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Proof of the extended Bloch-Messiah theorem in the thermal Hartree-Fock-Bogoliubov theory
Volume 247, number 2,3
PHYSICS LETTERS B
13 September 1990
Proof of the extended Bloch-Messiah theorem in the thermal Hartree-Fock-Bogoliubov theory K. Tanabe Department of Physics, Saitama University, Urawa, Saitama 338, Japan
and K. Sugawara-Tanabe Department of Physics, University of Tokyo, Tokyo 113, Japan Received 21 May 1990
It is shown that an application of the thermo field dynamics ( T F D ) to the thermal Hartree-Fock-Bogoliubov (THFB) theory leads to the extended form of the Bloch-Messiah theorem for the finite temperature formalism. The generalized density matrix defined on the enlarged operator space including tilded operators takes into account the thermal fluctuation of fermion number for the system under consideration.
1. The thermal Hartree-Foek-Bogoliubov (THFB) approximation
Straightforward extension of the Hartree-FockBogoliubov (HFB) approximation [1-3 ] within the statistical recipe gives the thermal HFB (THFB) approximation scheme for the many-body system thermally equilibrated at finite temperature [4-6]. The usefulness of the self-consistent solution to the THFB equation has been demonstrated for the description of the average properties of deformed nuclei in the excited region above the yrast line [4,7-10]. Furthermore, the thermal random phase approximation (TRPA) equation [11,12] based on the THFB solution has had success in the application to the giantdipole resonances built on hot rotating nuclei [1315 ]. It is shown that the TRPA equation is extended to take into account the effect of thermal fluctuations [ 16] by applying the thermo field dynamics (TFD) proposed by Takahashi and Umezawa [ 17,18 ]. The purpose of the present paper is to clarify a basic property of the THFB approximation that already includes the thermal fluctuation effect of particle 202
number, which can be directly demonstrated by the application of the TFD. First we briefly review the THFB approximation. In the variational derivation of the THFB equation [5], the variational parameters Aku, Bku, AT,u, BT,u and E u are introduced through the generalized Bogoliubov transformation
subject to WtW= WW*=I.
(2)
The trial density matrix is given by ~=
e x p ( - f l ~ u Euatuau) Tr[exp( - f l Z u a*uau) ] "
(3)
The standard form of the THFB equation is derived from the variational principle applied to the grand potential in case of a finite system like a nucleus, fiF= ~( ( B - 2 p 2 - 2 , N - O)Yx) - S T ) = 0 ,
(4)
Elsevier Science Publishers B.V. (North-Holland)
Volume 247, number 2,3
PHYSICS LETTERSB
S = - k T r ( f f l n g,),
(4 cont'd)
where < ) denotes the ensemble average,/~ is the microscopic hamiltonian and S the entropy. Three Lagrange multipliers 2p, 2n and o~ are determined by three constraints for ensemble averages of the proton (neutron) number operator 2 (fi/) and angular momentum operator a~x, = Z ,
= N ,
2. Application of the thermo field dynamics (TFD) In the thermo field dynamics ( T F D ) tilded operators t~u and a*u are introduced in order to enlarge the operator space. From now on this doubling is expressed by a® T-~a ,
(5)
Then, the single-particle density and the pair tensor are expressed as straightforward extensions of the zero-temperature formulas, Pkt = ( C~Ck ) = [B*( 1 - f ) Bt~+ AfA * ]kl ,
(6a)
Xkl = = [B*( 1 - f ) A t r +AfB* ]kt,
(6b)
(12)
with aul0> =~ul0>=0.
(13)
In this formalism the temperature-dependent vacuum is introduced by 10(/~)> = e -G 10>, G = ~ 0u(auOtu * "* - a~u a u ) ,
where
(14)
U
where
1 fu =- ( °t*u°tu) - exp([3Eu ) + 1
(7)
is the quasiparticle distribution function. The generalized density matrix defined in the same manner as the HFB approximation, tc+ I - p *
'
(8)
cannot be a projection, because R - R 2 remains finite
at T# 0, i.e. R-R2=
(9)
Wf( 1 -f) W +.
