Note on the Time Dependent Variational Approach with Quasi-Spin Squeezed State for Pairing Model

0.15 MB·English

by  Y. Tsue

Note on the Time Dependent Variational Approach
with Quasi-Spin Squeezed State for Pairing Model
Yasuhiko Tsue1 and Hideaki Akaike2
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1Physics Division, Faculty of Science, Kochi University, Kochi 780-8520,
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Japan
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2Department of Applied Science, Kochi University, Kochi 780-8520, Japan
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Abstract
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A simple many-fermion system in which there exists N identical fermions in a
i
X
singlesphericalorbitwithpairinginteractionistreatedbymeansofthetime-dependent
r
a
variational approach with a quasi-spin squeezed state with the aim of going beyond the
time-dependent Hartree-Fock or Bogoliubov theory. It is shown that the ground state
energy is reproduced well analytically in this approach.
1 typeset using TEX.cls Ver.0.9
PTP h i
§1. Introduction and Preliminary
The time-dependent variationalapproach withthe su(2)-coherent stateleads to thetime-
dependent Hartree-Fock (TDHF) or Bogoliubov (TDHB) theory for simple many-fermion
systems governed by the dynamical su(2)-algebra. In the previous paper,1) with the aim of
going beyond the TDHF approximation, we have investigated the dynamics of the Lipkin
modelbymeansofthetime-dependent variationalapproachwithaquasi-spinsqueezed state.
The quasi-spin squeezed state is a possible extension of the su(2)-coherent state.2),3) Thus,
the use of this extended trial state leads to an extended TDHF theory.4) In Ref.1), the
role of the quantum effects included in the quasi-spin squeezed state was analyzed in the
Lipkin model. Also, in Ref.5), the fermionic squeezed state was introduced for the O(4)
model with a schematic pairing plus quadrapole interaction and the role of quantum effects
were analyzed. The use of the squeezed state in the time-dependent variational approach
is one of possible candidates to go beyond the TDHF or TDHB approximation. However,
in the previous works, we have only investigated the role of the fluctuations included in the
squeezed state based on numerical results such as the ground state energy and dynamical
motion. Thus, itisdesirabletogiveaninvestigationbasedonananalyticalresults, especially,
for the approximate reproduction of the ground state energy in dynamical viewpoint in order
to clarify the role of quantum effects included in the quasi-spin squeezed state all the more.
In this paper, paying an attention to the role of the quantum fluctuation included in the
quasi-spin squeezed state and interesting to the dynamics of a system, we treat a simple
many-fermion system in which there exists N identical fermions in a single spherical orbit
with pairing interaction. The exact energy eigenvalue of the pairing model is known analyt-
ically. Thus, we can compare the result obtained in our squeezed state approach with the
exact one. The main aim of this paper is to give an analytical understanding for the role of
the quantum effects included in the quasi-spin squeezed state. Especially, the dynamics of
the variable representing quantum fluctuations is taken into account. As a result, it is shown
that, under a certain condition, the ground state energy is reproduced well analytically, like
1/N expansion method.
The single particle state is specified by a set of quantum number (j,m), where j and m
represent the magnitude of angular momentum of the single particle state and its projection
to the z-axis, respectively. Let us start with the following Hamiltonian:
G
Hˆ = ǫ cˆ†mcˆm − 4 (−)j−mcˆ†mcˆ†−m (−)j−m′cˆ−m′cˆm′ , (1.1)
m m m′
X X X
where ǫ and G represent the single particle energy and the force strength, respectively. The
operators cˆ and cˆ† are the fermion annihilation and creation operators with the quan-
m m
2
tum number m, which obey the anti-commutation relations: { cˆm , cˆ†m′ } = δmm′ and
{ cˆm , cˆm′ } = { cˆ†m , cˆ†m′ } = 0. We introduce the following operators:
1 1 1
Sˆ = cˆ† cˆ† , Sˆ = cˆ cˆ , Sˆ = ( cˆ† cˆ Ω) , (1.2)
+ 2 m m − 2 m m 0 2 m m −
m m m
X X X
e e
where cˆ = ( )j−mcˆ and Ω represents the half of the degeneracy: Ω = j + 1/2. These
m −m
−
operators compose the su(2)-algebra:
e
[Sˆ ,Sˆ ] = 2Sˆ , [Sˆ ,Sˆ ] = Sˆ . (1.3)
+ − 0 0 ± ±
±
Thus, these operators are called the quasi-spin operators.6),7) Then, the Hamiltonian (1.1)
can be rewritten in terms of the quasi-spin operators as
Hˆ = 2ǫ(Sˆ +S ) GSˆ Sˆ = ǫNˆ GSˆ Sˆ , (1.4)
0 j + − + −
− −
S = Ω/2 (= (j +1/2)/2) ,
j
where Nˆ represents the number operator:
Nˆ = cˆ† cˆ . (1.5)
m m
m
X
As is well known, the eigenstates and eigenvalues for this Hamiltonian are easily obtained.
