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# A new boson expansion theory utilizing a norm operator

Kimikazu Taniguchi<EMAIL_ADDRESS>Department of Medical Information
Science, Suzuka University of Medical Science, Suzuka 1001-1, Japan

###### Abstract

We propose a new boson expansion method using a norm operator. The small
parameter expansion, in which the boson approximation becomes the zeroth-order
approximation, requires the double commutation relations between phonon
operators that are not closed between the phonon excitation modes adopted as
boson excitations. This results in an infinite expansion regardless of whether
the type of the boson expansion is Hermitian or non-Hermitian. The small
parameter expansion does not hold when the commutation relations are closed.
The norm operator is expressed as a function of the number operator in the
physical subspace, which enables us to obtain substantially a finite boson
expansion regardless of the Hermitian or non-Hermitian type. We also point out
the problems of the conventional boson expansion methods. The normal-ordered
linked-cluster expansion theory has failed to refute Marshalek’s claim that
KT-1 and KT-2 are of chimerical boson expansion. The Dyson boson expansion
theory does not have exceptional superiority over other types.

xxxx, xxx

##  1 Introduction

Microscopic elucidation of the large-amplitude collective motion of atomic
nuclei remains one of the important and challenging tasks, and its achievement
requires developing a method to overcome small-amplitude-oscillation
approximations like the Tamm-Dancoff approximation and the random phase
approximation. The boson expansion theory is one of the methods going beyond
the small-amplitude-oscillation approximations KM91 .

The boson expansion theory was initially formulated by replacing fermion
quasi-particle pair operators with the boson polynomials that reproduce their
commutation relations BZ62 . Later, referring to the preceding work Usi60 ,
the boson expansion theory has been given as a mapping theory by utilizing a
one-to-one correspondence between the basis vector in fermion space and the
completely antisymmetric state vector in boson space MYT64 . The Holstein-
Primakoff and the Dyson Boson expansions have been formulated also in the same
way JDF71 . These formulations target all pair operator excitations in fermion
space.

Practical use, however, has not adopted all excitation modes but the
collective and, in case of need, some non-collective excitation modes of Tamm-
Dancoff type phonons. Initially, there were two methods: One is to construct
the mapping operator by the phonons with only the crucial excitation modes
LH75 ; HJJ76 , and the other is the method to pick up only these phonons and
seek boson expansions that reproduce their commutation relations KT76 . Thus
formulated boson expansion methods were used for elucidating the large-
amplitude collective motions such as the shape transitions of nuclei in the
transitional region KT76 ; TTT87 ; SK91 . The boson expansion method called
KT-2 KT76  formulated in the latter method was, however, claimed to result in
incorrect boson expansions Ma80a ; Ma80b  and reformulated into so-called KT-3
KT83  according to the former method. The Dyson boson expansion theory (DBET),
finite expansion of the non-Hermitian type, has been also formulated by the
former method Ta01 .

Seeming to establish the formulation and achieve certain results, the problems
remain with the boson expansion methods.

One is on the approximate treatment of the algebra of the Tamm-Dancoff type
phonons. The double commutators among the Tamm-Dancoff type phonons generally
do not close within partially selected excitation modes. Until now without
exception, the boson expansion methods with restriction of the phonon
excitation modes have used approximations that neglect the modes not selected
for the boson excitation modes. The normal-ordered linked-cluster expansion
theory (NOLCEXPT) KT83 ; SK88  neglects these in the inverse of the norm
matrices of the multi-phonon state vectors to obtain its boson expansions and
finally abandons all the still-remaining modes. DBET truncates the unselected
phonon operators by adopting the approximation named phonon-truncation
approximation TTT87 , which is also called closed-algebra approximation Ta01 .
Each of the approximations above is essential for NOLCEXPT and DBET. NOLCEXPT
adopts it for obtaining the same expansions as KT-2, and DBET to obtain the
finite expansions. These approximations all bring to make the double
commutators among the selected phonon operators closed. It is claimed that the
validity of this approximation has been verified for the specific nuclei, and
it is also shown that the norm of the multi-phonon state vector obtained under
this approximation rapidly approaches 0 as the number of phonon excitations
increases, which brings rapid convergence of the boson expansions HJJ76 ;
MTS81 . Such behavior of the norm is due to the effect of the Pauli exclusion
principle MTS81 . Its rapid decrease means that the effect is strong. On the
other hand, NOLCEXPT claims that its boson expansion is of a small parameter
expansion with good convergence. Therefore, in the fermion subspace spanned by
the multi-phonon state vectors with selected excitation modes, the effect of
the Pauli exclusion principle should be weak. If this is correct, then the
norms of the multi-phonon state vectors would not approach zero rapidly as the
number of phonon excitations increases. We should investigate the cause of
contradictory conclusions.

The other is about the phonon excitation number. Until now, for the multi-
phonon state vectors, used as the basis vectors of the fermion subspace to be
mapped, the sorts of the excitation modes have been limited, while the number
of phonon excitations has not KT83 ; SK88 ; Ta01 . Without restricting the
phonon excitation number, the eigenvalues of the norm matrices of the multi-
phonon state vectors become zero when the number of excitations becomes large
enough even with restricting the sorts of excitation modes. Nevertheless,
NOLCEXPT is formulated assuming that zero eigenvalues do not appear regardless
of the number of phonon excitationsKT83 ; SK88 . There is, however, no clear
explanation for the validity of this assumption.

We have proposed a boson-fermion expansion theory (BFEXP) TM90 ; TM91  as an
alternative to the boson expansion theory. The boson expansion theory treats
all the adopted phonon excitation modes as bosons, while BFEXP, in the zeroth
order approximation, represents only the phonons with collective excitation
modes as bosons and those with non-collective remains as original phonons. We
can derive the boson expansions from this method by extending the boson part
up to the non-collective modes necessary and depressing the fermion
excitations. Since the formulation of BFEXP has not used the approximation for
commutation relations among the phonon operators, it would be worthwhile to
formulate a new boson expansion method without the approximation for the
commutation relations among phonon operators and compare its boson expansions
with those derived from BFEXP.

In this article, we propose a new boson expansion theory, naming it the norm
operator method (NOM), which enables us to handle both Hermitian and non-
Hermitian types, the case with or without limiting the phonon excitation modes
and the number of excitations, and the contribution of the phonon excitation
modes which are neglected in the conventional boson expansion methods.

In section 2, we deal with the Tam-Dancoff type phonons, the multi-phonon
state vectors, and the ideal boson state vectors.

In section 3, we give a mapping utilizing a norm operator. As specific
examples, we deal with the case of mapping all modes of phonon excitations
with and without the restriction of the phonon excitation number and the
restricted case where the maximum number of phonon excitations is one.

Section 4 deals with the boson expansions. First, we confirm the conditions
for using the ideal boson state vectors and then give the formulae used in the
boson expansions. Next, we provide the conditions that boson expansions become
of a small parameter expansion, offer an order estimation method for the
expansion terms, perform the boson expansions, show that all types of mapping
of the small parameter expansion give infinite boson expansions, and provide
the boson expansions of the phonon operators and the scattering operators up
to terms that have not been obtained so far. We also deal with non-small
parameter boson expansions, where we obtain DBET and the boson expansions that
are finite and Hermite. Finally, we point out and stress the essential role of
the norm operator in the boson expansion method.

In section 5, we take up the conventional methods and point out their
problems.

Section 6 is a summary.

##  2 Fermion space and boson space

###  2.1 Tamm-Dancoff type phonon operators, scattering operators, and their
commutation relations

We introduce pair operators,

$X_{\mu}^{\dagger}=\displaystyle\sum_{\alpha<\beta}\psi_{\mu}(\alpha\beta)a_{\alpha}^{\dagger}a_{\beta}^{\dagger},$
(1a)
$X_{\mu}=\displaystyle\sum_{\alpha<\beta}\psi_{\mu}(\alpha\beta)a_{\beta}a_{\alpha},$
(1b)
$B_{q}=\sum_{\alpha\beta}\varphi_{q}(\alpha\beta)a_{\beta}^{\dagger}a_{\alpha},$
(2a) $B_{\bar{q}=}B_{q}^{\dagger}.$ (2b)

Here, $a_{\alpha}^{\dagger}$ and $a_{\alpha}$ are quasi-particle creation and
annihilation operators in a single-particle state $\alpha$. The coefficients
satisfy the following relations:

$\psi_{\mu}(\beta\alpha)=-\psi_{\mu}(\alpha\beta)$ (3a)
$\sum_{\alpha<\beta}\psi_{\mu}(\alpha\beta)\psi_{\mu^{\prime}}(\alpha\beta)=\delta_{\mu,\mu^{\prime}},$
(3b)
$\sum_{\mu}\psi_{\mu}(\alpha\beta)\psi_{\mu}(\alpha^{\prime}\beta^{\prime})=\delta_{\alpha,\alpha^{\prime}}\delta_{\beta,\beta^{\prime}}-\delta_{\alpha,\beta^{\prime}}\delta_{\beta,\alpha^{\prime}},$
(3c) $\varphi_{\bar{q}}(\alpha\beta)=\varphi_{q}(\beta\alpha).$ (4a)
$\displaystyle\sum_{\alpha\beta}\varphi_{q}(\alpha\beta)\varphi_{q^{\prime}}(\alpha\beta)=\delta_{q,q^{\prime}},$
(4b)
$\displaystyle\sum_{q}\varphi_{q}(\alpha\beta)\varphi_{q}(\alpha^{\prime}\beta^{\prime})=\delta_{\alpha,\alpha^{\prime}}\delta_{\beta,\beta^{\prime}}.$
(4c)

These are the most common orthogonal transformations of the quasi-particle
pairs $a_{\alpha}^{\dagger}a_{\beta}^{\dagger}$, $a_{\beta}a_{\alpha}$ and
$a_{\beta}^{\dagger}a_{\alpha}$. These are used to couple the angular momenta
of quasi-particles to those of the quasi-particle pairs. Some of $X_{\mu}$ and
$X_{\mu}^{\dagger}$ are composed by the further superposition of such pair
operators to reflect the dynamics into the selected phonons. Tamm-Dancoff
approximation or a similar approximation is usually applied to them for
identifying collective excitation modes and non-collective ones. Hereafter,
$X_{\mu}^{\dagger}$ and $X_{\mu}$ are called phonon creation and annihilation
operators, and $B_{q}$ is called a scattering operator.

The phonon and scattering operators satisfy the following commutation
relations:

$[X_{\mu},X_{\mu^{\prime}}^{\dagger}]=\delta_{\mu,\mu^{\prime}}-\sum_{q}\Gamma^{\mu\mu^{\prime}}_{q}B_{q},$
(5a)

$[B_{q},X_{\mu}^{\dagger}]=\sum_{\mu^{\prime}}\Gamma^{\mu\mu^{\prime}}_{q}X_{\mu^{\prime}}^{\dagger},$
(5b)

$[X_{\mu},B_{q}]=\sum_{\mu^{\prime}}\Gamma^{\mu^{\prime}\mu}_{q}X_{\mu^{\prime}},$
(5c)

where the definition of $\Gamma^{\mu\mu^{\prime}}_{q}$ is as follows:

$\Gamma^{\mu\mu^{\prime}}_{q}=\sum_{\alpha\beta}\varphi_{q}(\alpha\beta)\Gamma^{\mu\mu^{\prime}}_{\alpha\beta},\quad\Gamma^{\mu\mu^{\prime}}_{\alpha\beta}=\sum_{\gamma}\psi_{\mu}(\alpha\gamma)\psi_{\mu^{\prime}}(\beta\gamma).$
(6)

The following relation holds:

$\Gamma_{\bar{q}}^{\mu_{1}\mu_{2}}=\Gamma_{q}^{\mu_{2}\mu_{1}}.$ (7)

From Eqs. (5a) and (5b), we obtain

$[[X_{\mu_{1}},X_{\mu_{2}}^{\dagger}],X_{\mu_{3}}^{\dagger}]=-\sum_{\mu^{\prime}}Y(\mu_{1},\mu_{2},\mu_{3},\mu^{\prime})X_{\mu^{\prime}}^{\dagger},$
(8)

where the definition of $Y(\mu_{1}\mu_{2}\mu_{3}\mu_{4})$ is

$Y(\mu_{1}\mu_{2}\mu_{3}\mu_{4})=\sum_{q}\Gamma_{q}^{\mu_{1}\mu_{2}}\Gamma_{q}^{\mu_{3}\mu_{4}}=\sum_{\alpha\beta}\Gamma_{\alpha\beta}^{\mu_{1}\mu_{2}}\Gamma_{\alpha\beta}^{\mu_{3}\mu_{4}}.$
(9)

The following relation holds:

$\begin{array}[]{lll}Y(\mu_{1}\mu^{\prime}_{1}\mu^{\prime}_{2}\mu_{2})&=&Y(\mu_{2}\mu^{\prime}_{1}\mu^{\prime}_{2}\mu_{1})\\\
&=&Y(\mu_{1}\mu^{\prime}_{2}\mu^{\prime}_{1}\mu_{2})\\\
&=&Y(\mu^{\prime}_{1}\mu_{1}\mu_{2}\mu^{\prime}_{2}).\\\ \end{array}$ (10)

###  2.2 Multi-phonon and multi-boson state vectors

We divide the phonon excitation modes $\\{\mu\\}$ into two groups, $\\{t\\}$
and $\\{\bar{t}\\}$,and prepare the multi-phonon state vectors,

$|N;t\rangle\rangle=|t_{1},t_{2},\cdots,t_{N}\rangle\rangle=X_{t_{1}}^{\dagger}X_{t_{2}}^{\dagger}\cdots
X_{t_{N}}^{\dagger}|0\rangle\quad(0\leq N\leq N_{max}).$ (11)

$\\{t\\}$ usually consists of collective modes and some non-collective modes
if necessary, selected by the small amplitude approximation. We treat not only
these cases but also the case where all modes are adopted, that is,
$\\{t\\}=\\{\mu\\}$.

Next we introduce boson creation and annihilation operators, $b_{t}^{\dagger}$
and $b_{t^{\prime}}$, having the same indices as those of the multi-phonons,
$X_{t}^{\dagger}$ and $X_{t^{\prime}}$:

$[b_{t},b_{t^{\prime}}^{\dagger}]=\delta_{t,t^{\prime}}.$ (12)

The multi-boson states,

$|N;t))=|t_{1},t_{2},\cdots,t_{N}))=b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}\cdots
b_{t_{N}}^{\dagger}|0),$ (13)

are orthogonal to one another, and are normalized by their norms,

$\mathcal{N}_{B}(N;t)=((N:t|N;t)),$ (14)

such as

$|N;t)=|t_{1},t_{2},\cdots,t_{N})=\mathcal{N}_{B}(N;t)^{-1/2}|N;t)).$ (15)

They are so-called ideal boson state vectors.

##  3 Boson mapping

This section deals with boson mapping. We introduce a norm operator and
construct a mapping operator that can handle both Hermitian and non-Hermitian
types, both with and without limiting the types and number of phonon
excitation modes.