Note R
1®6~--,6l ,
IO>®t0>-,10>,
or/x//x//x//x//x//x/~+ 1) .
(Jx>=I
13 September 1990
=
(;)
f,
(') (.')
R A*
=
A*
(l-f).
(10)
sinOu=fl/2-gu,
cosOu=(1-fu)~/2-gu
T r ( R - R 2) = Tr [f( 1 - f ) ] ,
( 11 )
which is simply the thermal fluctuation of fermion number. Therefore, the Bloch-Messiah theorem [ 19 ] proved with the relation R - R 2 = 0 seems not to hold for the THFB scheme as pointed out in ref. [ 5 ]. This means that the pair tensor x does not take a canonical form even in the quasiparticle picture in which the density p is diagonalized.
(15)
Hence, the quasiparticle operators o~u and ~u are related to the new operators flu and flu by the special Bogoliubov transformation, °tu = eG flu e - a =guflu + gu/~*u,
(16a)
t2u = e G / ~ e-G=guffu--gufltu.
(16b)
Note that
/~u Io(/~) > --/~,~ I 0 (/~) > = o .
(17)
An essential feature of the TFD is that the ensemble average of any operator 0 is expressed in terms of the vacuum expectation value, i.e. <0> = T r ( f f 0 ) =
Because of (2), the trace of (9) becomes
.
<0(/~) 1010(/~) >.
(18)
Thus, the single-particle operators ck and gk are related to flu and flu by
ct
=I,V
,
(19)
with
(20) 203
Volume 247, number 2,3
- B*g
Ag ] '
PHYSICS LETTERS B
-B*g
Vtfiv=(P; iP pOip ) ,
AgJ' (20cont'd)
WW*= W* gT=I.
Vt~V=(K?i Q x;iQ).
(27)
(21)
Since the whole formalism is developed on the vacuum 10(fl) ) defined for a given temperature, a parallel of this extended form of the THFB scheme holds to the HFB scheme. As a direct consequence, the extended single-particle density matrix and pair tensor are to be defined by
fi=B.Bt~=(2p
13 September 1990
P) =fit,
The hermitean matrices p + iP and p - iP can be diagonalized by two different unitary transformations u and v, but have a common set of real eigenvalues, i.e.
(;* vO,)(P; iP
pOip)(;
0v)
heed~r
(22a) with
.)
22b,
where p and K are the same as defined by (6). In (22) new quantities P and Q are given by
P= Ag~Bt~- B*ggA*= - p,, Q =AggA t~_ B*g~B*= Q'~.
( 23a ) ( 23b )
Thus, the generalized density matrix R of (8) is replaced by the extended form, with/~ and g in (22), A=
~,
1-if*
24,
'
A-A2=0 .
(25)
This suggests the possibility of the extension of the Bloch-Messiah theorem to the THFB approximation.
Applying the unitary transformation
~, ein/4 1
( e in/4
v,= ~ \e-i":4
From (25) we can show that
[U*(~c--iQ)V*]kr(hkk--hrr) = 0 , [v*(~+iQ)u*]~t(h~r-h,) = 0 .
204
(30b)
Hence, we have seen that, if the hkk are not degenerate, all the elements [u*(x--iQ)V*]kr and [v*(KiQ)u* ];t vanish except for
[U*( ~c--iQ )v* ]k~=--tk~, [V*(~:+iQ)u*]~k--t;k= --tk~.