Then, the ground state energy can be obtained by setting the seniority number being zero:8)
1 N 2
E = ǫN GNΩ 2 + . (1.6)
0
− 4 − Ω Ω
(cid:18) (cid:19)
Next, we review the coherent state approach to this pairing model, which is identical
with the BCS approximation to the pairing model consisting of the single energy level. The
su(2)-coherent state is given as
φ(α) = exp(fSˆ f∗Sˆ ) 0 ,
+ −
| i − | i
0 = S = S ,S = S , (Sˆ 0 = 0) . (1.7)
j 0 j −
| i | − i | i
We impose the canonicity condition:9)
∂ 1 ∂ 1
φ(α) φ(α) = ξ∗ , φ(α) φ(α) = ξ . (1.8)
h |∂ξ| i 2 h |∂ξ∗| i −2
A possible solution of the above canonicity condition is presented as
f f∗
ξ = 2S sin f , ξ∗ = 2S sin f . (1.9)
j j
f | | f | |
| | | |
p p
3
Thus, the expectation values of Hamiltonian Hˆ and number operator Nˆ are easily calculated
and given as
1
φ(α) Hˆ φ(α) = ǫ 2 ξ 2 G 2S ξ 2 1 ξ 4 E ,
j
h | | i · | | − | | − − 2S | | ≡
(cid:18) (cid:18) j(cid:19) (cid:19)
φ(α) Nˆ φ(α) = 2 ξ 2 N . (1.10)
h | | i | | ≡
If total particle number conserves, that is, N =constant, then, the energy expectation value
E is obtained as a function of N :
1 N N
E = ǫN GNΩ 2 + . (1.11)
− 4 − Ω Ω2
(cid:18) (cid:19)
Compared the approximate ground state energy in (1.11) with the exact energy eigenvalue
in (1.6), the last terms in the parenthesis of the right-hand side are not identical, which has
order of 1/Ω if the order of the magnitudes of N and Ω is comparable. The main aim of this
paper is to show that the last term, 2/Ω in (1.6), can be recovered by taking into account
of the dynamics in the time-dependent variational approach with the quasi-spin squeezed
state.