The norm operator is defined as

$\hat{Z}=\sum_{N=0}^{N_{max}}\hat{Z}(N),$ (16a)
$\begin{array}[]{lll}\hat{Z}(N)&=&\displaystyle\sum_{tt^{\prime}}|N,t)\langle
N;t|N;t^{\prime}\rangle(N;t^{\prime}|\\\
&=&\displaystyle\sum_{t_{1}\leq\cdots\leq
t_{N}}\sum_{t^{\prime}_{1}\leq\cdots\leq t^{\prime}_{N}}|t_{1}\cdots
t_{N})\langle t_{1}\cdots t_{N}|t^{\prime}_{1}\cdots
t_{N}\rangle(t^{\prime}_{1}\cdots t^{\prime}_{N}|,\end{array}$ (16b)

where

$|N;t\rangle=\mathcal{N}_{B}(N;t)^{-1/2}|N;t\rangle\rangle.$ (17)

This norm operator is a modified one of the previously introduced KT83  by
adding the restriction $N_{max}$, which allows us to constrain the number of
phonon excitations and corresponding boson excitations. $\hat{Z}(N)$ satisfies
the eigenequation,

$\hat{Z}(N)|N;a)=z_{a}(N)|N;a),$ (18)

where $|N;a)$ is a normalized eigenvector and $z_{a}(N)$ is an eigenvalue. The
eigenvalues $z_{a}(N)$ become positive or zero and $a_{0}$ represents
$z_{a_{0}}(N)=0$. Using these, we obtain the spectral decomposition of
$\hat{Z}(N)$ as

$\hat{Z}(N)=\sum_{a\neq a_{0}}|N;a)z_{a}(N)(N;a|.$ (19)

Functions of $\hat{Z}(N)$ are defined by

$f(\hat{Z}(N))=\sum_{a\neq a_{0}}|N;a)f(z_{a}(N))(N;a|,$ (20)

and we obtain

$f(\hat{Z})=\sum_{N=0}^{N_{max}}f(\hat{Z}(N)).$ (21)

Introducing $u_{a}^{t}(N)=(N;t|N;a)$, then Eq. (18) becomes the eigenequation
of the multi-phonon norm matrix,

$\sum_{t^{\prime}}\langle N;t|N;t^{\prime}\rangle
u_{a}^{t^{\prime}}(N)=z_{a}(N)u_{a}^{t}(N),$ (22)

The eigenvectors are orthonomalized as

$\displaystyle\sum_{t}u_{a}^{t}(N)u_{a^{\prime}}^{t}(N)=\delta_{a,a^{\prime}},$
(23a) and satisfy the completeness relations
$\displaystyle\sum_{a}u_{a}^{t}(N)u_{a}^{t^{\prime}}(N)=\delta_{t,t^{\prime}}.$
(23b)

Using the norm operator $\hat{Z}$, we introduce the mapping operator $U_{\xi}$
as

$U_{\xi}=\hat{Z}^{\xi-\frac{1}{2}}\widetilde{U},$ (24)

where $\widetilde{U}$ is a mapping operator whose definition is as follows:

$\widetilde{U}=\sum_{N=0}^{N_{max}}{\widetilde{U}}(N),$ (25a)
$\begin{array}[]{lll}{\widetilde{U}}(N)&=&\displaystyle\sum_{t}|N;t)\langle
N;t|\\\ &=&\displaystyle\sum_{t_{1}\leq t_{2}\leq\cdots\leq
t_{N}}|t_{1}t_{2}\cdots t_{N})\langle t_{1}t_{2}\cdots t_{N}|,\end{array}$
(25b)

which satisfies the following relations,

$\widetilde{U}{\widetilde{U}}^{\dagger}=\hat{Z},$ (26a)
${\widetilde{U}}(N){\widetilde{U}}(N)^{\dagger}=\hat{Z}(N).$ (26b)

${\widetilde{U}}(N)$ is also expressed as

${\widetilde{U}}(N)=\sum_{a\neq a_{0}}z_{a}^{\frac{1}{2}}|N;a)\langle N;a|,$
(27)

where

$|N;a\rangle=z_{a}^{-\frac{1}{2}}(N)\sum_{t}u_{a}^{t}(N)|N;t\rangle\qquad(a\neq
a_{0}).$ (28)

$|N;a\rangle$ become orthonormalized basis vectors of the fermion subspace
spanned by $|N;t\rangle$. Using $|N:a\rangle$ and $|N:a)$, the mapping
operator is expressed as

$U_{\xi}=\sum_{N=0}^{N_{max}}U_{\xi}(N);\quad U_{\xi}(N)=\sum_{a\neq
a_{0}}z_{a}(N)^{\xi}|N;a)\langle N;a|.$ (29)

The following relations are satisfied:

$U_{-\xi}^{\dagger}U_{\xi}=\hat{T}_{F},\qquad
U_{\xi}U_{-\xi}^{\dagger}=\hat{T}_{B},$ (30)

where

$\hat{T}_{F}=\sum_{N=0}^{N_{max}}\hat{T}_{F}(N);\qquad\hat{T}_{F}(N)=\displaystyle\sum_{a\neq
a_{0}}|N;a\rangle\langle N;a|,$ (31)
$\hat{T}_{B}=\sum_{N=0}^{N_{max}}\hat{T}_{B}(N);\quad\hat{T}_{B}(N)=\sum_{a\neq
a_{0}}|N;a)(N;a|.$ (32)

In addition, we define the following operators,

$\breve{1}_{B}=\sum_{N=0}^{N_{max}}\hat{1}_{B}(N);\qquad\hat{1}_{B}(N)=\sum_{t}|N;t)(N;t|.$
(33)

If $\hat{Z}(N)$ has even one zero eigenvalue, then
$\hat{T}_{B}(N)\neq\hat{1}_{B}(N)$ and hence $\hat{T}_{B}\neq\breve{1}_{B}$.
Otherwise, they match one another.

The state vectors and operators of fermion space are mapped onto those of
boson subspace as

$|\psi^{\prime})_{\xi}=U_{\xi}|\psi^{\prime}\rangle,\qquad{}_{-\xi}(\psi|=\langle\psi|U_{-\xi}^{\dagger},$
(34a) $(O_{F})_{\xi}=U_{\xi}O_{F}U_{-\xi}^{\dagger}.$ (34b)

These satisfy the following relations:

$|\psi^{\prime})_{\xi}=\left\\{{}_{\xi}(\psi^{\prime}|\right\\}^{\dagger},\qquad{}_{-\xi}(\psi|=\left\\{|\psi)_{-\xi}\right\\}^{\dagger},$
(35a) $(O_{F})_{-\xi}=\left\\{(O_{F}^{\dagger})_{\xi}\right\\}^{\dagger}.$
(35b)

The mapping is of the Hermitian type when $\xi=0$ and, in other cases, of the
non-Hermitian type.

A one-to-one correspondence exists between the fermion subspace projected by
$\hat{T}_{F}$ and the boson subspace by $\hat{T}_{B}$. For the state vectors,
$|\psi\rangle$ and $|\psi^{\prime}\rangle$, which belong to the fermion
subspace projected by $\hat{T}_{F}$,

$\begin{array}[]{lll}\langle\psi|O_{F}|\psi^{\prime}\rangle&=&\langle\psi|\hat{T}_{F}O_{F}\hat{T}_{F}|\psi^{\prime}\rangle\\\
&=&\langle\psi|U_{-\xi}^{\dagger}U_{\xi}O_{F}U_{-\xi}^{\dagger}U|\psi^{\prime}\rangle\\\
&=&{}_{-\xi}(\psi|(O_{F})_{\xi}|\psi^{\prime})_{\xi},\end{array}$ (36)

that is, the matrix element of the fermion subspace becomes equal to that of
the corresponding boson subspace. The boson subspace corresponding to the
fermion subspace projected by $\hat{T}_{F}$ is called the physical subspace,
and the boson state vectors belonging to that space are called the physical
state vector. The projection operator of the physical subspace is
$\hat{T}_{B}$.

The relation

${}_{\xi}(\psi|(O_{F})_{-\xi}|\psi^{\prime})_{-\xi}={}_{-\xi}(\psi|(O_{F})_{\xi}|\psi^{\prime})_{\xi}$
(37)

holds, therefore it is sufficient to treat the case $\xi\geq 0$.

The mapping of the product of the fermion operators does not generally result
in the product of the mapped fermion operators. That is

$(O_{F}O^{\prime}_{F})_{\xi}\neq(O_{F})_{\xi}(O^{\prime}_{F})_{\xi},$ (38)

and therefore, the commutation relations of the fermion operators are mapped
as

$([O_{F},O^{\prime}_{F}])_{\xi}=(O_{F}O^{\prime}_{F})_{\xi}-(O^{\prime}_{F}O_{F})_{\xi}\neq[(O_{F})_{\xi},(O^{\prime}_{F})_{\xi}],$
(39)

while under the approximation $O_{F}O^{\prime}_{F}\approx
O_{F}\hat{T}_{F}O^{\prime}_{F}$,

$(O_{F}O^{\prime}_{F})_{\xi}\approx(O_{F})_{\xi}(O^{\prime}_{F})_{\xi},$ (40)

and

$([O_{F},O^{\prime}_{F}])_{\xi}\approx[(O_{F})_{\xi},(O^{\prime}_{F})_{\xi}]$
(41)

hold. The conventional practical boson expansion methods use this
approximation. If this approximation holds, it is sufficient to map the phonon
and scattering operators, otherwise, it becomes necessary to obtain the
mapping of the product of these fermion operators.

We denote the mapping of $\widetilde{U}$ as

$\widetilde{|\psi)}=\widetilde{U}|\psi\rangle,\qquad\widetilde{(\psi|}=\langle\psi|{\widetilde{U}}^{\dagger},$
(42a) $\widetilde{O_{F}}=\widetilde{U}O_{F}{\widetilde{U}}^{\dagger}.$ (42b)

The mapping of Eqs. (34) is expressed as

$|\psi^{\prime})_{\xi}=\hat{Z}^{\xi-\frac{1}{2}}\widetilde{|\psi^{\prime})},\qquad{}_{-\xi}(\psi|=\widetilde{(\psi|}\hat{Z}^{-\xi-\frac{1}{2}},$
(43a)
$(O_{F})_{\xi}=\hat{Z}^{\xi-\frac{1}{2}}\widetilde{O_{F}}\hat{Z}^{-\xi-\frac{1}{2}},$
(43b)

which makes it clear that the different treatment of the norm operator in the
mapping operator produces another type of mapping. The mapping of Eqs. (34) is
also expressed as

$|\psi^{\prime})_{\xi}=\hat{Z}^{\xi}|\psi^{\prime})_{0},\qquad{}_{-\xi}(\psi|={}_{0}(\psi|\hat{Z}^{-\xi},$
(44a) $(O_{F})_{\xi}=\hat{Z}^{\xi}(O_{F})_{0}\hat{Z}^{-\xi},$ (44b)

The mapping of $\xi=0$ being of the Hermitian type and that of $\xi\neq 0$
being of the non-Hermitian type transform one another by the similarity
transformation operator that becomes the power of the norm operator $\hat{Z}$.

###  3.1 The case where all the phonon excitation modes are adopted as the
boson excitation modes

Hereafter, we attach $(A)$ such as $\hat{Z}^{(A)}$ in the case that we
introduce boson operators corresponding to all phonon excitation modes for no
confusion.

We start with the following,

$\begin{array}[]{c}\displaystyle\sum_{\mu_{1}\leq\cdots\leq\mu_{N}}|\mu_{1},\cdots,\mu_{N}\rangle\langle\mu_{1},\cdots,\mu_{N}|=\displaystyle\frac{1}{N!}\sum_{\mu_{1}\cdots,\mu_{N}}|\mu_{1},\cdots,\mu_{N}\rangle\rangle\langle\langle\mu_{1},\cdots,\mu_{N}|\\\
=\displaystyle\frac{1}{2^{2N}N!}\sum_{\mu_{1}\cdots,\mu_{N}}\sum_{\begin{array}[]{c}\alpha_{1}\beta_{1}\cdots\alpha_{N}\beta_{N}\\\
\alpha^{\prime}_{1}\beta^{\prime}_{1}\cdots\alpha^{\prime}_{N}\beta^{\prime}_{N}\end{array}}\psi_{\mu_{1}}(\alpha_{1}\beta_{1})\psi_{\mu_{1}}(\alpha^{\prime}_{1}\beta^{\prime}_{1})\cdots\psi_{\mu_{N}}(\alpha_{N}\beta_{N})\psi_{\mu_{N}}(\alpha^{\prime}_{N}\beta^{\prime}_{N})\\\
|\alpha_{1}\beta_{1}\cdots\alpha_{N}\beta_{N}\rangle\langle\alpha_{1}\beta_{1}\cdots\alpha_{N}\beta_{N}|\\\
=\displaystyle\frac{1}{2^{N}N!}\sum_{\alpha_{1}\beta_{1}\cdots\alpha_{N}\beta_{N}}|\alpha_{1}\beta_{1}\cdots\alpha_{N}\beta_{N}\rangle\langle\alpha_{1}\beta_{1}\cdots\alpha_{N}\beta_{N}|\\\
=\displaystyle\frac{(2N)!}{2^{N}N!}\sum_{\alpha_{1}\beta_{1}\leq\cdots\leq\alpha_{N}\beta_{N}}|\alpha_{1}\beta_{1}\cdots\alpha_{N}\beta_{N}\rangle\langle\alpha_{1}\beta_{1}\cdots\alpha_{N}\beta_{N}|,\end{array}$
(45)

where

$|\alpha_{1}\beta_{1}\cdot\alpha_{N}\beta_{N}\rangle=a_{\alpha_{1}}^{\dagger}a_{\beta_{1}}^{\dagger}\cdots
a_{\alpha_{N}}^{\dagger}a_{\beta_{N}}^{\dagger}|0\rangle,$ (46)

and we use that the function $f(t_{1},\cdots,t_{N})$, which is completely
symmetric for the argument, satisfies the following KKS91 :

$\sum_{t_{1}\leq\cdots\leq
t_{N}}f(t_{1},\cdots,t_{N})=\sum_{t_{1},\cdots,t_{N}}\frac{\mathcal{N}_{B}(t_{1},\cdots,t_{N})}{N!}f(t_{1},\cdots,t_{N}).$
(47)

From above, we obtain the following relation,

$\sum_{\mu_{1}\leq\cdots\leq\mu_{N}}|\mu_{1},\cdots,\mu_{N}\rangle\langle\mu_{1},\cdots,\mu_{N}|=(2N-1)!!\hat{1}_{F}^{(A)}(N),$
(48)

where

$\hat{1}_{F}^{(A)}(N)=\sum_{\alpha_{1}\beta_{1}\leq\cdots\leq\alpha_{N}\beta_{N}}|\alpha_{1}\beta_{1}\cdots\alpha_{N}\beta_{N}\rangle\langle\alpha_{1}\beta_{1}\cdots\alpha_{N}\beta_{N}|;\quad(2N-1)!!=\frac{(2N)!}{2^{N}N!}.$
(49)

Let $\mathbf{Z}^{(A)}(N)$ be a matrix composed of the matrix element
$\langle\mu^{\prime}_{1},\cdots\mu^{\prime}_{N}|\mu_{1}\cdots\mu_{N}\rangle$,
we obtain, from this relation,

$\mathbf{Z}^{(A)}(N)^{2}=(2N-1)!!\mathbf{Z}^{(A)}(N),$ (50)

which indicates that the eigenvalues of this matrix are $(2N-1)!!$ or $0$.
Zero eigenvalues appear even at $N=2$ KT83 , and so do in the case $N\geq 2$.
From this relation we obtain

$\hat{Z}^{(A)}(N)^{2}=(2N-1)!!\hat{Z}^{(A)}(N),$ (51)

and

$\left(\hat{Z}^{(A)}\right)^{2}=(2\hat{N}_{B}^{(A)}-1)!!Z^{(A)};\quad\hat{N}_{B}^{(A)}=\sum_{\mu}b_{\mu}^{\dagger}b_{\mu}.$
(52)

The case N=2 in Eq. (50) is equivalent to the following relations KT76 ,

$\sum_{\mu\mu^{\prime}}Y(\mu^{\prime}_{1}\mu\mu^{\prime}\mu^{\prime}_{2})Y(\mu_{1}\mu\mu^{\prime}\mu_{2})=4((\mu^{\prime}_{1}\mu^{\prime}_{2}|\mu_{1}\mu_{2}))-2Y(\mu^{\prime}_{1}\mu_{1}\mu_{2}\mu^{\prime}_{2}).$
(53)

We introduce the following operators,

$b_{\alpha\beta}=\sum_{\mu}\psi_{\mu}(\alpha\beta)b_{\mu},\quad
b_{\alpha\beta}^{\dagger}=\sum_{\mu}\psi_{\mu}(\alpha\beta)b_{\mu}^{\dagger},$
(54)

which satisfies the commutation relations,

$[b_{\alpha^{\prime}\beta^{\prime}},b_{\alpha\beta}^{\dagger}]=\delta_{\alpha^{\prime}\alpha}\delta_{\beta^{\prime}\beta}-\delta_{\alpha^{\prime}\beta}\delta_{\beta^{\prime}\alpha}.$
(55)

Using these operators,
$\widetilde{U}^{(A)}(N)=\displaystyle\sum_{\mu}|N;\mu)\langle N;\mu|$ are
rewritten as

$\widetilde{U}^{(A)}(N)=\sqrt{(2N-1)!!}\sum_{\alpha_{1}<\beta_{1}<\cdots<\alpha_{N}<\beta_{N}}|\alpha_{1}\beta_{1}\cdots\alpha_{N}\beta_{N})_{M}\
\langle\alpha_{1}\beta_{1}\cdots\alpha_{N}\beta_{N}|,$ (56)

where

$|\alpha_{1}\beta_{1}\cdots\alpha_{N}\beta_{N})_{M}=\frac{1}{\sqrt{(2N-1)!!}}{\sum_{P}}^{\prime}(-)^{P}b_{\alpha_{1}\beta_{1}}^{\dagger}\cdots
b_{\alpha_{N}\beta_{N}}^{\dagger}|0),$ (57)

and $\displaystyle{\sum_{P}}^{\prime}$ means the summation so that the states
on the left side become totally antisymmetric MYT64 . From these, we obtain

$\begin{array}[]{lll}\hat{Z}^{(A)}(N)&=&\widetilde{U}^{(A)(N)}\widetilde{U}^{(A)}(N)^{\dagger}\\\
&=&\displaystyle(2N-1)!!\displaystyle\sum_{\alpha_{1}<\beta_{1}<\cdots\alpha_{N}\beta_{N}}|\alpha_{1}\beta_{1}\cdots\alpha_{N}\beta_{N})_{M}{}_{M}(\alpha_{1}\beta_{1}\cdots\alpha_{N}\beta_{N}|,\end{array}$
(58)

which is the spectral decomposition of $\hat{Z}^{(A)}(N)$ and indicates that
the eigenvectors of the eigenvalue $(2N-1)!!$ are
$|\alpha_{1}\beta_{1}\cdots\alpha_{N}\beta_{N})_{M}$. We also obtain

$\begin{array}[]{lll}\hat{T}_{F}^{(A)}&=&\displaystyle\sum_{N=0}^{N_{max}}\hat{T}_{F}^{(A)}(N),\\\
\hat{T}_{F}^{(A)}(N)&=&\hat{1}_{F}(N),\\\
\quad\hat{1}_{F}(N)&=&\displaystyle\sum_{\alpha_{1}<\beta_{1}<\cdots\alpha_{N}\beta_{N}}|\alpha_{1}\beta_{1}\cdots\alpha_{N}\beta_{N}\rangle\langle\alpha_{1}\beta_{1}\cdots\alpha_{N}\beta_{N}|,\end{array}$
(59)
$\begin{array}[]{lll}\hat{T}_{B}^{(A)}&=&\displaystyle\sum_{N=0}^{N_{max}}\hat{T}_{B}(N),\\\
\hat{T}_{B}^{(A)}(N)&=&\displaystyle\sum_{\alpha_{1}<\beta_{1}<\cdots\alpha_{N}\beta_{N}}|\alpha_{1}\beta_{1}\cdots\alpha_{N}\beta_{N})_{M}{}_{M}(\alpha_{1}\beta_{1}\cdots\alpha_{N}\beta_{N}|.\end{array}$
(60)

$\hat{Z}^{(A)}$ is written as

$\hat{Z}^{(A)}=(2\hat{N}_{B}^{(A)}-1)!!\hat{T}_{B}^{(A)}.$ (61)

The mapping operator is given as

$\begin{array}[]{lll}U_{\xi}^{(A)}&=&\displaystyle\sum_{N=0}^{N_{max}}U_{\xi}(N),\\\
U_{\xi}^{(A)}(N)&=&\displaystyle\\{(2N-1)!!\\}^{\xi}\sum_{\alpha_{1}<\beta_{1}<\cdots<\alpha_{N}<\beta_{N}}|\alpha_{1}\beta_{1}\cdots\alpha_{N}\beta_{N})_{M}\
\langle\alpha_{1}\beta_{1}\cdots\alpha_{N}\beta_{N}|.\end{array}$ (62)

If we set $\xi=0$ and $N_{max}\rightarrow\infty$, this mapping becomes the MYT
mapping MYT64 , from which we obtain the boson expansions of Holstein and
Primakoff, and if we take $\xi=\pm 1$, they become mapping operators for the
Dyson boson expansions JDF71 . Taking $N_{max}\rightarrow\infty$,
$U_{\xi}^{(A)}$ maps the whole fermion space that consists of even numbers of
quasi-particles to the boson subspace.

