g(b−b^†)]. (3.99)

We derive the exponent operator T by T = [H, H₀]

= [b^†b+g(b^†+b), b^†b]

= g(b−b^†). (3.100)

One can confirm this fact as follows: First, we calculate the transformation of the Hamil- tonian.

H˜ ≡ U⁻¹HU

= H+ [H, T] + 1

2![[H, T], T] + 1

3![[[H, T], T], T] +· · ·.

(3.101) Here we have

[H, T] = −g(b^†+b)−2g², (3.102) 1

2![[H, T], T] = g², (3.103)

1

n![· · ·[H, T], T],· · ·] = 0 (forn ≥3). (3.104) Then we have

H˜ =b^†b−g². (3.105)

Next, we calculate the transformation of boson operators.

˜b ≡ U⁻¹bU

= b+ [b, T] + 1

2![[b, T], T] + 1

3![[[b, T], T], T] +· · ·

= b−g, (3.106)

˜b^† ≡ U⁻¹b^†U

= b^†+ [b^†, T] + 1

2![[b^†, T], T] + 1

3![[[b^†, T], T], T] +· · ·

= b^†−g. (3.107)

Finaly we see

H˜ = ˜b^†˜b. (3.108)

We calculate the ground state for the transformed boson ˜b,˜b.

3.3.2 Method for solving of ω in model-space projective approach

We choose the vacuum state of the boson operator as the P space, and the states with more-than-one bosons as theQ space. Then we have the projection operator to the boson vacuum and the projection operator to the boson-existing sybspace, respectively

P ≡ |00|, (3.109)

Q ≡

n=0

|nn|, (3.110)

P + Q= 1, P² =P, Q² =Q, P Q=QP = 0. (3.111) We calculate the mapping operator for the anharmonic boson system in two ways.

3.3.3 method-1:solution for the decoupling equation for ω

First, we calculate the mapping operator ω by solving the decoupling equation. We put

ω =f(b^†, g)P, (3.112)

where f(b^†, g) is an arbitrary function of b^†. We may require that

f(0, g) = 0, (3.113)

f(b^†, g) → 0 as g →0, (3.114)

f(b^†,0) = 0. (3.115)

This relation requires thatf(b^†, g) is expanded in terms of b^† as

f(b^†, g) =cb^†+d(b^†)²+ higher order (c, d: constant). (3.116) Then we notice that

0|[b, f(b^†, g)]|0=c. (3.117)