§2. Quasi-spin squeezed state for pairing model
In this section, the quasi-spin squeezed state is introduced following to Ref.2). First, we
introduce the following operators:
Sˆ Sˆ
Aˆ† = + , Aˆ = − , Nˆ = 2(Sˆ +S ) , (2.1)
0 j
2S 2S
j j
where Nˆ is identical withpthe number operpator (1.5). Then, the commutation relations can
be expressed as
Nˆ
[Aˆ,Aˆ†] = 1 , [Nˆ,Aˆ] = 2Aˆ , [Nˆ,Aˆ†] = 2Aˆ† . (2.2)
− 2S −
j
Using the boson-like operator Aˆ†, the su(2)-coherent state in (1.7) can be recast into
1 α∗α 2Sj
φ(α) = exp(αAˆ†) 0 , Φ(α∗α) = 1+ , (2.3)
| i Φ(α∗α) | i 2S
(cid:18) j (cid:19)
where α is related to fpin (1.7) as α = 2S (f/ f ) tan f . The state φ(α) in (2.3) is a
j
· | | · | | | i
vacuum state for the Bogoliubov-transformed operator aˆ :
p m
aˆ = Ucˆ V( )j−mcˆ† , aˆ φ(α) = 0 . (2.4)
m m − − −m m| i
4
The coefficients U and V are given as
1 α 1
U = , V = , U2 + V 2 = 1 . (2.5)
1+α∗α/(2S ) 2S 1+α∗α/(2S ) | |
j j j
Ofcourse,aˆpandaˆ† arefermionannihpilationpandcreationoperatorsandtheanti-commutation
m m
relations are satisfied. By using the above Bogoliubov-transformed operators, we introduce
the following operators:
1 1
Bˆ† = aˆ† aˆ† , Bˆ = aˆ aˆ , Mˆ = aˆ† aˆ . (2.6)
8S m m 8S m m m m
j m j m m
X X X
e e
Then, the state φ(pα) satisfies p
| i
Bˆ φ(α) = Mˆ φ(α) = 0 . (2.7)
| i | i
Further, the commutation relations are as follows:
Mˆ
[Bˆ,Bˆ†] = 1 , [Mˆ,Bˆ] = 2Bˆ , [Mˆ,Bˆ†] = 2Bˆ† . (2.8)
− 2S −
j
The quasi-spin squeezed state can be constructed onthe su(2)-coherent state φ(α) by using
| i
the above boson-like operator Bˆ† as is similar to the ordinary boson squeezed state:
1 1
ψ(α,β) = exp βBˆ†2 φ(α) ,
| i Ψ(β∗β) 2 | i
(cid:18) (cid:19)
p[Sj] (2k 1)!! 2k−1 p
Ψ(β∗β) = 1+ − 1 ( β 2)k . (2.9)
(2k)!! − 2S | |
k=1 p=1 (cid:18) j(cid:19)
X Y
We call the state ψ(α,β) the quasi-spin squeezed state.
| i
We can easily calculate the expectation values for various operators with respect to the
quasi-spin squeezed state. The expectation values can be expressed in terms of the canonical
variables which are introduced through the canonicity conditions. The same results derived
in the following are originally given in the Lipkin model in Ref.10) at the first time and
these results were used in Ref.1) to analyze the effects of quantum fluctuations in the Lipkin
model. For the quasi-spin squeezed state ψ(α,β) in (2.9), the following expression is useful
| i
:
Ψ′(β∗β)1
ψ(α,β) ∂ ψ(α,β) = (β∗∂ β β∂ β∗)
h | z| i Ψ(β∗β) 2 z − z
2β∗β Ψ′(β∗β) 1 1
+ 1 (α∗∂ α α∂ α∗) ,
− S Ψ(β∗β) · 1+α∗α/(2S )2 z − z
(cid:18) j (cid:19) j
∂Ψ(u)