###  3.2 The case where the maximum phonon excitation number is 1

In the case where the maximum phonon excitation number is 1, $|0\rangle$ and
$|\mu\rangle$ become orthonormal bases of the fermion space of even-quasi-
particle excitations up to the two-quasi-particle excitations, which
correspond to $|0)$ and $|\mu)$, respectively. The mapping operator becomes as

$U_{\xi}=\widetilde{U}=|0)\langle 0|+\sum_{\mu}|\mu)\langle\mu|,$ (63)

and $\hat{Z}=\breve{1}_{B}$. As a result, the mapping operator has no
dependence on $\xi$, and the mapping becomes of the Hermitian type. The
projection operator onto the fermion subspace to be mapped is given by

$\hat{T}_{F}=|0\rangle\langle 0|+\sum_{\mu}|\mu\rangle\langle\mu|.$ (64a) The
projection operator onto the physical subspace, which has a one-to-one
correspondence to the fermion subspace, becomes as
$\hat{T}_{B}=\breve{1}_{B}.$ (64b)

This indicates that the ideal boson states are the physical state vectors,
with one-to-one correspondences to the fermion state vectors.

The following relations,

$X_{\mu}U_{-\xi}^{\dagger}=|0\rangle(\mu|=U_{-\xi}^{\dagger}b_{t}\breve{1}_{B},$
(65a) $U_{\xi}X_{\mu}^{\dagger}=|\mu)\langle
0|=\breve{1}_{B}b_{\mu}^{\dagger}U_{\xi},$ (65b)
$B_{q}U_{-\xi}^{\dagger}=\sum_{\mu^{\prime}}\sum_{\mu}\Gamma^{\mu^{\prime}\mu}_{q}|\mu\rangle(\mu^{\prime}|,$
(65c)

hold, and we obtain

$(X_{\mu})_{\xi}=|0)(\mu|=\breve{1}_{B}(X_{\mu})_{B}\breve{1}_{B}=(X_{\mu})_{B}\breve{1}_{B},\quad(X_{\mu})_{B}=b_{\mu},$
(66a)
$(X_{\mu}^{\dagger})_{\xi}=|\mu)(0|=\breve{1}_{B}(X_{\mu}^{\dagger})_{B}\breve{1}_{B}=\breve{1}_{B}(X_{\mu}^{\dagger})_{B},\quad(X_{\mu}^{\dagger})_{B}=b_{\mu}^{\dagger},$
(66b)
$\begin{array}[]{c}(B_{q})_{\xi}=\displaystyle\sum_{\mu^{\prime}}\sum_{\mu}\Gamma^{\mu^{\prime}\mu}_{q}|\mu)(\mu^{\prime}|=\breve{1}_{B}(B_{q})_{B}\breve{1}_{B}=\breve{1}_{B}(B_{q})_{B}=(B_{q})_{B}\breve{1}_{B},\\\
(B_{q})_{B}=\displaystyle\sum_{\mu\mu^{\prime}}\Gamma^{\mu^{\prime}\mu}_{q}b_{\mu}^{\dagger}b_{\mu^{\prime}}.\end{array}$
(66c)

The product of the operators becomes as follows:

$(O_{F}X_{\mu})_{\xi}=(O_{F})_{\xi}(X_{\mu})_{\xi}=\breve{1}_{B}(O_{F})_{B}\breve{1}_{B}(X_{\mu})_{B}\breve{1}_{B}=\breve{1}_{B}(O_{F})_{B}(X_{\mu})_{B}\breve{1}_{B},$
(67a)
$(X_{\mu}^{\dagger}O_{F})_{\xi}=(X_{\mu}^{\dagger})_{\xi}(O_{F})_{\xi}=\breve{1}_{B}(X_{\mu}^{\dagger})_{B}\breve{1}_{B}(O_{F})_{B}\breve{1}_{B}=\breve{1}_{B}(X_{\mu}^{\dagger})_{B}(O_{F})_{B}\breve{1}_{B},$
(67b)

therefore we can obtain the mapping of the product of
$X_{\mu}^{\dagger}$,$X_{\mu}$, and $B_{q}$ by arranging them in normal order.

The commutation relations of $(X_{\mu}^{\dagger})_{B}$, $(X_{\mu})_{B}$, and
$(B_{q})_{B}$ become as follows:

$[(X_{\mu})_{B},(X_{\mu^{\prime}}^{\dagger})_{B}]=\delta_{\mu,\mu^{\prime}}$
(68a)

$[(B_{q})_{B},(X_{\mu}^{\dagger})_{B}]=\sum_{\mu^{\prime}}\Gamma^{\mu\mu^{\prime}}_{q}(X_{\mu^{\prime}}^{\dagger})_{B}.$
(68b)

$[(X_{\mu})_{B},(B_{q})_{B}]=\sum_{\mu^{\prime}}\Gamma^{\mu^{\prime}\mu}_{q}(X_{\mu^{\prime}})_{B},$
(68c)

which are equal to the results of the boson approximation.

From the above, when the maximum number of phonons is $1$, by arranging the
phonon creation and annihilation operators and the scattering operators in
normal order and replacing them with $(X_{\mu}^{\dagger})_{B}$,
$(X_{\mu})_{B}$, and $(B_{q})_{B}$, respectively, then the fermion subspace is
completely mapped onto the boson subspace projected by $\breve{1}_{B}$.

In this way, NOM establishes the boson approximation as the boson mapping
whose maximum phonon excitation number is 1.

##  4 Boson expansions

###  4.1 Formulae for the boson expansions

We give here the formulae used to obtain the boson expansions of the mapped
fermion operators. We utilize

$\begin{array}[]{lll}\displaystyle\widetilde{U}(N)&=&\displaystyle\sum_{t_{1}\leq
t_{2}\leq\cdots\leq t_{N}}|t_{1}t_{2}\cdots t_{N})\langle t_{1}t_{2}\cdots
t_{N}|\\\ &=&\displaystyle\sum_{t_{1}t_{2}\cdots
t_{N}}\frac{{\mathcal{N}}_{B}(t_{1}t_{2}\cdots t_{N})}{N!}|t_{1}t_{2}\cdots
t_{N})\langle t_{1}t_{2}\cdots t_{N}|\\\
&=&\displaystyle\sum_{t_{1}t_{2}\cdots t_{N}}\frac{1}{N!}|t_{1}t_{2}\cdots
t_{N}))\langle\langle t_{1}t_{2}\cdots t_{N}|\end{array}$ (69)

and obtain the following series of formulae:

$\widetilde{U}(N)X_{t^{\prime}}=(X_{t^{\prime}})_{D}\widetilde{U}(N+1)\quad(N\geq
0),$ (70a)
$\begin{array}[]{lll}\widetilde{U}(1)X_{t}^{\dagger}&=&(X_{t}^{\dagger})_{D}\widetilde{U}(0),\\\
\widetilde{U}(N+1)X_{t}^{\dagger}&=&(X_{t}^{\dagger})_{D}\widetilde{U}(N)\\\
&&-\displaystyle\frac{1}{2}\sum_{t_{1}t_{2}}\sum_{\bar{t}_{1}^{\prime}}Y(tt_{1}t_{2}\bar{t}^{\prime}_{1})b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}\widetilde{U}(N-1)X_{\bar{t}^{\prime}_{1}}\quad(N\geq
1),\end{array}$ (70b)

$\begin{array}[]{lll}\widetilde{U}(0)B_{q}&=&0,\\\
\widetilde{U}(N)B_{q}&=&(B_{q})_{D}\widetilde{U}(N)+\displaystyle\sum_{t}\sum_{\bar{t}^{\prime}}\Gamma_{q}^{\bar{t}^{\prime}t}b_{t}^{\dagger}\widetilde{U}(N-1)X_{\bar{t}^{\prime}}\quad(N\geq
1),\end{array}$ (70c)

$\begin{array}[]{lll}\widetilde{U}(1)X_{\bar{t}}^{\dagger}&=&0,\\\
\widetilde{U}(N+1)X_{\bar{t}}^{\dagger}&=&\displaystyle-\frac{1}{2}\sum_{t_{1}t_{2}}\sum_{t^{\prime}_{1}}Y(\bar{t}t_{1}t_{2}t^{\prime}_{1})b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}b_{t^{\prime}_{1}}\widetilde{U}(N)\\\
&&\displaystyle-\frac{1}{2}\sum_{t_{1}t_{2}}\sum_{\bar{t}^{\prime}_{1}}Y(\bar{t}t_{1}t_{2}\bar{t}^{\prime}_{1})b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}\widetilde{U}(N-1)X_{\bar{t}^{\prime}_{1}}\quad(N\geq
1),\end{array}$ (70d)

where

$(X_{t^{\prime}})_{D}=b_{t^{\prime}},$ (71a)
$(X_{t}^{\dagger})_{D}=b_{t}^{\dagger}-\frac{1}{2}\sum_{t_{1}t_{2}}\sum_{t^{\prime}_{1}}Y(tt_{1}t_{2}t^{\prime}_{1})b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}b_{t^{\prime}_{1}},$
(71b)
$(B_{q})_{D}=\sum_{t}\sum_{t^{\prime}}\Gamma_{q}^{t^{\prime}t}b_{t}^{\dagger}b_{t^{\prime}}.$
(71c)

Eqs. (71) are the same as the boson expansions derived by DBET.
$(B_{\bar{q}})_{D}^{\dagger}=(B_{q})_{D}$ holds.

From these formulae, we obtain

$\widetilde{X_{t^{\prime}}}(N)=(X_{t^{\prime}})_{D}\hat{Z}(N+1)\qquad(N\geq
0),$ (72a)

$\begin{array}[]{lll}\widetilde{X_{t}^{\dagger}}(0)&=&(X_{t}^{\dagger})_{D}\hat{Z}(0)\\\
\widetilde{X_{t}^{\dagger}}(N)&=&(X_{t}^{\dagger})_{D}\hat{Z}(N)-\displaystyle\frac{1}{2}\sum_{t_{1}t_{2}}\sum_{\bar{t}^{\prime}_{1}}Y(tt_{1}t_{2}\bar{t}^{\prime}_{1})b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}\widetilde{X_{\bar{t}^{\prime}_{1}}}(N-1)\qquad(N\geq
1),\end{array}$ (72b)

$\begin{array}[]{lll}\widetilde{B_{q}}(0)&=&0,\\\
\widetilde{B_{q}}(N)&=&(B_{q})_{D}\hat{Z}(N)+\displaystyle\sum_{t}\sum_{\bar{t}^{\prime}}\Gamma_{q}^{\bar{t}^{\prime}t}b_{t}^{\dagger}\widetilde{X}_{\bar{t}^{\prime}}(N-1)\qquad(N\geq
1),\end{array}$ (72c)

$\begin{array}[]{lll}\widetilde{X_{\bar{t}}^{\dagger}}(0)&=&0,\\\
\widetilde{X_{\bar{t}}^{\dagger}}(N)&=&-\displaystyle\frac{1}{2}\sum_{t_{1}t_{2}}\sum_{t^{\prime}_{1}}Y(\bar{t}t_{1}t_{2}t^{\prime}_{1})b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}b_{t^{\prime}_{1}}\hat{Z}(N)\\\
&-&\displaystyle\frac{1}{2}\sum_{t_{1}t_{2}}\sum_{\bar{t}^{\prime}_{1}}Y(\bar{t}t_{1}t_{2}\bar{t}^{\prime}_{1})b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}\widetilde{X_{\bar{t}^{\prime}_{1}}}(N-1)\qquad(N\geq
1),\end{array}$ (72d)

where we use the following diffinitions:
$\widetilde{X_{\mu}}(N)=\widetilde{U}(N)X_{\mu}\widetilde{U}(N+1)^{\dagger}$,
$\widetilde{B_{q}}(N)=\widetilde{U}(N)B_{q}\widetilde{U}(N)^{\dagger}$.
$\widetilde{X_{\mu}^{\dagger}}(N)=\left(\widetilde{X_{\mu}}(N)\right)^{\dagger}$.
$\widetilde{B_{q}^{\dagger}}(N)=\left({\widetilde{B}_{q}}(N)\right)^{\dagger}$
holds.

We can obtain the boson expansion of $\hat{Z}(N)$ by using

$\hat{Z}(N)=\frac{1}{N}\sum_{t}(X_{t}^{\dagger})_{D}\hat{Z}(N-1)b_{t}-\frac{1}{2N}\sum_{t_{1}t_{2}}\sum_{t^{\prime}_{1}}\sum_{\bar{t}^{\prime}}Y(t^{\prime}_{1}t_{1}t_{2}\bar{t}^{\prime})b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}\widetilde{X_{\bar{t}^{\prime}}}(N-2)b_{t^{\prime}_{1}}\quad(N\geq
2),$ (73)

which is derived by applying

$\widetilde{U}^{\dagger}(N)=\frac{1}{N}\sum_{t}X_{t}^{\dagger}\widetilde{U}^{\dagger}(N-1)b_{t}\qquad(N\geq
1)$ (74)

obtained from Eq. (69) to Eq. (26a), expressing $\hat{Z}(N)$ as

$\hat{Z}(N)=\frac{1}{N}\sum_{t}\widetilde{X_{t}^{\dagger}}(N-1)b_{t}\quad(N\geq
1),$ (75)

and substituting Eqs. (72b) into this. $\hat{Z}(N)$ up to $N=2$ are as
follows,

$\hat{Z}(0)=\hat{1}_{B}(0),$ (76a) $\hat{Z}(1)=\hat{1}_{B}(1),$ (76b)
$\hat{Z}(2)=\hat{1}_{B}(2)\left(\hat{1}_{B}-\frac{1}{4}\sum_{t_{1}t_{2}}\sum_{t^{\prime}_{1}t^{\prime}_{2}}Y(t^{\prime}_{2}t_{1}t_{2}t^{\prime}_{1})b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}b_{t^{\prime}_{1}}b_{t^{\prime}_{2}}\right)\hat{1}_{B}(2).$
(76c)

We use the following equation for the case $N=2$,

$b_{t}\hat{1}_{B}(N)=\hat{1}_{B}(N-1)b_{t}.$ (77)

Once $\widetilde{X_{\bar{t}^{\prime}}}(N)$,
$\widetilde{X_{\bar{t}}}^{\dagger}(N)$, and $\hat{Z}(N)$ are obtained from Eq.
(72d), the Hermitian conjugate of Eq. (72d), and Eq. (73),
$\widetilde{X_{t^{\prime}}}(N)$, $\widetilde{X_{t}}^{\dagger}(N)$, and
$\widetilde{B_{q}}(N)$ are given by substituting these into Eqs. (72).