ein/4 "~ e _ i ~ / 4 ] = r tr ' e - i~/4"~ ei~/4 j = v* ,
(;* v°,)(K°iQ KoiQ)(o* :(3. 12I= •
i)
(26)
~,:c~cj+½ E VOk:CfCfC:k, ijkl
(31)
(32)
(33)
is given by ( / 4 ) = T r (~p+ '~rp+ ~K*) ,
to d and if, we get
(30a)
This completes the proof of the extended form for the Bloch-Messiah theorem. In the THFB approximation, the ensemble average of the hamiltonian with two-body interactions
3. Extension of the Bloch-Messiah theorem
1 ( e -in/4
(29)
Therefore, the canonical form is attained also for the antisymmetric tensor ~ given by (27), i.e.
which satisfies
V= ~
hk/,=hff=h~k=h};.
where
(34)
Volume 247, number 2,3
PHYSICS LETTERS B
r,j= Y V,,j~l'k~, a,j, =½ Z V,j~Xk~, kl
(35)
References
hl
are the self-energy a n d the p a i r i n g gap m a t r i c e s expressed in the original single-particle picture. Since ( 2 8 ) a n d ( 32 ) yield, respectively,
p+iP=uhu*,
p-iP=vhv t ,
(36a)
x + i Q = v t u tr,
(36b)
and
x-iQ=utv
13 September 1990
tr,
it is o b v i o u s that the e x p e c t a t i o n value ( 3 4 ) c a n be expressed in t e r m s o f the d i a g o n a l i z e d o r d e r p a r a m eters h a n d t by the r e p l a c e m e n t
P = ½( u h u t + v h v *) , I¢= ½( Utvtr-F vtU `r ) .
(37)
I n c o n c l u s i o n , we have f o u n d the T H F B approxim a t i o n to be a s c h e m e that takes i n t o a c c o u n t the effect o f the t h e r m a l f l u c t u a t i o n o f particle n u m b e r . I n o r d e r to d e f i n e the correct f o r m o f the d e n s i t y m a t r i x a n d the pair tensor, the o p e r a t o r space s h o u l d be d o u b l e d b y the i n c l u s i o n o f t i l d e d operators. T h i s m u s t be o n e o f the s u p p o r t i n g e v i d e n c e s o f the T F D formalism.
[ 1 ] M. Baranger, Phys. Rev. 122 ( 1961 ) 992; 130 (1963) 1244. [2] H.J. Mang, Phys. Rep. 18 (1975) 325. [ 3 ] P. Ring and P. Schuck, The nuclear many-body problem (Springer, Berlin, 1980 ) Ch. 7, and references therein. [4] K. Tanabe and K. Sugawara-Tanabe, Phys. Lett. B 97 (1980) 337. [ 5 ] K. Tanabe, K. Sugawara-Tanabe and H.J. Mang, Nucl. Phys. A 357 (1981) 20. [6] A.L. Goodman, Nucl. Phys. A 352 ( 1981 ) 30. [ 7 ] K. Sugawara-Tanabe, K. Tanabe and H.J. Mang, Nucl. Phys. A 357 (1981) 45. [8] A.L. Goodman, Nucl. Phys. A 352 (1981) 45. [9] K. Tanabe and K. Sugawara-Tanabe, Nucl. Phys. A 390 (1982) 385. [ 10] A.L. Goodman, Phys. Rev. C 39 (1989) 2478. [ 11 ] J.L. Egido and P. Ring, Phys. Rev. C 25 (1982) 3239. [12] K. Tanabe and K. Sugawara-Tanabe, Phys. Lett. B 135 (1984) 353. [ 13 ] K. Tanabe and K. Sugawara-Tanabe, Prog. Theor. Phys. 76 (1986) 356. [ 14] K. Sugawara-Tanabe and K. Tanabe, Prog. Theor. Phys. 76 (1986) 1272. [15]K. Sugawara-Tanabe and K. Tanabe, Phys. Lett. B 192 (1987) 268. [16] K. Tanabe, Phys. Rev. C 37 (1988) 2802. [17]Y. Takahashi and H. U mezawa, Coll. Phenom. 2 (1975) 55. [ 18 ] H. Umezawa, H. Matsumoto and M. Tachiki, Thermo field dynamics and condensed states (North-Holland, Amsterdam, 1982). [ 19] C. Bloch and A. Messiah, Nucl. Phys. 39 (1962) 95.