Ψ′(u) , (2.10)
≡ ∂u
5
where ∂ = ∂/∂z. The canonicity conditions are imposed in order to introduce the sets of
z
canonical variables (X,X∗) and (Y,Y∗) as follows :
1 1
ψ(α,β) ∂X ψ(α,β) = X∗ , ψ(α,β) ∂X∗ ψ(α,β) = X ,
h | | i 2 h | | i −2
1 1
ψ(α,β) ∂Y ψ(α,β) = Y∗ , ψ(α,β) ∂Y∗ ψ(α,β) = Y . (2.11)
h | | i 2 h | | i −2
Possible solutions for X and Y are obtained as
X X∗
α = , α∗ = ,
1 X∗X 4Y∗Y 1 X∗X 4Y∗Y
− 2Sj − 2Sj − 2Sj − 2Sj
q Y Yq∗
β = , β∗ = , (2.12)
K(Y∗Y) K(Y∗Y)
where K(Y∗Y) is introdpuced and satisfies the rpelation
K(Y∗Y)Ψ(Y∗Y/K) = Ψ′(Y∗Y/K) . (2.13)
The expectation values for Bˆ, Bˆ†, Mˆ and the products of these operators are easily obtained
and are expressed in terms of the canonical variables as follows :
ψ(α,β) Bˆ ψ(α,β) = ψ(α,β) Bˆ† ψ(α,β) = 0 ,
h | | i h | | i
ψ(α,β) Mˆ ψ(α,β) = 4Y∗Y , (2.14a)
h | | i
ψ(α,β) Bˆ2 ψ(α,β) = 2Y K(Y∗Y) ,
h | | i
ψ(α,β) Bˆ†2 ψ(α,β) = 2Yp∗ K(Y∗Y) , (2.14b)
h | | i
2
ψ(α,β) Bˆ†Bˆ ψ(α,β) = 2(1p 1/(2S ))Y∗Y (Y∗Y)2 L(Y∗Y) ,
j
h | | i − − S ·
j
ψ(α,β) BˆBˆ† ψ(α,β) = ψ(α,β) Bˆ†Bˆ ψ(α,β) +(1 2Y∗Y/S ) , (2.14c)
j
h | | i h | | i −
ψ(α,β) Mˆ2 ψ(α,β) = 16Y∗Y(1+Y∗Y L(Y∗Y)) , (2.14d)
h | | i ·
where L(Y∗Y) is defined and satisfies
Ψ′′(β∗β)
K(Y∗Y)2 L(Y∗Y) = . (2.14e)
· Ψ(β∗β)
By using the relations between the original variables α and β and the canonical variables
X and Y, the coefficients of the Bogoliubov transformation (2.5), U and V, are expressed as
1 X∗X 4Y∗Y
U = − 2Sj − 2Sj , V = X 1 . (2.15)
q 1 4Y∗Y 2Sj 1 4Y∗Y
− 2Sj − 2Sj
q p q
6
Then, the operators Aˆ, Aˆ† and Nˆ, which are related to the quasi-spin operators Sˆ , Sˆ and
− +
Sˆ , respectively, in (2.1), can be expressed as
0
Mˆ
Aˆ = 2S UV 1 V2Bˆ† +U2Bˆ ,
j
− 2S −
j!
p
Mˆ
Aˆ† = 2S UV∗ 1 +U2Bˆ† V∗2Bˆ ,
j
− 2S −
j!
p
Mˆ
Nˆ = 4S V∗V 1 + 2S U(VBˆ† +V∗Bˆ)+Mˆ . (2.16)
j j
− 2S
j!
p
Thus, the expectation values for Aˆ, Aˆ†, Nˆ and the products of these operators are easily
obtained and are expressed in terms of the canonical variables. For example, from (2.1), the
expectation values of quasi-spin operators are derived as
ψ(α,β) Sˆ ψ(α,β) = X∗ 2S X∗X 4Y∗Y ,
+ j
h | | i − −
ψ(α,β) Sˆ ψ(α,β) = 2pS X∗X 4Y∗Y X ,
− j
h | | i − −
ψ(α,β) Sˆ ψ(α,β) = Xp∗X +2Y∗Y S (2.17)
0 j
h | | i −
and also the expectation value of the number operator in (1.5) is calculated as
N = ψ(α,β) Nˆ ψ(α,β) = 2X∗X +4Y∗Y . (2.18)
h | | i
The above expressions in (2.17) correspond to the Holstein-Primakoff boson realization for
the su(2)-algebra such as S = X∗ 2S X∗X and so on. Thus, we can conclude that the
+ j
−
variable Y 2 represents the quantum effect.