###  4.2 On the use of ideal boson state vectors

The effect of the Pauli exclusion principle is reflected generally in the
boson operators and the boson state vectors by the mapping. While, if we
restrict the types of phonon excitation modes and the number of phonon
excitations so that zero eigenvalues do not appear in the norm matrices of the
multiphonon state vectors, then

$\hat{T}_{B}=\breve{1}_{B},$ (78)

holds. In this case, the ideal boson state vectors $|N;t)$, which do not bear
the effect of the Pauli exclusion principle, become the physical state
vectors. As a result, all effects of the Pauli exclusion principle are fully
reflected in the mapped operators.

In order that the boson expansion method is practical, the phonon excitation
modes and the maximum number of excitations should be chosen so that the ideal
boson state vectors become the physical state vectors KT83 .

###  4.3 Boson expansions as a small parameter expansion

In this subsection, we obtain the norm operator and the other mapped operators
in the boson expansion being a small parameter expansion, where
$\Gamma_{q}^{\mu\mu^{\prime}}$ are regarded as of the order of magnitude
$O(\Gamma)$.

####  4.3.1 On the conditions for being a small parameter expansion and the
evaluation of the order of magnitude of each term of expansions

For realizing a small parameter expansion where the boson approximation
becomes the zeroth order approximation, $\hat{Z}\approx\breve{1}_{B}$ must
hold as the zeroth order approximation. For that purpose, it is necessary to
limit the type of mode and the number of phonon excitations in the mapping
operator so that zero eigenvalues do not appear in the norm matrices of the
multiphonon state vectors. This is the same condition for the ideal boson
state vectors to become physical. This condition is necessary but not
sufficient, however.

Denoting the matrix each element of which is $\langle
t^{\prime}_{1}t^{\prime}_{2}|t_{1}t_{2}\rangle$ as $\mathbf{Z}(N)$,
$\mathbf{Z}^{(A)}(N)$ is expressed as

$\mathbf{Z}^{(A)}(N)=\left(\begin{array}[]{@{\,}cc@{\,}}\mathbf{Z}(N)&\mathbf{W}(N)\\\
\mathbf{W}(N)^{T}&\mathbf{Z^{\prime}}(N).\end{array}\right),$ (79)

As shown in the appendix, if $\mathbf{W}(2)=\mathbf{0}(2)$, i.e.
$Y(t^{\prime}_{1}\bar{t}\mu t^{\prime}_{2})=0$, then
$\mathbf{W}(N)=\mathbf{0}(N)$ for $N\geq 3$. Hence, in this case, we obtain

$\mathbf{Z}^{(A)}(N)=\left(\begin{array}[]{@{\,}cc@{\,}}\mathbf{Z}(N)&\mathbf{0}(N)\\\
\mathbf{0}(N)^{T}&\mathbf{Z^{\prime}}(N)\end{array}\right).$ (80)

Substituting this into Eq.(50),

$\mathbf{Z}(N)^{2}=(2N-1)!!\mathbf{Z}(N),$ (81a)
$\mathbf{Z^{\prime}}(N)^{2}=(2N-1)!!\mathbf{Z^{\prime}}(N),$ (81b)

are obtained. From Eq. (81a), we obtain

$\hat{Z}(N)^{2}=(2N-1)!!\hat{Z}(N),$ (82)

from which we find

$\hat{Z}(N)=(2N-1)!!\hat{T}_{B}(N)=(2\hat{N}_{B}-1)!!\hat{T}_{B}(N),\quad\hat{N}_{B}=\sum_{t}b_{t}^{\dagger}b_{t},$
(83a) $\hat{Z}=(2\hat{N}_{B}-1)!!\hat{T}_{B}.$ (83b)

$\\{t\\}$ and $N_{max}$ are set so that no zero eigenvalue appears in
$\mathbf{Z}(N)$. It is $\mathbf{Z^{\prime}}(N)$ that has zero eigenvalues.
Therefore the eigenvalues of $\mathbf{Z}(N)$ are only $(2N-1)!!$.
$\hat{T}_{B}(N)=\hat{1}_{B}(N)$, and then $\hat{T}_{B}=\breve{1}_{B}$ hold.
Even so, $\hat{Z}\approx\breve{1}_{B}$ does not hold as the zeroth order
approximation, and the boson expansions can not be obtained as the small
parameter expansion. $\mathbf{W}(2)$ must not be a zero matrix if the small
parameter expansion holds.

We investigate the case of N=2 to establish the small parameter expansion and
an order evaluation of the terms in them. Substituting Eq. (79) into Eq. (50)
and taking $N=2$, we obtain

$\mathbf{Z}(2)^{2}+\mathbf{W}(2)\mathbf{W}(2)^{T}=3\mathbf{Z}(2),$ (84a)
$\mathbf{Z}(2)\mathbf{W}(2)+\mathbf{W}(2)\mathbf{Z^{\prime}}(2)=3\mathbf{W}(2),$
(84b)
$\mathbf{W}(2)^{T}\mathbf{W}(2)+\mathbf{Z^{\prime}}(2)^{2}=3\mathbf{Z^{\prime}}(2),$
(84c)

from which we derive

$\sum_{\mu\mu^{\prime}}Y(t^{\prime}_{1}\mu\mu^{\prime}t^{\prime}_{2})Y(t_{1}\mu\mu^{\prime}t_{2})=4((t^{\prime}_{1}t^{\prime}_{2}|t_{1}t_{2}))-2Y(t^{\prime}_{1}t_{1}t_{2}t^{\prime}_{2}),$
(85a)
$\sum_{\mu\mu^{\prime}}Y(t^{\prime}_{1}\mu\mu^{\prime}t^{\prime}_{2})Y(\bar{t}_{1}\mu\mu^{\prime}\mu_{1})+2Y(t^{\prime}_{1}\bar{t}_{1}\mu_{1}t^{\prime}_{2})=0,$
(85b)
$\sum_{\mu\mu^{\prime}}Y(\bar{t}^{\prime}_{1}\mu\mu^{\prime}\mu^{\prime}_{1})Y(\bar{t}_{1}\mu\mu^{\prime}\mu_{1})=4((\bar{t}^{\prime}_{1}\mu^{\prime}_{1}|\bar{t}_{1}\mu_{1}))-2Y(\bar{t}^{\prime}_{1}\bar{t}_{1}\mu_{1}\mu^{\prime}_{1}),$
(85c)

Since $\mathbf{Z}^{(A)}(2)$ has zero eigenvalues KT83 , these relations
include some parts where the small parameter expansion breaks down.
$Y(\mu_{1}\mu_{2}\mu_{3}\mu_{4})\sim O(\Gamma^{2})$ should hold. Therefore if
$\mu$-sums do not affect the evaluation of the order of magnitude, these
equations have discrepancies in the order of magnitude of each term. The naive
evaluation does not hold, and we must correctly evaluate the case where we
take $\mu$-sum.

We choose $\\{t\\}$ so that the small parameter expansion holds in any
situation.
$\displaystyle\sum_{tt^{\prime}}Y(t_{1}tt^{\prime}t_{2})Y(t^{\prime}_{1}tt^{\prime}t^{\prime}_{2})$
should, then, be estimated as $O(\Gamma^{4})$. To find out more about
$\bar{t}$-sum, we take up
$\displaystyle\sum_{\mu}Y(t_{1}t_{2}t_{3}\mu)\Gamma_{q}^{\mu t_{4}}$. Because
we choose $\\{t\\}$ so that
$\sum_{t}Y(t_{1}t_{2}t_{3}t)\Gamma_{q}^{tt_{4}}\sim O(\Gamma^{3})$ hold, then
we obtain

$\sum_{\mu}Y(t_{1}t_{2}t_{3}\mu)\Gamma_{q}^{\mu
t_{4}}=\sum_{\bar{t}}Y(t_{1}t_{2}t_{3}\bar{t})\Gamma_{q}^{\bar{t}t_{4}}+O(\Gamma^{3}).$
(86)

While

$\sum_{\mu}Y(t_{1}t_{2}t_{3}\mu)\Gamma_{q}^{\mu
t_{4}}=\sum_{q^{\prime}q^{\prime\prime}}\sum_{\alpha\beta\gamma}\varphi_{q}(\alpha\beta)\varphi_{q^{\prime}}(\gamma\alpha)\varphi_{q^{\prime\prime}}(\gamma\alpha)(\Gamma_{q^{\prime}}^{t_{1}t_{2}}\Gamma_{q^{\prime\prime}}^{t_{3}t_{4}}+\Gamma_{q^{\prime}}^{t_{1}t_{3}}\Gamma_{q^{\prime\prime}}^{t_{2}t_{4}}),$
(87)

holds, which indicates that the order of the right-hand side is
$O(\Gamma^{2})$. Therefore the estimation of $\bar{t}$-sum should become as

$\sum_{\bar{t}}Y(t_{1}t_{2}t_{3}\bar{t})\Gamma_{q}^{\bar{t}t_{4}}\sim
O(\Gamma^{2}).$ (88)

This indicates that if we take a single $\bar{t}$-sum, we should estimate its
magnitude by one order lower. Based on this evaluation, we evaluate

$\sum_{t\bar{t}}Y(t_{1}t\bar{t}t_{2})Y(t^{\prime}_{1}t\bar{t}t^{\prime}_{2})\sim
O(\Gamma^{3}),$ (89)

By applying these order evaluations to Eqs. (85), we obtain

$\sum_{\bar{t}\bar{t}^{\prime}}Y(\mu^{\prime}_{1}\bar{t}\bar{t}^{\prime}\mu^{\prime}_{2})Y(\mu_{1}\bar{t}\bar{t}^{\prime}\mu_{2})=4((\mu^{\prime}_{1}\mu^{\prime}_{2}|\mu_{1}\mu_{2}))-2Y(\mu^{\prime}_{1}\mu_{1}\mu_{2}\mu^{\prime}_{2})+O(\Gamma^{3}),$
(90a)
$\sum_{\bar{t}\bar{t}^{\prime}}Y(t^{\prime}_{1}\bar{t}\bar{t}^{\prime}t^{\prime}_{2})Y(\bar{t}_{1}\bar{t}\bar{t}^{\prime}\mu_{1})+2Y(t^{\prime}_{1}\bar{t}_{1}\mu_{1}t^{\prime}_{2})=O(\Gamma^{3}),$
(90b)
$\sum_{\bar{t}\bar{t}^{\prime}}Y(\bar{t}^{\prime}_{1}\bar{t}\bar{t}^{\prime}\mu^{\prime}_{1})Y(\bar{t}_{1}\bar{t}\bar{t}^{\prime}\mu_{1})=4((\bar{t}^{\prime}_{1}\mu^{\prime}_{1}|\bar{t}_{1}\mu_{1}))-2Y(\bar{t}^{\prime}_{1}\bar{t}_{1}\mu_{1}\mu^{\prime}_{1})+O(\Gamma^{3}).$
(90c)

We can identify that the parts where the double $\bar{t}$-sums are performed
across two coefficients are responsible for the failure of the small parameter
expansion. Eqs. (90) become conditions for the small parameter expansion to
hold.

####  4.3.2 Boson expansions of mapped operators as the small parameter
expansion

Here we perform the boson expansions of the mapped operators as the small
parameter expansion.

Eq. (43b) indicates that we can derive the boson expansions of $(O_{F})_{\xi}$
from those of the norm operator $\hat{Z}$ and $\widetilde{O_{F}}$. We give the
terms of the boson expansions up to the order of magnitude $O(\Gamma^{4})$.

From Eq. (72d), its Hermitian conjugate, and Eq. (73), we find the recurrence
formulae for obtaining the boson expansions of $\hat{Z}(N)$,
$\widetilde{X_{\bar{t}^{\prime}}}(N)$, and
$\widetilde{X_{\bar{t}}^{\dagger}}(N)$ up to the desired order of magnitude.
These recurrence formulae generate no parts where double $\bar{t}$-sums are
performed across two coefficients in the expansions, which makes it possible
to avoid convergence difficulty caused by them. The recurrence formulae of
$\hat{Z}(N)$ are as follows:

$\hat{Z}(N)=\sum_{k=1}^{4}\hat{Z}^{(k)}(N)+O(\Gamma^{5});\quad\hat{Z}^{(k)}(N)\sim
O(\Gamma^{k}),$ (91a)
$\hat{Z}^{(0)}(N)=\frac{1}{N}\sum_{t}b_{t}^{\dagger}\hat{Z}^{(0)}(N-1)b_{t},$
(91b) $\hat{Z}^{(1)}(N)=0,$ (91c)

$\begin{array}[]{lll}\hat{Z}^{(2)}(N)&=&\displaystyle\frac{1}{N}\sum_{t}b_{t}^{\dagger}\hat{Z}^{(2)}(N-1)b_{t}\\\
&&-\displaystyle\frac{1}{2N}\sum_{t_{1}t_{2}}\sum_{t^{\prime}_{1}t^{\prime}_{2}}Y(t^{\prime}_{2}t_{1}t_{2}t^{\prime}_{1})b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}b_{t^{\prime}_{1}}\hat{Z}^{(0)}(N-1)b_{t^{\prime}_{2}},\end{array}$
(91d)
$\begin{array}[]{lll}\hat{Z}^{(3)}(N)&=&\displaystyle\frac{1}{N}\sum_{t}b_{t}^{\dagger}\hat{Z}^{(3)}(N-1)b_{t}\\\
&&+\displaystyle\frac{1}{4N}\sum_{t_{1}t_{2}t_{3}}\sum_{t^{\prime}_{1}t^{\prime}_{2}t^{\prime}_{3}}\sum_{\bar{t}}Y(t^{\prime}_{3}t_{1}t_{2}\bar{t})Y(\bar{t}t^{\prime}_{1}t^{\prime}_{2}t_{3})b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}\hat{Z}^{(0)}(N-1)b_{t_{3}}^{\dagger}b_{t^{\prime}_{1}}b_{t^{\prime}_{2}}b_{t^{\prime}_{3}},\end{array}$
(91e)

$\begin{array}[]{c}\hat{Z}^{(4)}(N)=\displaystyle\frac{1}{N}\sum_{t}b_{t}^{\dagger}\hat{Z}^{(4)}(N-1)b_{t}-\displaystyle\frac{1}{2N}\sum_{t_{1}t_{2}}\sum_{t^{\prime}_{1}t^{\prime}_{2}}Y(t^{\prime}_{2}t_{1}t_{2}t^{\prime}_{1})b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}b_{t^{\prime}_{1}}\hat{Z}^{(2)}(N-1)b_{t^{\prime}_{2}}\\\
-\displaystyle\frac{1}{8N}\sum_{t_{1}t_{2}t_{3}t_{4}}\sum_{t^{\prime}_{1}t^{\prime}_{2}t^{\prime}_{3}t^{\prime}_{4}}\sum_{\bar{t}\bar{t}^{\prime}}Y(t^{\prime}_{4}t_{1}t_{2}\bar{t})Y(\bar{t}t^{\prime}_{2}t^{\prime}_{3}\bar{t}^{\prime})Y(\bar{t}^{\prime}t_{3}t_{4}t^{\prime}_{1})b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}b_{t_{3}}^{\dagger}b_{t_{4}}^{\dagger}b_{t^{\prime}_{1}}\hat{Z}^{(0)}(N-1)b_{t^{\prime}_{2}}b_{t^{\prime}_{3}}b_{t^{\prime}_{4}}.\end{array}$
(91f)

The solution of Eq,(91b) is easily obtained as

$\hat{Z}^{(0)}(N)=\frac{1}{N!}\sum_{t_{1}t_{2}\cdots
t_{N}}b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}\cdots
b_{t_{N}}^{\dagger}\hat{Z}(0)b_{t_{1}}b_{t_{2}}\cdots
b_{t_{N}}=\hat{1}_{B}(N).$ (92)

Substituting it into Eq. (91d) and using Eq. (77), we obtain

$\begin{array}[]{lll}\hat{Z}^{(2)}(N)&=&\displaystyle\frac{1}{N}\sum_{t}b_{t}^{\dagger}\hat{Z}^{(2)}(N-1)b_{t}\\\
&&-\displaystyle\frac{1}{2N}\sum_{t_{1}t_{2}}\sum_{t^{\prime}_{1}t^{\prime}_{2}}Y(t^{\prime}_{2}t_{1}t_{2}t^{\prime}_{1})b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}\hat{1}_{B}(N-2)b_{t^{\prime}_{1}}b_{t^{\prime}_{2}}.\end{array}$
(93a) For finding the solution, assuming it as
$\hat{Z}^{(2)}(N)=y^{(2)}(N)\sum_{t_{1}t_{2}}\sum_{t^{\prime}_{1}t^{\prime}_{2}}Y(t^{\prime}_{2}t_{1}t_{2}t^{\prime}_{1})b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}\hat{1}_{B}(N-2)b_{t^{\prime}_{1}}b_{t^{\prime}_{2}},$
(93b) substituting this into the recurrence formula, and using
$\sum_{t}b_{t}^{\dagger}\hat{1}_{B}(N)b_{t}=(N+1)\hat{1}_{B}(N+1),$ (93c) we
find $y^{(2)}(N)=\frac{N-2}{N}y^{(2)}(N-1)-\frac{1}{2N}.$ (93d)
$y^{(2)}(N)=-1/4$ is the solution, and we obtain
$\hat{Z}^{(2)}(N)=-\frac{1}{4}\sum_{t_{1}t_{2}}\sum_{t^{\prime}_{1}t^{\prime}_{2}}Y(t^{\prime}_{2}t_{1}t_{2}t^{\prime}_{1})b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}\hat{1}_{B}(N-2)b_{t^{\prime}_{1}}b_{t^{\prime}_{2}}.$
(93e)

