A Possible Boson Realization of Generalized Lipkin Model for Many-Fermion SystemThe su(M + 1)-Algebraic Model in Non-Symmetric Boson Representation
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Progress of Theoretical Physics, Vol. 105, No. 6, June 2001
A Possible Boson Realization of Generalized Lipkin Model
for Many-Fermion System
The su(M + 1)-Algebraic Model
in Non-Symmetric Boson Representation
Atsushi Kuriyama, Constança Providência,∗ João da Providência,∗
Yasuhiko Tsue∗∗ and Masatoshi Yamamura
Faculty of Engineering, Kansai University, Suita 564-8680, Japan
∗ Departamento de Fisica, Universidade de Coimbra, P-3000 Coimbra, Portugal
∗∗ Department of Material Science, Kochi University, Kochi 780-8520, Japan
(Received November 17, 2000)
On the basis of the formalism proposed by three of the present authors (A. K., J. P. & M.
Y.), the generalized Lipkin model consisting of (M + 1) single-particle levels is investigated.
This model is essentially a kind of the su(M + 1)-algebraic model and, in contrast to the
conventional treatment, the case in which fermions are partially occupied in each level is
discussed. The scheme for obtaining the orthogonal set for the irreducible representation is
presented.
§1.
Introduction and preliminaries
The Lipkin model, 1) which was proposed by Lipkin, Meshkov and Glick in 1965,
has played a particular role in schematic studies of collective motions in manyfermion system. This is a kind of shell model: Under a certain interaction, many
fermions move in two single-particle levels with the same degeneracy. This is essentially one kind of the su(2)-algebraic model and it has also contributed to the
schematic understanding of finite temperature effects in many-fermion system. 2) On
the other hand, the Lipkin model has been generalized to the case of many singleparticle levels, for example, the case of three levels is the most popular and it is
a kind of the su(3)-algebraic model. The investigations based on this model have
been made not only for collective motion 3) but also for finite temperature effects in
many-fermion system. 4)
The generators of the su(2) algebra, which we denote (S+ , S− , S0 ), play a central
role in the Lipkin model. These are expressed in terms of the bilinear forms of particle
∗ ) and hole operators (β
, β∗ ):
1m , α
1m
operators (α
m
m
S+ =
m
∗ ∗
1m
α
βm
,
1 ∗
1m α
1m −
S0 =
α
2
S− =
Ω−
m
m
1m ,
βm
α
m
∗
βm
βm
.
(1.1a)
Here, Ω denotes the degeneracy of the single-particle levels. The definitions of the
other notations will be given in §2. For later convenience, we use (S1 , S1 , S11 ) for the
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Kuriyama, Providência, da Providência, Tsue and Yamamura
su(2) generators defined as
S1 = S+ ,
S1 = S− ,
S11 = 2S0 .
(1.1b)
In addition to the above generators, there exists one operator, which we denote N
as
= Ω−
N
m
∗
βm
+
βm
∗
1m
1m .
α
α
(1.2)
m
denotes the total fermion number and it obeys
The operator N
, S1 = N
, S1 = N
, S1 = 0 .
N
1
(1.3)
Conventionally, the orthogonal set for the Lipkin model is obtained by operating
with S1 successively on the state |m which obeys the condition
S1 |m = 0 ,
S11 |m = −σ1 |m .
(1.4)
The state |m is the eigenstate of the Casimir operator of the su(2) algebra:
1
Γsu(2) |m = σ1 (σ1 + 2)|m ,
2
1 1 2
Γsu(2) = S1 S1 + S1 S1 +
S
.
2 1
(1.5)
The quantity σ1 /2 denotes the magnitude of the quasi-spin. Further, the relation
:
(1.3) supports that |m is also the eigenstate of N
|m = N |m .
N
(1.6)
By operating with S1 for (σ1 + σ0 )/2 times on |m, we have
(σ1 +σ0 )/2
|m ,
|(γ); N, σ1 , σ0 = S1
|m = |(γ); N, σ1 .