p
| |
The model Hamiltonian (1.4) can be expressed in terms of the fermion number operator
Nˆ and the boson-like operators Aˆ and Aˆ† as
Hˆ = ǫNˆ 2S GAˆ†Aˆ . (2.19)
j
−
Thus, the expectation value of this Hamiltonian is easily obtained. We denote it as H :
sq
H = ψ(α,β) Hˆ ψ(α,β)
sq
h | | i
n2 4
= ǫ(2n +4n ) 2GS + X n +n2 L n2
X Y − j X 2S (2S 4n ) − S2XY Y Y − Y
(cid:26) j j − Y j
2 1(cid:2) n2 L(cid:3)
n K cos(2θ θ )+2 1 XY 1 n Y · ,
Y X Y Y
−S XY − − S − 2S − S
j (cid:20) j (cid:21)(cid:20)(cid:18) j(cid:19) j (cid:21)(cid:27)
p (2.20)
7
where we introduce the action-angle variables instead of (X,X∗) and (Y,Y∗) as
X = √n e−iθX , X∗ = √n eiθX ,
X X
Y = √n e−iθY , Y∗ = √n eiθY (2.21)
Y Y
and and are defined as
X Y
n2 2n n 1
= n X Y X , = . (2.22)
X (cid:18) X − 2Sj − Sj (cid:19) Y 1 2nY 2
− S
(cid:18) j (cid:19)
The dynamics of this system can be investigated approximately by determining the time-
dependence of the canonical variables (X,X∗) and (Y,Y∗) or (n ,θ ) and (n ,θ ). The
X X Y Y
time-dependence of these canonical variables is derived from the time-dependent variational
principle :
δ ψ(α,β) i∂ Hˆ ψ(α,β) dt = 0 . (2.23)
t
h | − | i
Z
§3. Time evolution of variational state
In the su(2)-coherent state approximation, the expectation value of the Hamiltonian is
calculated as
n n2
H = φ(α) Hˆ φ(α) = 2ǫn 2GS n X + X . (3.1)
ch h | | i X − j X − 2S 4S2
(cid:26) j j (cid:27)
In this approximation, namely, usual time-dependent Hartree-Bogoliubov approximation,
the canonical equations of motion derived from the time-dependent variational principle
have the following forms:
∂H n n
˙ ch X X
θ = = 2ǫ 2GS 1 ,
X ∂n − j − S − 2S2
X (cid:18) j j (cid:19)
∂H
n˙ = ch = 0 . (3.2)
X
− ∂θ
X
The solutions of the above equations of motion are easily obtained as
n n
0 0
θ (t) = 2 ǫ GS 1 t+θ ,
X − j − S − 2S2 X0
(cid:20) (cid:18) j j (cid:19)(cid:21)
n (t) = n (constant) . (3.3)
X 0
On the other hand, in the quasi-spin squeezed state approximation, the equations of
motion derived from (2.23) are written as
∂H
˙ sq
θ =
X
∂n
X
8
∂ n 4 ∂
= 2ǫ 2GS X + X + X [n +n2L n2 ]
− j ∂n S (2S 4n ) S2 · ∂n ·Y · Y Y − Y
(cid:26) X j j − Y j X
2 ∂ ∂
n K X cos(2θ θ ) 2 X Y L ,
Y X Y
−S · ∂n − − · ∂n S ·
j X X j (cid:27)
∂H p
n˙ = sq = 8G n K sin(2θ θ ) , (3.4)
X Y X Y
− ∂θ XY −
X
∂H p
˙ sq
θ =
Y
∂n
Y
∂ n2 4 ∂ ∂
= 4ǫ 2GS X + X ·Y + X + Y [n +n2 L n2 ]
− j ∂n 2S3 S2 · ∂n Y ∂n X ·Y · Y Y − Y
(cid:26) Y j j (cid:20) Y Y (cid:21)
4 ∂L
+ [1+2n L+n2 2n ]
S2 ·XY · Y Y ∂n − Y
j Y
1 ∂K
(K +n ) cos(2θ θ )
Y X Y
−S √n K ∂n XY −
j Y Y
2 ∂ ∂
n K X + Y cos(2θ θ )
Y X Y
−S ∂n Y ∂n X −
j (cid:20) Y Y (cid:21)
2 p∂ ∂ 1 n2
X + Y 1 n Y L
Y
−S ∂n Y ∂n X · − 2S − S
j (cid:20) Y Y (cid:21) (cid:20)(cid:18) j(cid:19) j (cid:21)
1 1 ∂L
+2 1 XY 1 2n L+n2 ,
− S · − 2S − S Y Y ∂n