Following the same procedure in order, we can obtain the solution of the
recurrence formulae for each order of magnitude. Organizing the solutions
obtained in this way using Eq. (77), we finally obtain

$\hat{Z}(N)=\mathcal{\hat{Z}}\hat{1}_{B}(N)=\hat{1}_{B}(N)\mathcal{\hat{Z}}=\hat{1}_{B}(N)\mathcal{\hat{Z}}\hat{1}_{B}(N),$
(94a)
$\mathcal{\hat{Z}}=\mathcal{\hat{Z}}^{(0)}+\mathcal{\hat{Z}}^{(2)}+\mathcal{\hat{Z}}^{(3)}+\mathcal{\hat{Z}}^{(4)}+O(\Gamma^{5}),$
(94b) $\mathcal{\hat{Z}}^{(0)}=\hat{1}_{B},$ (94c)
$\mathcal{\hat{Z}}^{(2)}=-\frac{1}{4}\sum_{t_{1}t_{2}}\sum_{t^{\prime}_{1}t^{\prime}_{2}}Y(t^{\prime}_{2}t_{1}t_{2}t^{\prime}_{1})b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}b_{t^{\prime}_{1}}b_{t^{\prime}_{2}},$
(94d)

$\mathcal{\hat{Z}}^{(3)}=\frac{1}{12}\sum_{t_{1}t_{2}t_{3}}\sum_{t^{\prime}_{1}t^{\prime}_{2}t^{\prime}_{3}}\sum_{\bar{t}}Y(t^{\prime}_{3}t_{1}t_{2}\bar{t})Y(\bar{t}t^{\prime}_{1}t^{\prime}_{2}t_{3})b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}b_{t_{3}}^{\dagger}b_{t^{\prime}_{1}}b_{t^{\prime}_{2}}b_{t^{\prime}_{3}},$
(94e)

$\begin{array}[]{lll}\mathcal{\hat{Z}}^{(4)}&=&\mathcal{\hat{Z}}_{in}^{(4)}+\mathcal{\hat{Z}}_{out}^{(4)}\\\
&\mathcal{\hat{Z}}_{in}^{(4)}&=\displaystyle\frac{1}{12}\sum_{t_{1}t_{2}t_{3}}\sum_{t^{\prime}_{1}t^{\prime}_{2}t^{\prime}_{3}}\sum_{t}Y(t^{\prime}_{1}t_{1}t_{2}t)Y(tt^{\prime}_{2}t^{\prime}_{3}t_{3})b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}b_{t_{3}}^{\dagger}b_{t^{\prime}_{1}}b_{t^{\prime}_{2}}b_{t^{\prime}_{3}}\\\
&&+\displaystyle\frac{1}{32}\sum_{t_{1}t_{2}t_{3}t_{4}}\sum_{t^{\prime}_{1}t^{\prime}_{2}t^{\prime}_{3}t^{\prime}_{4}}Y(t^{\prime}_{2}t_{1}t_{2}t^{\prime}_{1})Y(t^{\prime}_{4}t_{3}t_{4}t^{\prime}_{3})b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}b_{t_{3}}^{\dagger}b_{t_{4}}^{\dagger}b_{t^{\prime}_{1}}b_{t^{\prime}_{2}}b_{t^{\prime}_{3}}b_{t^{\prime}_{4}},\\\
&\mathcal{\hat{Z}}_{out}^{(4)}&=-\displaystyle\frac{1}{32}\sum_{t_{1}t_{2}t_{3}t_{4}}\sum_{t^{\prime}_{1}t^{\prime}_{2}t^{\prime}_{3}t^{\prime}_{4}}\sum_{\bar{t}\bar{t}^{\prime}}Y(t^{\prime}_{4}t_{1}t_{2}\bar{t})Y(\bar{t}t^{\prime}_{2}t^{\prime}_{3}\bar{t}^{\prime})Y(\bar{t}^{\prime}t_{3}t_{4}t^{\prime}_{1})b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}b_{t_{3}}^{\dagger}b_{t_{4}}^{\dagger}b_{t^{\prime}_{1}}b_{t^{\prime}_{2}}b_{t^{\prime}_{3}}b_{t^{\prime}_{4}}.\end{array}$
(94f)

From these results, we can easily find the norm operator as

$\begin{array}[]{lll}\hat{Z}&=&\mathcal{\hat{Z}}\breve{1}_{B}=\breve{1}_{B}\mathcal{\hat{Z}}=\breve{1}_{B}\mathcal{\hat{Z}}\breve{1}_{B},\\\
&\mathcal{\hat{Z}}&=\hat{1}_{B}+\mathcal{\hat{Y}},\\\
&&\mathcal{\hat{Y}}=\mathcal{\hat{Y}}_{in}+\mathcal{\hat{Y}}_{out},\\\
&&\quad\mathcal{\hat{Y}}_{in}=\mathcal{\hat{Z}}^{(2)}+\mathcal{\hat{Z}}_{in}^{(4)}+O(\Gamma^{5}),\\\
&&\quad\mathcal{\hat{Y}}_{out}=\mathcal{\hat{Z}}^{(3)}+\mathcal{\hat{Z}}_{out}^{(4)}+O(\Gamma^{5}).\end{array}$
(95)

The $\xi$-th power of $\hat{Z}$ becomes

$\begin{array}[]{l}\hat{Z}^{\xi}=\mathcal{\hat{Z}}^{\xi}\breve{1}_{B}=\breve{1}_{B}\mathcal{\hat{Z}}^{\xi},\\\
\quad\mathcal{\hat{Z}}^{\xi}=\hat{1}_{B}+\xi\mathcal{\hat{Y}}+\displaystyle\frac{1}{2}\xi(\xi-1)\mathcal{\hat{Y}}^{2}+O(\Gamma^{6}).\end{array}$
(96)

Once $\hat{Z}(N)$ is known, we can obtain $\widetilde{X_{\bar{t}}}(N)$ from
the following recurrence formula derived from Eqs. (72d),

$\begin{array}[]{lll}\widetilde{X_{\bar{t}^{\prime}}}(N)&=&-\displaystyle\frac{1}{2}\sum_{t_{1}}\sum_{t^{\prime}_{1}t^{\prime}_{2}}Y(\bar{t}^{\prime}t^{\prime}_{1}t^{\prime}_{2}t_{1})\hat{Z}(N)b_{t_{1}}^{\dagger}b_{t^{\prime}_{1}}b_{t^{\prime}_{2}}\\\
&&+\displaystyle\frac{1}{4}\sum_{t_{1}t_{2}}\sum_{t^{\prime}_{1}t^{\prime}_{2}t^{\prime}_{3}}\sum_{\bar{t}}Y(\bar{t}^{\prime}t^{\prime}_{2}t^{\prime}_{3}\bar{t})Y(\bar{t}t_{1}t_{2}t^{\prime}_{1})b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}b_{t^{\prime}_{1}}\hat{Z}(N-1)b_{t^{\prime}_{2}}b_{t^{\prime}_{3}}\\\
&&+\displaystyle\frac{1}{4}\sum_{t_{1}t_{2}}\sum_{t^{\prime}_{1}t^{\prime}_{2}}\sum_{\bar{t}}\sum_{\bar{t}^{\prime}_{1}}Y(\bar{t}^{\prime}t^{\prime}_{1}t^{\prime}_{2}\bar{t})Y(\bar{t}t_{1}t_{2}\bar{t}^{\prime}_{1})b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}\widetilde{X_{\bar{t}^{\prime}_{1}}}(N-2)b_{t^{\prime}_{1}}b_{t^{\prime}_{2}},\end{array}$
(97)

and find the solutions as

$\widetilde{X_{\bar{t}^{\prime}}}(N)=(X_{\bar{t}^{\prime}})_{L}\mathcal{\hat{Z}}\hat{1}_{B}(N+1)=\hat{1}_{B}(N)(X_{\bar{t}^{\prime}})_{L}\mathcal{\hat{Z}},$
(98a)
$(X_{\bar{t}^{\prime}})_{L}=(X_{\bar{t}^{\prime}})_{L}^{(2)}+(X_{\bar{t}^{\prime}})_{L}^{(3)}+(X_{\bar{t}^{\prime}})_{L}^{(4)}+O(\Gamma^{5}),$
(98b)
$\begin{array}[]{lll}(X_{\bar{t}^{\prime}})_{L}^{(2)}&=&-\displaystyle\frac{1}{2}\sum_{t_{1}}\sum_{t^{\prime}_{1}t^{\prime}_{2}}Y(\bar{t}^{\prime}t^{\prime}_{1}t^{\prime}_{2}t_{1})b_{t_{1}}^{\dagger}b_{t^{\prime}_{1}}b_{t^{\prime}_{2}},\\\
(X_{\bar{t}^{\prime}})_{L}^{(3)}&=&\displaystyle\frac{1}{4}\sum_{t_{1}t_{2}}\sum_{t^{\prime}_{1}t^{\prime}_{2}t^{\prime}_{3}}\sum_{\bar{t}}Y(\bar{t}^{\prime}t^{\prime}_{2}t^{\prime}_{3}\bar{t})Y(\bar{t}t_{1}t_{2}t^{\prime}_{1})b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}b_{t^{\prime}_{1}}b_{t^{\prime}_{2}}b_{t^{\prime}_{3}},\\\
(X_{\bar{t}^{\prime}})_{L}^{(4)}&=&[\mathcal{\hat{Z}}^{(2)},(X_{\bar{t}^{\prime}})_{L}^{(2)}]\\\
&&-\displaystyle\frac{1}{8}\sum_{t_{1}t_{2}t_{3}}\sum_{t^{\prime}_{1}t^{\prime}_{2}t^{\prime}_{3}t^{\prime}_{4}}\sum_{\bar{t}\bar{t}^{\prime\prime}}Y(\bar{t}^{\prime}t^{\prime}_{1}t^{\prime}_{2}\bar{t})Y(\bar{t}t_{1}t_{2}\bar{t}^{\prime\prime})Y(\bar{t}^{\prime\prime}t^{\prime}_{3}t^{\prime}_{4}t_{3})b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}b_{t_{3}}^{\dagger}b_{t^{\prime}_{1}}b_{t^{\prime}_{2}}b_{t^{\prime}_{3}}b_{t^{\prime}_{4}},\end{array}$
(98c)
$\begin{array}[]{l}[\mathcal{\hat{Z}}^{(2)},(X_{\bar{t}^{\prime}})_{L}^{(2)}]=-\displaystyle\frac{1}{4}\sum_{t_{1}}\sum_{t^{\prime}_{1}t^{\prime}_{2}}\sum_{tt^{\prime}}Y(\bar{t}^{\prime}tt^{\prime}t_{1})Y(t^{\prime}_{1}tt^{\prime}t^{\prime}_{2})b_{t_{1}}^{\dagger}b_{t^{\prime}_{1}}b_{t^{\prime}_{2}}\\\
\qquad-\displaystyle\frac{1}{4}\sum_{t_{1}t_{2}}\sum_{t^{\prime}_{1}t^{\prime}_{2}t^{\prime}_{3}}\sum_{t}\left\\{2Y(t_{1}t^{\prime}_{1}t^{\prime}_{2}t)Y(t\bar{t}^{\prime}t_{2}t^{\prime}_{3})-Y(\bar{t}^{\prime}t^{\prime}_{1}t^{\prime}_{2}t)Y(tt_{1}t_{2}t^{\prime}_{3})\right\\}b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}b_{t^{\prime}_{1}}b_{t^{\prime}_{2}}b_{t^{\prime}_{3}}.\end{array}$
(98d)

From these, we obtain

$\widetilde{X_{\bar{t}^{\prime}}}=\sum_{N=0}^{N_{max}-1}\widetilde{X_{\bar{t}^{\prime}}}(N)=(X_{\bar{t}^{\prime}})_{L}\mathcal{\hat{Z}}\breve{1}_{B}=\breve{1}_{B}^{(-1)}(X_{\bar{t}^{\prime}})_{L}\mathcal{\hat{Z}}=\breve{1}_{B}(X_{\bar{t}^{\prime}})_{L}\mathcal{\hat{Z}}\breve{1}_{B},$
(99)

where

$\breve{1}_{B}^{(\Delta N)}=\sum_{N=0}^{N_{max}+\Delta N}\hat{1}_{B}(N).$
(100)

$\breve{1}_{B}^{(0)}=\breve{1}_{B}$ and
$\breve{1}_{B}\breve{1}_{B}^{(-1)}=\breve{1}_{B}^{(-1)}\breve{1}_{B}=\breve{1}_{B}^{(-1)}$
hold. Organizing the Hermitian conjugate of Eq. (98a), we find

$\begin{array}[]{lll}\widetilde{X_{\bar{t}}^{\dagger}}(N)&=&((X_{\bar{t}})_{L}\mathcal{\hat{Z}}\hat{1}_{B}(N+1))^{\dagger}=\hat{1}_{B}(N+1)\mathcal{\hat{Z}}(X_{\bar{t}})_{L}^{\dagger}\\\
&=&\mathcal{\hat{Z}}(X_{\bar{t}})_{L}^{\dagger}\hat{1}_{B}(N)=\left\\{\mathcal{\hat{Z}}(X_{\bar{t}})_{L}^{\dagger}\mathcal{\hat{Z}}^{-1}\right\\}\mathcal{\hat{Z}}\hat{1}_{B}(N)\\\
&=&(X_{\bar{t}}^{\dagger})_{L}\mathcal{\hat{Z}}\hat{1}_{B}(N),\end{array}$
(101a) where
$(X_{\bar{t}}^{\dagger})_{L}=\mathcal{\hat{Z}}(X_{\bar{t}})_{L}^{\dagger}\mathcal{\hat{Z}}^{-1},$
(101b)
$(X_{\bar{t}}^{\dagger})_{L}=(X_{\bar{t}}^{\dagger})_{L}^{(2)}+(X_{\bar{t}}^{\dagger})_{L}^{(3)}+(X_{\bar{t}}^{\dagger})_{L}^{(4)}+O(\Gamma^{5}),$
(101c)
$\begin{array}[]{lll}(X_{\bar{t}}^{\dagger})_{L}^{(2)}&=&((X_{\bar{t}})_{L}^{(2)})^{\dagger},\quad(X_{\bar{t}}^{\dagger})_{L}^{(3)}=((X_{\bar{t}})_{L}^{(3)})^{\dagger},\\\
(X_{\bar{t}}^{\dagger})_{L}^{(4)}&=&((X_{\bar{t}})_{L}^{(4)})^{\dagger}+[\mathcal{\hat{Z}}^{(2)},((X_{\bar{t}})_{L}^{(2)})^{\dagger}]\\\
&=&-\displaystyle\frac{1}{8}\sum_{t_{1}t_{2}t_{3}t_{4}}\sum_{t^{\prime}_{1}t^{\prime}_{2}t^{\prime}_{3}}\sum_{\bar{t}^{\prime}\bar{t}^{\prime\prime}}Y(\bar{t}t_{1}t_{2}\bar{t}^{\prime})Y(\bar{t}^{\prime}t^{\prime}_{1}t^{\prime}_{2}\bar{t}^{\prime\prime})Y(\bar{t}^{\prime\prime}t_{3}t_{4}t^{\prime}_{3})b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}b_{t_{3}}^{\dagger}b_{t_{4}}^{\dagger}b_{t^{\prime}_{1}}b_{t^{\prime}_{2}}b_{t^{\prime}_{3}},\end{array}$
(101d)

and obtain

$\widetilde{X_{\bar{t}}^{\dagger}}=\sum_{N=0}^{N_{max}-1}\widetilde{X_{\bar{t}}^{\dagger}}(N)=(X_{\bar{t}}^{\dagger})_{L}\mathcal{\hat{Z}}\breve{1}_{B}^{(-1)}=\breve{1}_{B}(X_{\bar{t}}^{\dagger})_{L}\mathcal{\hat{Z}}=\breve{1}_{B}(X_{\bar{t}}^{\dagger})_{L}\mathcal{\hat{Z}}\breve{1}_{B}.$
(102)

In this way, we can obtain $(X_{\bar{t}^{\prime}})_{L}$ and
$(X_{\bar{t}}^{\dagger})_{L}$ as infinite expansions.

Dealing with the terms up to $O(\Gamma^{4})$, we have found that
$\hat{Z}(N)=\mathcal{\hat{Z}}\hat{1}_{B}(N)=\hat{1}_{B}(N)\mathcal{\hat{Z}}$,
$\widetilde{X_{\bar{t}^{\prime}}}(N)=\widetilde{X_{\bar{t}^{\prime}}}\hat{1}_{B}(N+1)=\hat{1}_{B}(N)\widetilde{X_{\bar{t}^{\prime}}}$,
and
$\widetilde{X_{\bar{t}}^{\dagger}}(N)=\hat{1}_{B}(N+1)\widetilde{X_{\bar{t}}^{\dagger}}=\widetilde{X_{\bar{t}}^{\dagger}}\hat{1}_{B}(N)$.
Oppositely, assuming that these hold for any $N$, and substituting them into
Eq. (72d) and Eq. (73), we can find the relational expressions for
$\mathcal{\hat{Z}}$, $\widetilde{X_{\bar{t}^{\prime}}}$, and
$\widetilde{X_{\bar{t}}^{\dagger}}$, and solve these for each order of
magnitude, then we obtain the same results. This result suggests that the $N$
dependency of these operators found up to $O(\Gamma^{4})$ generally holds.