(1.7)
Here, (γ) denotes a set of the quantum numbers additional to those related to the
tell us that |m can be specified by the
su(2) algebra. The definitions of S11 and N
∗β
∗ α
and
1m
1m ,
quantum numbers n and n1 which are the eigenvalues of m βm
mα
m
respectively:
(1.8)
|m = |(γ); n, n1 .
The state |m is called the minimum weight state, but, in this paper, we call it
the intrinsic state in the meaning analogous to the rotational model. The relation
between (N, σ1 ) and (n, n1 ) is given as
N = Ω − n + n1 ,
1
n = Ω − (N + σ1 ) ,
2
σ1 = Ω − n − n1 ,
1
n1 = (N − σ1 ) .
2
(1.9a)
(1.9b)
A Possible Boson Realization of Generalized Lipkin Model
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Since 0 ≤ Ω − n ≤ Ω and 0 ≤ n1 ≤ Ω, we have the following relations:
if 0 ≤ N ≤ Ω ,
if Ω ≤ N ≤ 2Ω ,
0 ≤ σ1 ≤ N ,
0 ≤ σ1 ≤ 2Ω − N .
(1.10a)
(1.10b)
The above is an outline of the Lipkin model and it is easily generalized to the su(M +
1)-algebraic model in the (M + 1) single-particle levels with the same degeneracy,
which will be discussed in §3.
Our present interest is concerned with the explicit determination of |m in terms
of the fermion operators. With the use of the quasi-fermion operators obeying certain
constraints, 5) one can, in principle, construct |m, but, it may be difficult to apply
this idea to the generalized case given in §3. However, one case is simply realized:
n = n1 = 0, i.e., N = σ1 = Ω. In this case, |m corresponds to the vacuum of
1m and βm
1m |m = βm
α
(α
|m = 0) and it means that one level is fully occupied
by the fermions, i.e., the closed shell system. The case of the generalized Lipkin
model is in the same situation as that in the su(2) model. Except the case of the
finite temperature effects, 2) many of the investigations based on the Lipkin model
are restricted to the case n = n1 = 0, that is, the use of the orthogonal set obtained
by operating with S1 on the state |(γ); n = 0, n1 = 0 (= |(γ); N = Ω, σ1 = Ω). In
Ref. 2), all the values of σ1 permitted, which are shown in the relation (1.10a), are
taken into account for the system with N = Ω. The case of the su(3) Lipkin model 3)
is also in the situation similar to the above except for finite temperature effects. 4)
As is well known, there exist two forms for the boson realization of the Lipkin
model. One is the Holstein-Primakoff representation 6) and the other the Schwinger
representation. 7) In the former, the three generators are expressed as
S1 = C ∗ ·
s1 − C ∗ C ,
S1 =
s1 − C ∗ C · C ,
S11 = 2C ∗ C − s1 .
(1.11)
C
∗ ) denote boson operators and the relation between s1 and σ1 in Eq. (1.5)
Here, (C,
is given later. The state |s1 , s0 ) is constructed in the form
|s1 , s0 ) = (S1 )(s1 +s0 )/2 |m) = (C ∗ )(s1 +s0 )/2 |0) ,
|m) = |0) . (C|0)
= 0)
(1.12)
The Schwinger representation gives the following form for the generators:
Ŝ 1 = â∗1 b̂ ,
Ŝ1 = b̂∗ â1 ,
Ŝ11 = â∗1 â1 − b̂∗ b̂ .
(1.13)
Here, (â1 , â∗1 ) and (b̂, b̂∗ ) denote two kinds of bosons. The state |s1 , s0 is constructed
in the form
|s1 , s0 = (Ŝ 1 )(s1 +s0 )/2 |m = (â∗1 )(s1 +s0 )/2 (b̂∗ )(s1 −s0 )/2 |0 ,
|m = (b̂∗ )s1 |0 . (â1 |0 = b̂|0 = 0)
(1.14)
In a certain case mentioned below, the Holstein-Primakoff representation is easily
generalized and it is called the symmetric representation. 8)
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Kuriyama, Providência, da Providência, Tsue and Yamamura
The above two boson r (...truncated)