205
PHYSICS LETTERS B
13 September 1990
Proof of the extended Bloch-Messiah theorem in the thermal Hartree-Fock-Bogoliubov theory K. Tanabe Department of Physics, Saitama University, Urawa, Saitama 338, Japan
and K. Sugawara-Tanabe Department of Physics, University of Tokyo, Tokyo 113, Japan Received 21 May 1990
It is shown that an application of the thermo field dynamics ( T F D ) to the thermal Hartree-Fock-Bogoliubov (THFB) theory leads to the extended form of the Bloch-Messiah theorem for the finite temperature formalism. The generalized density matrix defined on the enlarged operator space including tilded operators takes into account the thermal fluctuation of fermion number for the system under consideration.
1. The thermal Hartree-Foek-Bogoliubov (THFB) approximation
Straightforward extension of the Hartree-FockBogoliubov (HFB) approximation [1-3 ] within the statistical recipe gives the thermal HFB (THFB) approximation scheme for the many-body system thermally equilibrated at finite temperature [4-6]. The usefulness of the self-consistent solution to the THFB equation has been demonstrated for the description of the average properties of deformed nuclei in the excited region above the yrast line [4,7-10]. Furthermore, the thermal random phase approximation (TRPA) equation [11,12] based on the THFB solution has had success in the application to the giantdipole resonances built on hot rotating nuclei [1315 ]. It is shown that the TRPA equation is extended to take into account the effect of thermal fluctuations [ 16] by applying the thermo field dynamics (TFD) proposed by Takahashi and Umezawa [ 17,18 ]. The purpose of the present paper is to clarify a basic property of the THFB approximation that already includes the thermal fluctuation effect of particle 202
number, which can be directly demonstrated by the application of the TFD. First we briefly review the THFB approximation. In the variational derivation of the THFB equation [5], the variational parameters Aku, Bku, AT,u, BT,u and E u are introduced through the generalized Bogoliubov transformation
subject to WtW= WW*=I.
(2)
The trial density matrix is given by ~=
e x p ( - f l ~ u Euatuau) Tr[exp( - f l Z u a*uau) ] "
(3)
The standard form of the THFB equation is derived from the variational principle applied to the grand potential in case of a finite system like a nucleus, fiF= ~( ( B - 2 p 2 - 2 , N - O)Yx) - S T ) = 0 ,
(4)
Elsevier Science Publishers B.V. (North-Holland)
Volume 247, number 2,3
PHYSICS LETTERSB
S = - k T r ( f f l n g,),
(4 cont'd)
where < ) denotes the ensemble average,/~ is the microscopic hamiltonian and S the entropy. Three Lagrange multipliers 2p, 2n and o~ are determined by three constraints for ensemble averages of the proton (neutron) number operator 2 (fi/) and angular momentum operator a~x,
2. Application of the thermo field dynamics (TFD) In the thermo field dynamics ( T F D ) tilded operators t~u and a*u are introduced in order to enlarge the operator space. From now on this doubling is expressed by a® T-~a ,
(5)
Then, the single-particle density and the pair tensor are expressed as straightforward extensions of the zero-temperature formulas, Pkt = ( C~Ck ) = [B*( 1 - f ) Bt~+ AfA * ]kl ,
(6a)
Xkl =
(6b)
(12)
with aul0> =~ul0>=0.
(13)
In this formalism the temperature-dependent vacuum is introduced by 10(/~)> = e -G 10>, G = ~ 0u(auOtu * "* - a~u a u ) ,
where
(14)
U
where
1 fu =- ( °t*u°tu) - exp([3Eu ) + 1
(7)
is the quasiparticle distribution function. The generalized density matrix defined in the same manner as the HFB approximation, tc+ I - p *
'
(8)
cannot be a projection, because R - R 2 remains finite
at T# 0, i.e. R-R2=
(9)
Wf( 1 -f) W +.