(cid:20) j (cid:21) (cid:20)(cid:18) j(cid:19) j (cid:20) Y (cid:21)(cid:21)(cid:27)
∂H
n˙ = sq = 4G n K sin(2θ θ ) (3.5)
Y Y X Y
− ∂θ − XY −
Y
p
1.4 nnX 5 µX
Y 0 µ
1.2 Y
-5
nY 1 -10
,X µY
n 0.8 ,X -15
µ
-20
0.6
-25
0.4
-30
0.2 -35
0 2e+006 4e+006 6e+006 8e+006 1e+007 1.2e+007 1.4e+007 1.6e+007 1.8e+007 2e+007 0 2e+006 4e+006 6e+006 8e+006 1e+007 1.2e+0071.4e+0071.6e+0071.8e+007 2e+007
time time
Fig. 1. Thetimeevolution ofn (t)andn (t) Fig. 2. The time evolution of θ (t) and θ (t)
X Y X Y
is plotted in the case of the quasi-spin is plotted in the case of the quasi-spin
squeezed state approach. The parame- squeezed state approach. The parame-
ters are taken as G = 1.2, ǫ = 1.0 and ters are taken as G = 1.2, ǫ = 1.0 and
Ω = 2N = 8. The initial values are Ω = 2N = 8. The initial values are
n (t = 0) = 1.3, θ (t = 0) = 0.0, n (t = 0) = 1.3, θ (t = 0) = 0.0,
X X X X
n (t = 0) = 0.35 and θ (t = 0) = 0.0. n (t = 0) = 0.35 and θ (t = 0) = 0.0.
Y Y Y Y
9
Here, the number conservation is satisfied:
N˙ = 2n˙ +4n˙ = 0 . (3.6)
X Y
The time evolution of n (t) and n (t) in Fig.1 and θ (t) and θ (t) in Fig.2 is plotted
X Y X Y
with appropriate initial conditions. The parameters used here are G = 1.2, ǫ = 1.0 and
Ω = 2N = 8. In the su(2)-coherent state approach, n is constant of motion. However,
X
quasi-spin squeezed state approach, n and n oscillate with antiphase, while the total
X Y
particle number is conserved.
§4. Dynamical approach to the ground state energy
Hereafter, weassumethat Y 2 (= n ) 1because Y 2 meansthequantumfluctuations.
Y
| | ≪ | |
Then, K(Y∗Y) and L(Y∗Y) defined in (2.13) and (2.14e), respectively, can be evaluated by
the expansion with respect to Y 2. As a result, we obtain
| |
1 1 7 9
K(Y∗Y) = 1 + 1 + Y∗Y + .
2 − 2S − 2S (2S )2 ···
(cid:18) j(cid:19) (cid:18) j j (cid:19)
1 2 3
L(Y∗Y) = 3 1 1
1 1 − 2S − 2S
− 2Sj (cid:18) j(cid:19)(cid:18) j(cid:19)
48 1 2 3 4
1 1 1 Y∗Y + . (4.1)
−2S (1 1 )2 − 2S − 2S − 2S ···
j − 2Sj (cid:18) j(cid:19)(cid:18) j(cid:19)(cid:18) j(cid:19)
Then, the expectation values for the Hamiltonian, the time-derivative and the number op-
erator can be expressed as
n2
H = ǫ(2n +4n ) G 2S n n2 + X
sq X Y − j X − X 2S
(cid:20) j
1 n
2√2 1 1 X n √n cos(2θ θ )
X Y X Y
− − 2S − 2S −
s j (cid:18) j(cid:19)
10 2 8
+2 2S 1 4n + n + n2 n2 n +O(n3/2) ,
j − − X 2S X 2S X − (2S )2 X Y Y
(cid:18) j j j (cid:19) (cid:21)
(4.2a)
ψ(α,β) i∂ ψ(α,β) = (n θ˙ +n θ˙ ) , (4.2b)
t X X Y Y
h | | i
N = ψ(α,β) Nˆ ψ(α,β) = 2n +4n . (4.2c)
X Y
h | | i
From the time-dependent variational principle (2.23) or (3.4) and (3.5), the following equa-
tions of motion are derived under the above-mentioned approximation :
∂H n 1 n
θ˙ = sq 2ǫ G 2 S n + X √2 1 1 X √n cos(2θ θ )
X j X Y X Y
∂n ≈ − · − 2S − − 2S − S −
X (cid:20) j s j(cid:18) j (cid:19)
10