Applying the above results to Eqs. (72) and summing up $N$, we obtain

$\begin{array}[]{lll}\widetilde{X_{t^{\prime}}}&=&(X_{t^{\prime}})_{L}\mathcal{\hat{Z}}\breve{1}_{B}=\breve{1}_{B}^{(-1)}(X_{t^{\prime}})_{L}\mathcal{\hat{Z}}=\breve{1}_{B}(X_{t^{\prime}})_{L}\mathcal{\hat{Z}}\breve{1}_{B},\\\
(X_{t^{\prime}})_{L}&=&(X_{t^{\prime}})_{D},\end{array}$ (103a)
$\begin{array}[]{lll}\widetilde{X_{t}^{\dagger}}&=&\breve{1}_{B}(X_{t}^{\dagger})_{L}\mathcal{\hat{Z}}=(X_{t}^{\dagger})_{L}\mathcal{\hat{Z}}\breve{1}_{B}^{(-1)}=\breve{1}_{B}(X_{t}^{\dagger})_{L}\mathcal{\hat{Z}}\breve{1}_{B},\\\
(X_{t}^{\dagger})_{L}&=&(X_{t}^{\dagger})_{D}+(X_{t}^{\dagger})_{out}\\\
(X_{t}^{\dagger})_{out}&=&-\displaystyle\frac{1}{2}\sum_{t_{1}t_{2}}\sum_{\bar{t}^{\prime}_{1}}Y(tt_{1}t_{2}\bar{t}^{\prime}_{1})b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}(X_{\bar{t}^{\prime}_{1}})_{L},\end{array}$
(103b)

$\begin{array}[]{lll}\widetilde{B_{q}}&=&(B_{q})_{L}\mathcal{\hat{Z}}\breve{1}_{B}=\breve{1}_{B}(B_{q})_{L}\mathcal{\hat{Z}}=\breve{1}_{B}(B_{q})_{L}\mathcal{\hat{Z}}\breve{1}_{B},\\\
(B_{q})_{L}&=&(B_{q})_{D}+(B_{q})_{out},\\\
(B_{q})_{out}&=&\displaystyle\sum_{t}\sum_{\bar{t}^{\prime}}\Gamma_{q}^{\bar{t}^{\prime}t}b_{t}^{\dagger}(X_{\bar{t}^{\prime}})_{L}.\end{array}$
(103c)

While, from $B_{q}=B_{\bar{q}}^{\dagger}$,
$\widetilde{B_{q}}=\widetilde{B_{\bar{q}}}^{\dagger}$ and we find another
expression for $\widetilde{B_{q}}$ as

$\begin{array}[]{lll}\widetilde{B_{q}}&=&\breve{1}_{B}\mathcal{\hat{Z}}(B_{\bar{q}})_{L}^{\dagger}=\mathcal{\hat{Z}}(B_{\bar{q}})_{L}^{\dagger}\breve{1}_{B}=\breve{1}_{B}\mathcal{\hat{Z}}(B_{\bar{q}})_{L}^{\dagger}\breve{1}_{B},\\\
(B_{\bar{q}})_{L}^{\dagger}&=&(B_{q})_{D}+(B_{\bar{q}})_{out}^{\dagger},\\\
(B_{\bar{q}})_{out}^{\dagger}&=&\displaystyle\sum_{t}\sum_{\bar{t}}\Gamma_{q}^{t^{\prime}\bar{t}}(X_{\bar{t}})_{L}^{\dagger}b_{t^{\prime}},\end{array}$
(104)

where we use $(B_{q})_{D}=(B_{\bar{q}})_{D}^{\dagger}$. Using two types of
expressions for $\widetilde{B_{q}}$, we obtain

$\breve{1}_{B}[(B_{q})_{D},\mathcal{\hat{Z}}]\breve{1}_{B}=\breve{1}_{B}\\{\mathcal{\hat{Z}}(B_{\bar{q}})_{out}^{\dagger}-(B_{q})_{out}\mathcal{\hat{Z}}\\}\breve{1}_{B}.$
(105)

From Eq. (24) and Eq. (96), we can express the mapping operator as

$U_{\xi}=\mathcal{\hat{Z}}^{\xi-\frac{1}{2}}\widetilde{U},$ (106)

and Eq. (43) becomes as follows:

$|\psi^{\prime})_{\xi}=\mathcal{\hat{Z}}^{\xi-\frac{1}{2}}\widetilde{|\psi^{\prime})},\qquad{}_{-\xi}(\psi|=\widetilde{(\psi|}\mathcal{\hat{Z}}^{-\xi-\frac{1}{2}},$
(107a)
$(O_{F})_{\xi}=\mathcal{\hat{Z}}^{\xi-\frac{1}{2}}\widetilde{O_{F}}\mathcal{\hat{Z}}^{-\xi-\frac{1}{2}},$
(107b)

If $O_{F}$ is a phonon creation, a phonon annihilation, or a scattering
operator,
$\widetilde{O_{F}}=\breve{1}_{B}(O_{F})_{L}\mathcal{\hat{Z}}\breve{1}_{B}$
holds. Therefore the mapped $O_{F}$ can be expressed as

$(O_{F})_{\xi}=\breve{1}_{B}(O_{F})_{B(\xi)}\breve{1}_{B};\quad(O_{F})_{B(\xi)}=\mathcal{\hat{Z}}^{\xi-\frac{1}{2}}(O_{F})_{L}\mathcal{\hat{Z}}^{-\xi+\frac{1}{2}},$
(108)

and

${}_{-\xi}(\psi|(O_{F})_{\xi}|\psi^{\prime})_{\xi}={}_{-\xi}(\psi|(O_{F})_{B(\xi)}|\psi^{\prime})_{\xi}$
(109)

holds. Therefore we can regard $(O_{F})_{\xi}$ as $(O_{F})_{B(\xi)}$ in the
physical subspace. The boson expansions of $(O_{F})_{B(\xi)}$ become infinite
expansions for an arbitrary $\xi$ because those of $(O_{F})_{L}$ become
infinite expansions.

For $\xi\neq 0$, the boson expansions become of the non-Hermitian type. In the
case of $\xi=\frac{1}{2}$, $(O_{F})_{\xi(\frac{1}{2})}=(O_{F})_{L}$ holds.

For $\xi=0$, the boson expansions become the Hermitian type and can be derived
using

$\begin{array}[]{lll}(O_{F})_{B(0)}&=&\mathcal{\hat{Z}}^{-\frac{1}{2}}(O_{F})_{L}\mathcal{\hat{Z}}^{\frac{1}{2}}\\\
&=&(O_{F})_{L}+\frac{1}{2}[(O_{F})_{L},\mathcal{\hat{Y}}]-\frac{3}{8}\mathcal{\hat{Y}}[(O_{F})_{L},\mathcal{\hat{Y}}]-\frac{1}{8}[(O_{F})_{L},\mathcal{\hat{Y}}]\mathcal{\hat{Y}}+O(\Gamma^{6}).\end{array}$
(110)

The boson expansions of the phonon creation and annihilation operators and the
scattering operators are as follows:

$(X_{t^{\prime}})_{B(0)}=(X_{t^{\prime}})_{B(0)in}+(X_{t^{\prime}})_{B(0)out},$
(111a)
$(X_{t^{\prime}})_{B(0)in}=b_{t^{\prime}}+(X_{t^{\prime}})_{B(0)in}^{(2)}+(X_{t^{\prime}})_{B(0)in}^{(4)}+O(\Gamma^{5}),$
(111b)
$(X_{t^{\prime}})_{B(0)in}^{(2)}=-\frac{1}{4}\sum_{t_{1}}\sum_{t^{\prime}_{1}t^{\prime}_{2}}Y(t^{\prime}t^{\prime}_{1}t^{\prime}_{2}t_{1})b_{t_{1}}^{\dagger}b_{t^{\prime}_{1}}b_{t^{\prime}_{2}},$
(111c)
$\begin{array}[]{r}(X_{t^{\prime}})_{B(0)in}^{(4)}=-\displaystyle\frac{1}{32}\sum_{t_{1}}\sum_{t^{\prime}_{1}t^{\prime}_{2}}\sum_{tt^{\prime\prime}}Y(t^{\prime}tt^{\prime\prime}t^{\prime}_{1})Y(t^{\prime}_{2}tt^{\prime\prime}t^{\prime}_{1})b_{t_{1}}^{\dagger}b_{t^{\prime}_{1}}b_{t^{\prime}_{2}}\\\
+\displaystyle\frac{1}{96}\sum_{t_{1}t_{2}}\sum_{t^{\prime}_{1}t^{\prime}_{2}t^{\prime}_{3}}\sum_{t}\\{2Y(t^{\prime}_{1}t_{1}t^{\prime}t)Y(tt^{\prime}_{2}t^{\prime}_{3}t_{2})\\\
-5Y(t^{\prime}t^{\prime}_{1}t^{\prime}_{2}t)Y(tt_{1}t_{2}t^{\prime}_{3})\\}b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}b_{t^{\prime}_{1}}b_{t^{\prime}_{2}}b_{t^{\prime}_{3}},\end{array}$
(111d)
$(X_{t^{\prime}})_{B(0)out}=(X_{t^{\prime}})_{B(0)out}^{(3)}+(X_{t^{\prime}})_{B(0)out}^{(4)}+O(\Gamma^{5})$
(111e)
$\begin{array}[]{r}(X_{t^{\prime}})_{B(0)out}^{(3)}=\displaystyle\frac{1}{24}\sum_{t_{1}t_{2}}\sum_{t^{\prime}_{1}t^{\prime}_{2}t^{\prime}_{3}}\sum_{\bar{t}}\\{2Y(t^{\prime}_{1}t_{1}t^{\prime}\bar{t})Y(\bar{t}t^{\prime}_{2}t^{\prime}_{3}t_{2})\\\
+Y(t^{\prime}_{1}t_{1}t_{2}\bar{t})Y(\bar{t}t^{\prime}_{2}t^{\prime}_{3}t^{\prime})\\}b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}b_{t^{\prime}_{1}}b_{t^{\prime}_{2}}b_{t^{\prime}_{3}},\end{array}$
(111f)
$\begin{array}[]{r}(X_{t^{\prime}})_{B(0)out}^{(4)}=-\displaystyle\frac{1}{16}\sum_{t_{1}t_{2}t_{3}}\sum_{t^{\prime}_{1}t^{\prime}_{2}t^{\prime}_{3}t^{\prime}_{4}}\sum_{\bar{t}\bar{t}^{\prime}}Y(t^{\prime}_{1}t^{\prime}t_{1}\bar{t})Y(\bar{t}t^{\prime}_{2}t^{\prime}_{3}\bar{t}^{\prime})Y(\bar{t}^{\prime}t_{2}t_{3}t^{\prime}_{4})\\\
b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}b_{t_{3}}^{\dagger}b_{t^{\prime}_{1}}b_{t^{\prime}_{2}}b_{t^{\prime}_{3}}b_{t^{\prime}_{4}}.\end{array}$
(111g)
$(X_{\bar{t}^{\prime}})_{B(0)}=(X_{\bar{t}^{\prime}})_{B(0)}^{(2)}+(X_{\bar{t}^{\prime}})_{B(0)}^{(3)}+(X_{\bar{t}^{\prime}})_{B(0)}^{(4)}+O(\Gamma^{5}),$
(112a)
$(X_{\bar{t}^{\prime}})_{B(0)}^{(2)}=(X_{\bar{t}^{\prime}})_{L}^{(2)},\quad(X_{\bar{t}^{\prime}})_{B(0)}^{(3)}=(X_{\bar{t}^{\prime}})_{L}^{(3)},$
(112b)
$(X_{\bar{t}^{\prime}})_{B(0)}^{(4)}=(X_{\bar{t}^{\prime}})_{L}^{(4)}-\frac{1}{2}[\mathcal{\hat{Z}}^{(2),}(X_{\bar{t}^{\prime}})_{L}^{(2)}].$
(112c)
$\begin{array}[]{lll}(B_{q})_{B(0)}&=&(B_{q})_{L}+\frac{1}{2}\mathcal{\hat{Z}}\\{(B_{\bar{q}})_{out}{}^{\dagger}-(B_{q})_{out}\\}+O(\Gamma^{5}),\\\
&=&(B_{q})_{B(0)in}+(B_{q})_{B(0)out},\end{array}$ (113a)
$(B_{q})_{B(0)in}=(B_{q})_{D},$ (113b)
$(B_{q})_{B(0)out}=(B_{q})_{B(0)out}^{(2)}+(B_{q})_{B(0)out}^{(3)}+(B_{q})_{B(0)out}^{(4)}+O(\Gamma^{5}),$
(113c)
$(B_{q})_{B(0)out}^{(k)}=\frac{1}{2}\\{(B_{q})_{out}^{(k)}+(B_{\bar{q}})_{out}^{(k)}{}^{\dagger}\\}\quad(k=2,3),$
(113d)
$(B_{q})_{B(0)out}^{(4)}=\frac{1}{2}\\{(B_{q})_{out}^{(4)}+(B_{\bar{q}})_{out}^{(4)}{}^{\dagger}\\}+\frac{1}{2}\mathcal{\hat{Z}}^{(2)}\\{(B_{\bar{q}})_{out}^{(2)}{}^{\dagger}-(B_{q})_{out}^{(2)}\\},$
(113e)
$\begin{array}[]{r}\frac{1}{2}\mathcal{\hat{Z}}^{(2)}\\{(B_{\bar{q}})_{out}^{(2)}{}^{\dagger}-(B_{q})_{out}^{(2)}\\}=\displaystyle\frac{1}{8}\sum_{t_{1}t_{2}}\sum_{t^{\prime}_{1}t^{\prime}_{2}}\sum_{tt^{\prime}}\sum_{\bar{t}}\\{\Gamma_{q}^{t^{\prime}_{1}\bar{t}}Y(\bar{t}tt^{\prime}t^{\prime}_{2})-\Gamma_{q}^{\bar{t}t}Y(\bar{t}t^{\prime}_{1}t^{\prime}_{2}t^{\prime})\\}Y(tt_{1}t_{2}t^{\prime})\\\
b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}b_{t^{\prime}_{1}}b_{t^{\prime}_{2}}\\\
+\displaystyle\frac{1}{8}\sum_{t_{1}t_{2}t_{3}}\sum_{t^{\prime}_{1}t^{\prime}_{2}t^{\prime}_{3}}\sum_{t}\sum_{\bar{t}}\\{2\Gamma_{q}^{t^{\prime}_{1}\bar{t}}Y(\bar{t}t_{1}tt^{\prime}_{2})-\Gamma_{q}^{\bar{t}t}Y(\bar{t}t^{\prime}_{1}t^{\prime}_{2}t_{1})-\Gamma_{q}^{\bar{t}t_{1}}Y(\bar{t}t^{\prime}_{1}t^{\prime}_{2}t)\\}\\\
Y(tt_{1}t_{2}t^{\prime}_{3})b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}b_{t_{3}}^{\dagger}b_{t^{\prime}_{1}}b_{t^{\prime}_{2}}b_{t^{\prime}_{3}}\\\
+\displaystyle\frac{1}{16}\sum_{t_{1}t_{2}t_{3}t_{4}}\sum_{t^{\prime}_{1}t^{\prime}_{2}t^{\prime}_{3}t^{\prime}_{4}}\sum_{\bar{t}}\\{\Gamma_{q}^{t^{\prime}_{1}\bar{t}}Y(\bar{t}t_{1}t_{2}t^{\prime}_{2})-\Gamma_{q}^{\bar{t}t_{1}}Y(\bar{t}t^{\prime}_{1}t^{\prime}_{2}t_{2})\\}Y(t^{\prime}_{4}t_{3}t_{4}t^{\prime}_{3})\\\
b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}b_{t_{3}}^{\dagger}b_{t_{4}}^{\dagger}b_{t^{\prime}_{1}}b_{t^{\prime}_{2}}b_{t^{\prime}_{3}}b_{t^{\prime}_{4}}.\end{array}$
(113f)

Here, we use Eq. (105) to find $(B_{q})_{B(0)}$. From Eq. (103c),
$(B_{q})_{out}^{(k)}=\displaystyle\sum_{t}\sum_{\bar{t}^{\prime}}\Gamma_{q}^{\bar{t}^{\prime}t}b_{t}^{\dagger}(X_{\bar{t}^{\prime}})_{L}^{(k)}$.