Note R
1®6~--,6l ,
IO>®t0>-,10>,
or/x//x//x//x//x//x/~+ 1) .
(Jx>=I
13 September 1990
=
(;)
f,
(') (.')
R A*
=
A*
(l-f).
(10)
sinOu=fl/2-gu,
cosOu=(1-fu)~/2-gu
T r ( R - R 2) = Tr [f( 1 - f ) ] ,
( 11 )
which is simply the thermal fluctuation of fermion number. Therefore, the Bloch-Messiah theorem [ 19 ] proved with the relation R - R 2 = 0 seems not to hold for the THFB scheme as pointed out in ref. [ 5 ]. This means that the pair tensor x does not take a canonical form even in the quasiparticle picture in which the density p is diagonalized.
(15)
Hence, the quasiparticle operators o~u and ~u are related to the new operators flu and flu by the special Bogoliubov transformation, °tu = eG flu e - a =guflu + gu/~*u,
(16a)
t2u = e G / ~ e-G=guffu--gufltu.
(16b)
Note that
/~u Io(/~) > --/~,~ I 0 (/~) > = o .
(17)
An essential feature of the TFD is that the ensemble average of any operator 0 is expressed in terms of the vacuum expectation value, i.e. <0> = T r ( f f 0 ) =
Because of (2), the trace of (9) becomes
.
<0(/~) 1010(/~) >.
(18)
Thus, the single-particle operators ck and gk are related to flu and flu by
ct
=I,V
,
(19)
with
(20) 203
Volume 247, number 2,3
- B*g
Ag ] '
PHYSICS LETTERS B
-B*g
Vtfiv=(P; iP pOip ) ,
AgJ' (20cont'd)
WW*= W* gT=I.
Vt~V=(K?i Q x;iQ).
(27)
(21)
Since the whole formalism is developed on the vacuum 10(fl) ) defined for a given temperature, a parallel of this extended form of the THFB scheme holds to the HFB scheme. As a direct consequence, the extended single-particle density matrix and pair tensor are to be defined by
fi=B.Bt~=(2p
13 September 1990
P) =fit,
The hermitean matrices p + iP and p - iP can be diagonalized by two different unitary transformations u and v, but have a common set of real eigenvalues, i.e.
(;* vO,)(P; iP
pOip)(;
0v)
heed~r
(22a) with
.)
22b,
where p and K are the same as defined by (6). In (22) new quantities P and Q are given by
P= Ag~Bt~- B*ggA*= - p,, Q =AggA t~_ B*g~B*= Q'~.
( 23a ) ( 23b )
Thus, the generalized density matrix R of (8) is replaced by the extended form, with/~ and g in (22), A=
~,
1-if*
24,
'
A-A2=0 .
(25)
This suggests the possibility of the extension of the Bloch-Messiah theorem to the THFB approximation.
Applying the unitary transformation
~, ein/4 1
( e in/4
v,= ~ \e-i":4
From (25) we can show that
[U*(~c--iQ)V*]kr(hkk--hrr) = 0 , [v*(~+iQ)u*]~t(h~r-h,) = 0 .
204
(30b)
Hence, we have seen that, if the hkk are not degenerate, all the elements [u*(x--iQ)V*]kr and [v*(KiQ)u* ];t vanish except for
[U*( ~c--iQ )v* ]k~=--tk~, [V*(~:+iQ)u*]~k--t;k= --tk~.
ein/4 "~ e _ i ~ / 4 ] = r tr ' e - i~/4"~ ei~/4 j = v* ,
(;* v°,)(K°iQ KoiQ)(o* :(3. 12I= •
i)
(26)
~,:c~cj+½ E VOk:CfCfC:k, ijkl
(31)
(32)
(33)
is given by ( / 4 ) = T r (~p+ '~rp+ ~K*) ,
to d and if, we get
(30a)
This completes the proof of the extended form for the Bloch-Messiah theorem. In the THFB approximation, the ensemble average of the hamiltonian with two-body interactions
3. Extension of the Bloch-Messiah theorem
1 ( e -in/4
(29)
Therefore, the canonical form is attained also for the antisymmetric tensor ~ given by (27), i.e.