Finally, we deal with the product of operators. Let $O_{F}$ and
$O^{\prime}_{F}$ be the phonon creation, annihilation operators, or scattering
operators, respectively, we can derive the boson expansions of their product
as

$(O_{F}O^{\prime}_{F})_{B(\xi)}=\mathcal{\hat{Z}}^{\xi-\frac{1}{2}}\widetilde{O_{F}O^{\prime}_{F}}\mathcal{\hat{Z}}^{-\xi-\frac{1}{2}}.$
(114)

If
$\widetilde{O_{F}O^{\prime}_{F}}=\breve{1}_{B}(O_{F})_{L}\breve{1}_{B}(O^{\prime}_{F})_{L}\mathcal{\hat{Z}}\breve{1}_{B}$
holds, we obtain

$(O_{F}O^{\prime}_{F})_{B(\xi)}=(O_{F})_{B(\xi)}(O^{\prime}_{F})_{B(\xi)},$
(115)

and if Eq. (40) holds,

$(O_{F}O^{\prime}_{F})_{B(\xi)}\approx(O_{F})_{B(\xi)}(O^{\prime}_{F})_{B(\xi)}.$
(116)

In the case that Eq. (115) and Eq. (116) hold, it is sufficient to obtain only
the boson expansions of the basic fermion pair operators. Conventional
practical boson expansion methods have used, as a matter of course, the
approximation of Eq. (116). Eq. (114) makes it possible to judge whether this
approximation is good or bad. We present $\widetilde{O_{F}O^{\prime}_{F}}$ in
the appendix.

We finally point out that $\bar{t}$-sum does not need to sum all $\bar{t}$ in
the following cases.

$[[X_{\tau_{1}},X_{\tau_{2}}^{\dagger}],X_{\tau_{3}}^{\dagger}]\approx-\sum_{\tau^{\prime}}Y(\tau_{1},\tau_{2},\tau_{3},\tau^{\prime})X_{\tau^{\prime}}^{\dagger},$
(117)

are satisfied for the phonon excitation modes $\\{\tau\\}$, and $\\{\tau\\}$
contains $\\{t\\}$ and is set up such that the small parameter expansion
holds, then $\bar{t}$ not contained in $\\{\tau\\}$ can be neglected in
$\bar{t}$-sum. An example is a case where $\\{\tau\\}$ contains a sufficient
variety of phonon excitation modes and

$\sum_{\tau}\psi_{\tau}(\alpha\beta)\psi_{\tau}(\alpha^{\prime}\beta^{\prime})\approx\delta_{\alpha\alpha^{\prime}}\delta_{\beta\beta^{\prime}}-\delta_{\alpha\beta^{\prime}}\delta_{\beta\alpha^{\prime}}$
(118)

are satisfied. In this case,

$a_{\alpha}^{\dagger}a_{\beta}^{\dagger}\approx\sum_{\tau}\psi_{\tau}(\alpha\beta)$
(119)

are satisfied, therefore,

$X_{\bar{\tau}}^{\dagger}\approx 0$ (120)

are satisfied, and Eq. (117) are derived. In this case, however, $\\{\tau\\}$
cannot be regarded as $\\{t\\}$ because $\\{\tau\\}$ contains so sufficient
variety of phonon excitation modes that $\hat{Z}$ includes zero eigenvalues.

###  4.4 Boson expansions in the case where the double commutators of the
phonon operators are closed

In this section, we treat the boson expansions in the case where the double
commutators of Eq. (8) are closed in $\\{t\\}$.

If $Y(t^{\prime}_{1}t\bar{t}t^{\prime}_{2})=0$, then the double commutators of
Eq. (8) are closed in $\\{t\\}$. For further analysis, we denote more
concretely $\mathbf{W}(2)$ and $\mathbf{Z^{\prime}}(2)$ in Eq. (79) as
follows,

$\mathbf{W}(2)=\left(\mathbf{W}^{(1)}(2)\ \mathbf{W}^{(2)}(2)\right),$ (121a)
$\mathbf{Z^{\prime}}(N)=\left(\begin{array}[]{@{\,}cc@{\,}}\mathbf{Z}^{(1)}(2)&\mathbf{Z^{\prime}}^{(3)}(2)\\\
\mathbf{Z^{\prime}}^{(3)}(2)^{T}&\mathbf{Z^{\prime}}^{(2)}(2)\end{array}\right),$
(121b)

where $\mathbf{W}^{(1)}(2)$ is what becomes a zero matrix when
$Y(t^{\prime}_{1}t\bar{t}t^{\prime}_{2})=0$. Substituting this into Eq. (84b),
then $\mathbf{W}^{(2)}(2)\mathbf{Z^{\prime}}^{(3)}(2)^{T}$ becomes a zero
matrix, and we obtain

$\sum_{\bar{t}\bar{t}^{\prime}}Y(t^{\prime}_{1}\bar{t}\bar{t}^{\prime}t^{\prime}_{2})Y(t_{1}\bar{t}\bar{t}^{\prime}\bar{t}_{1})=0,$
(122)

which indicates that Eqs. (90) do not hold. It also indicates that if
$Y(t^{\prime}_{1}t\bar{t}t^{\prime}_{2})=0$, then
$Y(t^{\prime}_{1}\bar{t}_{1}\bar{t}_{2}t^{\prime}_{2})=0$ should be satisfied.
$\mathbf{W}(2)=\mathbf{0}(2)$ holds, and $|t_{1}t_{2}\rangle$ and
$|\bar{t}\mu\rangle$ become orthogonal. Therefore, if the double commutators
of Eq. (8) are closed in $\\{t\\}$, the boson expansions are not obtained as
the small parameter expansion.

Starting with Eqs. (72) and applying $Y(t^{\prime}_{1}\bar{t}\mu
t^{\prime}_{2})=0$ to them, we derive
$\widetilde{X}_{\bar{t}}^{\dagger}(N)=0$, from which we obtain
$\widetilde{X}_{\bar{t}}=0$ and $\widetilde{X}_{\bar{t}}^{\dagger}=0$.
Therefore, $(X_{\bar{t}^{\prime}})_{\xi}=0$ and
$(X_{\bar{t}}^{\dagger})_{\xi}=0$ hold. Inversely, if
$(X_{\bar{t}^{\prime}})_{\xi}=0$ and $(X_{\bar{t}}^{\dagger})_{\xi}=0$ hold,
then $Y(t^{\prime}_{1}\bar{t}\mu t^{\prime}_{2})=0$ should hold, that is
$|t^{\prime}_{1}t^{\prime}_{2}\rangle$ and $|\bar{t}\mu\rangle$ should be
orthogonal because

$Y(t^{\prime}_{1}\bar{t}\mu
t^{\prime}_{2})=((t^{\prime}_{1}t^{\prime}_{2}|\bar{t}\mu))-\langle\langle
t^{\prime}_{1}t^{\prime}_{2}|\bar{t}\mu\rangle\rangle,$ (123)

and

$\langle\langle t^{\prime}_{1}t^{\prime}_{2}|\bar{t}\mu\rangle\rangle=\langle
0|X_{t^{\prime}_{1}}X_{t^{\prime}_{2}}X_{\bar{t}}^{\dagger}X_{\mu}|0\rangle=(0|(X_{t^{\prime}_{1}})_{\xi}(X_{t^{\prime}_{2}})_{\xi}(X_{\bar{t}}^{\dagger})_{\xi}(X_{\mu})_{\xi}|0)$
(124)

hold. It is a necessary and sufficient condition for
$(X_{\bar{t}^{\prime}})_{\xi}=0$ and $(X_{\bar{t}}^{\dagger})_{\xi}=0$ that
$|t^{\prime}_{1}t^{\prime}_{2}\rangle$ and $|\bar{t}\mu\rangle$ are
orthogonal. We also obtain
$\widetilde{X}_{t^{\prime}}=(X_{t^{\prime}})_{D}\hat{Z}$,
$\widetilde{X}_{t}^{\dagger}=(X_{t}^{\dagger})_{D}\hat{Z}$, and
$\widetilde{B}_{q}=(B_{q})_{D}\hat{Z}$. Therefore,

$(O_{F})_{\xi}=\hat{Z}^{\xi-\frac{1}{2}}(O_{F})_{D}\hat{Z}^{-\xi+\frac{1}{2}},$
(125)

where $O_{F}$ is $X_{t^{\prime}}$, $X_{t}^{\dagger}$, or $B_{q}$.

From Eq, (2b) and Eq. (72c), $[(B_{q})_{D},\hat{Z}(N)]=0$ holds, then

$[(B_{q})_{D},\hat{Z}]=0.$ (126)

Hence

$(B_{q})_{\xi}=(B_{q})_{D}\hat{T}_{B}=\hat{T}_{B}(B_{q})_{D}=\hat{T}_{B}(B_{q})_{D}\hat{T}_{B}$
(127)

holds for any $\xi$.

For $O_{F}$ and $O^{\prime}_{F}$ being the phonon operators or the scattering
operators, respectively, we obtain

$(O_{F}O^{\prime}_{F})_{\xi}=\hat{Z}^{\xi-\frac{1}{2}}(O_{F})_{D}(O^{\prime}_{F})_{D}\hat{Z}^{-\xi+\frac{1}{2}},$
(128)

In addition, the following,

$(O_{F}O^{\prime}_{F})_{\xi}=(O_{F})_{\xi}(O^{\prime}_{F})_{\xi},$ (129)

is satisfied if $\hat{T}_{B}$ becomes $\breve{1}_{B}$ and
$O_{F}O^{\prime}_{F}$ is normal ordered because
$\hat{Z}^{\xi-\frac{1}{2}}\hat{Z}^{-\xi+\frac{1}{2}}=\breve{1}_{B}$ and
$\breve{1}_{B}(X_{t}^{\dagger})_{D}=\breve{1}_{B}(X_{t}^{\dagger})_{D}\breve{1}_{B}$,
$(X_{t^{\prime}})_{D}\breve{1}_{B}=\breve{1}_{B}(X_{t^{\prime}})\breve{1}_{B}$,
and Eq. (126) are satisfied.

Eq. (73) becomes

$\hat{Z}(N)=\frac{1}{N}\sum_{t}(X_{t}^{\dagger})_{D}\hat{Z}(N-1)b_{t}\quad(N\geq
2).$ (130)

The solution of Eq. (130) should be given by Eq. (83a). While Eq. (130) can be
solved directly in the case that $\mathbf{Z}(2)$ has no zero eigenvalues. In
this case,
$Y(t^{\prime}_{1}t_{1}t_{2}t^{\prime}_{2})=-2((t^{\prime}_{1}t^{\prime}_{2}|t_{1}t_{2}))$
holds. Therefore,

$(X_{t}^{\dagger})_{D}=b_{t}^{\dagger}(2\hat{N}_{B}+1),$ (131)

from which we obtain

$\hat{Z}(N)=\frac{(2N-1)}{N}\sum_{t}b_{t}^{\dagger}\hat{Z}(N-1)b_{t}.$ (132)

$\hat{Z}(2)=3\hat{1}_{B}(2)$, and if $\hat{Z}(N-1)=(2N-3)!!\hat{1}_{B}(N-1)$,
then $\hat{Z}(N-1)=(2N-1)!!\hat{1}_{B}(N)$. These match Eq. (83a) in the case
that $\hat{T}_{B}(N)=\hat{1}_{B}(N)$ holds.

Eq. (130) can be also solved formally as

$\hat{Z}(N)=\frac{1}{N!}\sum_{t_{1}\cdots
t_{N}}(X_{t_{1}}^{\dagger})_{D}\cdots(X_{t_{N}}^{\dagger})_{D}|0)(0|b_{t_{1}}\cdots
b_{t_{N}}.$ (133)

From Eq. (133), we find the relation,

$\hat{Z}(N)b_{t}^{\dagger}=(X_{t}^{\dagger})_{D}\hat{Z}(N-1).$ (134)

While, from Eq. (83a) and Eq. (130), we obtain

$(2N-1)\hat{T}_{B}(N)b_{t}^{\dagger}=(X_{t}^{\dagger})_{D}\hat{T}_{B}(N-1).$
(135)

The mapped operators are given as follows,

$(O_{F})_{\xi}=\hat{T}_{B}(O_{F})_{B(\xi)}\hat{T}_{B},$ (136a)
$(O_{F})_{B(\xi)}=\left\\{(2\hat{N}_{B}-1)!!\right\\}^{\xi-\frac{1}{2}}(O_{F})_{D}\left\\{(2\hat{N}_{B}-1)!!\right\\}^{-\xi+\frac{1}{2}}$
(136b)

From Eq. (128), we obtain

$(O_{F}O^{\prime}_{F})_{B(\xi)}=\left\\{(2\hat{N}_{B}-1)!!\right\\}^{\xi-\frac{1}{2}}(O_{F})_{D}(O^{\prime}_{F})_{D}\left\\{(2\hat{N}_{B}-1)!!\right\\}^{-\xi+\frac{1}{2}}.$
(137)

The difference due to $\xi$ is renormalized to the boson excitation number,
and the remaining boson expansions are the same as those of the DBET, which
are finite. Therefore, we can substantially treat all types of boson
expansions as finite expansions when the concerning states are the physical
states that are eigenstates of the boson number operator. If $\xi=0$, we
obtain finite boson expansions of the Hermitian type.

In the case that $O_{F}$ preserves the number of quasi-particles, then
$(O_{F})_{D}$ preserves the number of bosons, and the norm operator parts
cancel out completely. As a result, we obtain finite boson expansions for any
$\xi$ such as

$(O_{F})_{B(\xi)}=(O_{F})_{D},$ (138)

from which we also derive

$(O_{F})_{B(0)}=(O_{F})_{B(\xi)}.$ (139)

Even if $O^{(1)}_{F}$ and $O^{(2)}_{F}$ do not necessarily preserve the quasi-
particle number, respectively, if $O_{F}=O^{(1)}_{F}O^{(2)}_{F}$ preserves the
quasi-particle number,Eq. (137) enables us to derive

$(O^{(1)}_{F}O^{(2)}_{F})_{B(0)}=(O^{(1)}_{F}O^{(2)}_{F})_{B(\xi)}=(O^{(1)}_{F})_{D}(O^{(2)}_{F})_{D}.$
(140)

Hence $(O_{F})_{D}$ and $(O^{(1)}_{F})_{D}(O^{(2)}_{F})_{D}$ are regarded as a
finite boson expansion of the Hermitian type.

For $\xi=\frac{1}{2}$, the norm operator does not appear in the mapped fermion
operators, and we can obtain the boson expansions as follows,

$(O_{F})_{B(\frac{1}{2})}=(O_{F})_{D},$ (141)
$(O_{F}O^{\prime}_{F})_{B(\frac{1}{2})}=(O_{F})_{B(\frac{1}{2})}(O^{\prime}_{F})_{B(\frac{1}{2})}=(O_{F})_{D}(O^{\prime}_{F})_{D}.$
(142)

We obtain the finite expansions of DBET. From Eqs. (44), it is straitforward
to proof that Hermitian treatmentTa01  holds exactly for the eigenvector of
$\hat{Z}$, $|N;a)$. Therefore, when $\hat{T}_{B}=\hat{1}_{B}$ holds, it is
applied precisely for the ideal boson state vector, $|N:t)$.

For $\xi=0$, the boson mapping becomes of the Hermite type. We can obtain
mapped operators as follows,

$\begin{array}[]{lll}(X_{t}^{\dagger})_{B(0)}&=&\left\\{(2\hat{N}_{B}-1)!!\right\\}^{-\frac{1}{2}}(X_{t}^{\dagger})_{D}\left\\{(2\hat{N}_{B}-1)!!\right\\}^{\frac{1}{2}},\\\
&=&(X_{t}^{\dagger})_{D}\displaystyle\frac{1}{\sqrt{1+2\hat{N}_{B}}}=b_{t}^{\dagger}\sqrt{1+2\hat{N}_{B}},\end{array}$
(143a)
$(B_{q})_{B(0)}=\sum_{tt^{\prime}}\Gamma_{q}^{t^{\prime}t}b_{t}^{\dagger}b_{t^{\prime}}.$
(143b)

Here we use the relation,

$\hat{T}_{B}b_{t}^{\dagger}(2\hat{N}_{B}+1)=(X_{t}^{\dagger})_{D}\hat{T}_{B},$
(144)

obtained from Eq. (135) for the derivation of $(X_{t}^{\dagger})_{B(0)}$. The
scattering operators are expressed as finite expansions in the physical
subspace. The phonon operators do not become of the small parameter expansion
whose zeroth-order approximation becomes the boson approximation. The boson
approximation holds only when the phonon excitation number does not exceed
one.

###  4.5 On the role of the norm operator

In this subsection, based on the results obtained so far, we summarize the
role the norm operator plays in the boson expansion method. What is important
is the relation between the norm operator consisting of all kinds of phonons
and the norm operator constituting the boson mapping operator.