which satisfies
V= ~
hk/,=hff=h~k=h};.
where
(34)
Volume 247, number 2,3
PHYSICS LETTERS B
r,j= Y V,,j~l'k~, a,j, =½ Z V,j~Xk~, kl
(35)
References
hl
are the self-energy a n d the p a i r i n g gap m a t r i c e s expressed in the original single-particle picture. Since ( 2 8 ) a n d ( 32 ) yield, respectively,
p+iP=uhu*,
p-iP=vhv t ,
(36a)
x + i Q = v t u tr,
(36b)
and
x-iQ=utv
13 September 1990
tr,
it is o b v i o u s that the e x p e c t a t i o n value ( 3 4 ) c a n be expressed in t e r m s o f the d i a g o n a l i z e d o r d e r p a r a m eters h a n d t by the r e p l a c e m e n t
P = ½( u h u t + v h v *) , I¢= ½( Utvtr-F vtU `r ) .
(37)
I n c o n c l u s i o n , we have f o u n d the T H F B approxim a t i o n to be a s c h e m e that takes i n t o a c c o u n t the effect o f the t h e r m a l f l u c t u a t i o n o f particle n u m b e r . I n o r d e r to d e f i n e the correct f o r m o f the d e n s i t y m a t r i x a n d the pair tensor, the o p e r a t o r space s h o u l d be d o u b l e d b y the i n c l u s i o n o f t i l d e d operators. T h i s m u s t be o n e o f the s u p p o r t i n g e v i d e n c e s o f the T F D formalism.
[ 1 ] M. Baranger, Phys. Rev. 122 ( 1961 ) 992; 130 (1963) 1244. [2] H.J. Mang, Phys. Rep. 18 (1975) 325. [ 3 ] P. Ring and P. Schuck, The nuclear many-body problem (Springer, Berlin, 1980 ) Ch. 7, and references therein. [4] K. Tanabe and K. Sugawara-Tanabe, Phys. Lett. B 97 (1980) 337. [ 5 ] K. Tanabe, K. Sugawara-Tanabe and H.J. Mang, Nucl. Phys. A 357 (1981) 20. [6] A.L. Goodman, Nucl. Phys. A 352 ( 1981 ) 30. [ 7 ] K. Sugawara-Tanabe, K. Tanabe and H.J. Mang, Nucl. Phys. A 357 (1981) 45. [8] A.L. Goodman, Nucl. Phys. A 352 (1981) 45. [9] K. Tanabe and K. Sugawara-Tanabe, Nucl. Phys. A 390 (1982) 385. [ 10] A.L. Goodman, Phys. Rev. C 39 (1989) 2478. [ 11 ] J.L. Egido and P. Ring, Phys. Rev. C 25 (1982) 3239. [12] K. Tanabe and K. Sugawara-Tanabe, Phys. Lett. B 135 (1984) 353. [ 13 ] K. Tanabe and K. Sugawara-Tanabe, Prog. Theor. Phys. 76 (1986) 356. [ 14] K. Sugawara-Tanabe and K. Tanabe, Prog. Theor. Phys. 76 (1986) 1272. [15]K. Sugawara-Tanabe and K. Tanabe, Phys. Lett. B 192 (1987) 268. [16] K. Tanabe, Phys. Rev. C 37 (1988) 2802. [17]Y. Takahashi and H. U mezawa, Coll. Phenom. 2 (1975) 55. [ 18 ] H. Umezawa, H. Matsumoto and M. Tachiki, Thermo field dynamics and condensed states (North-Holland, Amsterdam, 1982). [ 19] C. Bloch and A. Messiah, Nucl. Phys. 39 (1962) 95.
205