What kinds of boson expansions are derived is determined by the structure of
the norm operator when all modes are adopted, $\hat{Z}^{(A)}$. The structure
is determined by the introduced single-particle states, $\\{\alpha\\}$, and
the amplitudes of the Tam-Dancoff phonons, $\psi_{\mu}(\alpha\beta)$ .
$\hat{Z}^{(A)}$ is composed of the norm operator $\hat{Z}$, which is used for
mapping, and other operators as

$\hat{Z}^{(A)}=\hat{Z}+\hat{W}+\hat{W}^{\dagger}+\hat{Z},^{\prime}$ (145a)
where
$\hat{Z}=\breve{1}_{B}\hat{Z}^{(A)}\breve{1}_{B},\quad\hat{W}=\breve{1}_{B}\hat{Z}^{(A)}(\breve{1}_{B}^{(A)}-\breve{1}_{B}),\quad\hat{Z}^{\prime}=(\breve{1}_{B}^{(A)}-\breve{1}_{B})\hat{Z}^{(A)}(\breve{1}_{B}^{(A)}-\breve{1}_{B}),$
(145b)

The condition Eq. (52) is imposed on $\hat{Z}^{(A)}$, which is regardless of
how to take $\psi_{\mu}(\alpha\beta)$. $\hat{Z}$, $\hat{W}$, and
$\hat{Z}^{\prime}$ are determined so as to satisfy the condition Eq. (52).

The double commutation relations of the phonon operators that constitute
$\hat{Z}$ are generally not closed among them. $\hat{Z}$ must have eigenvalues
nearly to 1 for the small parameter expansion, which also allows the use of
ideal boson state vectors as physical. It is possible to directly check
whether this condition is satisfied because $\hat{Z}$ is specifically obtained
by the boson expansion assuming the small parameter expansion. Only this
condition is not, however, a sufficient condition for realizing the small
parameter expansion. Including not only $\hat{Z}$ but also $\hat{W}$ and
$\hat{Z}^{\prime}$ gives the necessary and sufficient condition. In this case,
it is not allowed to treat $\hat{W}$ as a zero operator, that is, to assume
that the double commutation relation of the phonon operators constituting
$\hat{Z}$ is closed because the small parameter expansion does not hold.

In the case that $\hat{W}$ can be regarded as zero, the boson expansions can
be substantially treated as finite expansions. The realization of this type of
practical boson expansion is more difficult because $\\{t\\}$ should selected
so that the ideal boson state vectors become physical and the dynamics are
reflected enough, as with the small parameter expansion, under the above
condition.

##  5 Comments on the conventional methods

Conventional practical boson expansion methods, without exception, discard the
phonon excitation modes that are not adopted for the boson excitation modes.
We call this procedure the non-adopted modes discarding (NAMD).

Since NAMD closes double commutators of phonon operators within the adopted
modes for those of bosons, it is incompatible with a small expansion. The
incompatibility between NAMD and the small parameter expansion has not been
considered in formulating the conventional practical boson expansion methods.

In the case that NADM is precisely applicable, DBET is formulated exactly,
which does not mean that DBET necessarily has its exceptional superiority
because we can substantially obtain finite expansions of the Hermite type by
treating the boson number operator parts appropriately.

In the case that the small parameter expansion is applicable, all the boson
expansions become infinite ones and include the terms neglected by NAMD.
Applying NADM to Eq. (111), Eq. (112), and Eq. (113), it is found that the
remaining terms up to $O(\Gamma^{2})$ coincide with those obtained by
NOLCEXPT. The order of magnitude of the neglected terms is also
$O(\Gamma^{2})$, respectively. NOLCEXPT obtains, with NADM, the terms only up
to $O(\Gamma^{2})$. On the other hand, the finite boson expansions of DBET are
obtained from$(O_{F})_{B(\frac{1}{2})}$ by applying NADM. The order of
magnitude of the neglected terms by NADM is also $O(\Gamma^{2})$ in the same
order as the smallest one of the non-neglected terms by NADM. In both NOLCEXPT
and DBET, NADM neglects terms of the order of magnitude that should be
adopted, which indicates that NAMD can not be used as a proper approximation
under the small parameter expansion.

The investigation so far makes it clear that the comment of NOLCEXPT KT83 ;
SK88  on NAMD is incorrect. NOLCEXPT claims that the scattering operators are
expressed as finite expansions. It is realized only when NADM is precisely or
well approximately applied and not when the small parameter expansion is
realized. Eqs. (143) indicate that it is impossible to express the phonon
operators as infinite normal-ordered small parameter expansions although the
scattering operators become finite. NOLCEXPT has failed to refute Marshalek’s
claim Ma80a ; Ma80b  that KT-1 KT72  and KT-2 KT76  are chimerical boson
expansions.

As already mentioned, Hermitian treatment becomes exact when NAMD becomes
exact. On the other hand, in the case that the small parameter expansion is
applicable, Hermitian treatment becomes an approximation, and it can be
generally evaluated using the norm operator by following the method of KTM99 .
It is concluded that Hermitian treatment holds as far as it is possible to
neglect terms of $O(\Gamma^{4})$.

Next, we comment on the problems related to a modified Marumori boson
expansion method HJJ76 ; MTS81 . The modified Marumori boson expansion method
concludes that NADM is good using a norm of a multi-phonon state vector
despite the small parameter expansion being available HJJ76 . The reason why
the conclusion is derived incorrectly is as follows: The norm of the multi-
phonon state vector is obtained from $\hat{Z}(N)$. Since the neglected terms
by NADM do not appear up to $O(\Gamma^{3})$ in $\hat{Z}(N)$, it is impossible
to evaluate whether NADM is a good approximation by investigating $\hat{Z}(N)$
only up to $O(\Gamma^{2})$ with the small parameter expansion. For
explanation, we adopt a case where $\\{t\\}$ consists of only one type of
excitation mode $c$. We define $|N\rangle$ and $|N)$ as

$|N\rangle=|\overbrace{c\cdots c}^{N}\rangle,\quad|N)=|\overbrace{c\dots
c}^{N}).$ (146)

They sattisfy $\langle N|N\rangle=(N|\hat{Z}(N)|N)$. Assuming the small
parameter expansion, and setting $\langle 2|2\rangle=1-\varepsilon$, then
$\varepsilon=\frac{1}{2}Y(cccc)\sim O(\Gamma^{2})$. Expressing $\langle
N|N\rangle$ derived by the small parameter expansion up to $O(\Gamma^{2})$ as
${\mathcal{N}}^{(2)}(N)$, we obtain
$\mathcal{N}^{(2)}(N)=1-N(N-1)\frac{\varepsilon}{2}+O(\Gamma^{3})$. On the
other hand, Eq. (130) enables us to obtain $\hat{Z}(N)$ with the sum of all
terms without the neglected ones by NADM. Expressing $\langle N|N\rangle$ thus
obtained as ${\mathcal{N}}^{(all)}(N)$,

${\mathcal{N}}^{(all)}(N)={\mathcal{N}}^{(all)}(N-1)\left(1+(\langle
2|2\rangle-1)(N-1)\right),$ (147)

holds, and we obtain
${\mathcal{N}}^{(all)}(N)=(1-(N-1)\varepsilon)(1-(N-2)\varepsilon)\cdots(1-\varepsilon)$.
As $N$ becomes large, the difference between both becomes prominent, which is,
however, only $2\varepsilon^{2}\sim O(\Gamma^{4})$ for $N=3$. It indicates
that the small parameter expansion is well applied for this case and that both
coincide well up to $O(\Gamma^{2})$ with the exact one. Therefore it is
impossible to judge whether NADM is good or not by the comparison of
${\mathcal{N}}^{(all)}(3)$ with the exact one. It is a wrong conclusion that
NADM holds well for $\varepsilon\approx 0.1$ HJJ76 , where the small parameter
expansion becomes possible. $\varepsilon$ should become approximately -2 for
NADM to become good. In addition, it is not the strong but the weak effect of
the Pauli exclusion principle that makes the small parameter expansion
possible. The comment on the convergence of the modified Marumori boson
expansions is mistaken.

Conventional practical boson expansion methods restrict, for mapping, only the
sort of phonon excitation modes and not the number of those. $\hat{Z}(N)$
necessarily has zero eigenvalues for large phonon excitation numbers even if
restricting the sorts of modes, which makes ideal boson state vectors
unphysical and the small parameter expansion impossible. We should restrict
the phonon excitation number beforehand. Nevertheless, NOLCEXPT does not
restrict the phonon excitation number. Instead, it treats, without a clear
basis, all the norm matrices of the multi-phonon state as having no zero
eigenvalues. Nevertheless, it gives the correct expansion terms up to
$O(\Gamma^{2})$ under NADM. The restriction of the phonon excitation number
beforehand gives a clear reason for it because we obtain

$\lim_{N_{max}\rightarrow\infty}(O_{F})_{\xi}=(O_{F})_{B(\xi)}$ (148)

from Eq. (108). It indicates that we can obtain the correct results of the
small parameter expansion without limiting the phonon excitation number
beforehand. Afterward, we should limit the boson excitation number.

As for BFEXP, by replacing the collective modes $\\{c\\}$ and the non-
collective modes $\\{n\\}$ to $\\{t\\}$ and $\\{\bar{t}\\}$, respectively, and
suppressing the fermion excitations, we can obtain the boson expansions from
BFEXP TM90 ; TM91 . The Hermitian-type boson expansions obtained thus agree
with those obtained from BFEXP by adopting the proper transformation TKM94 .
Further comparison requires the derivation of higher-order terms in BFEXP.

##  6 Summary

We have proposed a new boson expansion theory, the norm operator method, where
the norm operator plays a crucial role.

The different treatment of the norm operator determines the type of boson
expansions as Hermitian or non-Hermitian. The mapping operator limits the
number of phonon excitations in addition to the phonon excitation modes
beforehand to use the ideal boson state vectors as physical and avoid the
breakdown of the small parameter expansion whose zeroth-order approximation is
the boson approximation.

In the case that the closed algebraic approximation or phonon truncation
approximation holds, that is, the double commutation relations between the
phonons with excitation modes adopted as boson excitations are closed, the
small parameter expansion is not available. The norm operator is expressed as
a function of the boson number operator, which substantially makes all types
of boson expansions be of finite expansion.

The small parameter expansion is not compatible with the closed-algebra
approximation or the phonon truncation approximation. The contribution of the
phonon excitation modes neglected by the approximation makes the boson
expansion become infinite expansion regardless of whether it is of the
Hermitian type or not. We have obtained the higher-order terms of the boson
expansion not expanded so far in addition to the neglected by the
approximation.

Conventional practical boson expansion methods have used the closed-algebra
approximation or the phonon truncation approximation without recognizing its
playing role mentioned above, and the claims derived from this approximation
have no validity: The normal-ordered linked-cluster expansion theory has
failed to refute Marshalek’s claim that KT-1 and KT-2 are of the chimerical
boson expansion. The Dyson boson expansion theory does not have exceptional
superiority over the other types of boson expansions.

The boson-fermion expansion theory derives the same boson expansions with the
Hermitian-type boson expansions obtained here up to the next-to-leading order.
The boson-fermion expansion theory should derive higher-order expansion terms
for further comparison.

## References

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##  Appendix A formulae of the product of the pair operators

We denote $B_{q}$, $X_{t^{\prime\prime}}$, or $X_{\bar{t}^{\prime}}$ as
$O_{F}$. the following equations hold:

$\widetilde{X_{t^{\prime}}O_{F}}=\breve{1}_{B}(X_{t^{\prime}})_{L}B(O_{F})_{L}\mathcal{\hat{Z}}\breve{1}_{B}=\breve{1}_{B}(X_{t^{\prime}})_{L}B(O_{F})_{L}\mathcal{\hat{Z}}\breve{1}_{B},$
(149a)
$\widetilde{O_{F}^{\dagger}X_{t}^{\dagger}}=\breve{1}_{B}(O_{F}^{\dagger})_{L}(X_{t}^{\dagger})_{L}\mathcal{\hat{Z}}\breve{1}_{B}.$
(149b)

$\widetilde{X_{t}^{\dagger}X_{t^{\prime}}}=\breve{1}_{B}\left\\{(X_{t}^{\dagger})_{D}(X_{t^{\prime}})_{D}-\frac{1}{2}\sum_{t_{1}t_{2}}\sum_{\bar{t}^{\prime}}Y(tt_{1}t_{2}\bar{t}^{\prime}_{1})b_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}(X_{t^{\prime}})_{L}(X_{\bar{t}^{\prime}_{1}})_{L}\right\\}\mathcal{\hat{Z}}\breve{1}_{B}.$
(150)

$\widetilde{X_{\bar{t}}^{\dagger}X_{\bar{t}^{\prime}}^{\dagger}}=\breve{1}_{B}(X_{\bar{t}}^{\dagger})_{L}(X_{\bar{t}^{\prime}}^{\dagger})_{L}\mathcal{\hat{Z}}\breve{1}_{B}+O(\Gamma^{5}),$
(151a)

$\widetilde{X_{\bar{t}}X_{\bar{t}^{\prime}}}=\breve{1}_{B}(X_{\bar{t}})_{L}(X_{\bar{t}^{\prime}})_{L}\mathcal{\hat{Z}}\breve{1}_{B}+O(\Gamma^{5}).$
(151b)

$\widetilde{X_{\bar{t}}^{\dagger}X_{\bar{t}^{\prime}}}=\breve{1}_{B}(X_{\bar{t}}^{\dagger})_{L}(X_{\bar{t}^{\prime}})_{L}\mathcal{\hat{Z}}\breve{1}_{B}+O(\Gamma^{5}).$
(152)

$\widetilde{B_{q}X_{\mu^{\prime}}}=(B_{q})_{D}\widetilde{X_{\mu^{\prime}}}+\sum_{t}\sum_{\bar{t}^{\prime}}\Gamma_{q}^{\bar{t}^{\prime}t}b_{t}^{\dagger}\widetilde{X_{\bar{t}^{\prime}}X_{\mu^{\prime}}},$
(153a)
$\widetilde{X_{\mu}^{\dagger}B_{q}}=\widetilde{X_{\mu}^{\dagger}}(B_{q})_{D}+\sum_{t^{\prime}}\sum_{\bar{t}}\Gamma_{q}^{t^{\prime}\bar{t}}\widetilde{X_{\mu}^{\dagger}X_{\bar{t}}^{\dagger}}b_{t^{\prime}}.$
(153b)

##  Appendix B Proof of $\mathbf{W}(N)=\mathbf{0}(N)$ for $N\geq 3$

$\hat{Z}(N)$ is related to $\hat{Z}^{(A)}(N)$ as

$\hat{Z}(N)=\breve{1}_{B}\hat{Z}^{(A)}(N)\breve{1}_{B}=\hat{1}_{B}(N)\hat{Z}^{(A)}(N)\hat{1}_{B}(N).$
(154)

Introducing

$\begin{array}[]{lll}\hat{Z}^{\prime}(N)&=&(\breve{1}_{B}^{(A)}-\breve{1}_{B})\hat{Z}^{(A)}(N)(\breve{1}_{B}^{(A)}-\breve{1}_{B})\\\
&=&(\hat{1}_{B}^{(A)}(N)-\hat{1}_{B}(N))\hat{Z}^{(A)}(N)(\hat{1}_{B}^{(A)}(N)-\hat{1}_{B}(N)),\end{array}$
(155)

and

$\hat{W}(N)=\breve{1}_{B}\hat{Z}^{(A)}(\breve{1}_{B}^{(A)}-\breve{1}_{B})=\hat{1}_{B}(N)\hat{Z}^{(A)}(\hat{1}_{B}^{(A)}(N)-\hat{1}_{B}(N)),$
(156)

we obtain

$\hat{Z}(N)^{(A)}=\hat{Z}(N)+\hat{W}(N)+\hat{W}(N)^{\dagger}+\hat{Z}^{\prime}(N).$
(157)

Eq. (79) expresses this relation as those of the matrices where
$\mathbf{Z}^{(A)}(N)$, $\mathbf{Z}(N)$, $\mathbf{Z^{\prime}}(N)$, and
$\mathbf{W}(N)$ are matrices representing $\hat{Z}^{(A)}(N)$, $\hat{Z}(N)$,
$\hat{Z}^{\prime}(N)$, and $\hat{W}(N)$, respectively.

If $Y(t^{\prime}_{1}\bar{t}\mu t^{\prime}_{2})=0$, then $\hat{W}(2)=0$. On the
other hand, from Eq. (73), we obtain

$\hat{Z}^{(A)}(N)=\frac{1}{N}\sum_{\mu}(X_{\mu}^{\dagger})_{D}\hat{Z}(N-1)^{(A)}b_{\mu}.$
(158)

If $\hat{W}(N-1)=0$, then

$\hat{Z}^{(A)}(N)=\frac{1}{N}\sum_{\mu}(X_{\mu}^{\dagger})_{D}Z(N-1)b_{\mu}+\frac{1}{N}\sum_{\mu}(X_{\mu}^{\dagger})_{D}\hat{Z}^{\prime}(N-1)b_{\mu}.$
(159)

On the othrer hand,
$\hat{1}_{B}(N-1)b_{\mu}(\hat{1}_{B}^{(A)}(N)-\hat{1}_{B}(N))=0$ and
$\hat{1}_{B}(N)(X_{\mu}^{\dagger})_{D}(\hat{1}_{B}^{(A)}(N-1)-\hat{1}_{B}(N-1))=0$
hold. Therefore if $\hat{W}(N-1)=0$, then $\hat{W}(N)=0$. That is
$\hat{W}(N)=0$ for $N\geq 3$, and then $\mathbf{W}(N)=\mathbf{0}(N)$ for
$N\geq 3$.