On the nuclear Meissner effect

Volume 251, n u m b e r 3

PHYSICS LETTERS B

22 November 1990

On the nuclear Meissner effect J.L. Egido Departamento de Fisica Te6rica, Universidad A u t 6 n o m a de Madrid, 17-28049 Madrid, Spain

and P. R i n g P h y s i k d e p a r t m e n t der Technischen Universit?it Mfinchen, W-8046 Garching, F R G

Received 29 June 1990; revised manuscript received 10 September 1990

We briefly c o m m e n t on the nuclear Meissner effect in view o f a recent criticism of Sugawara-Tanabe et al. We show that the criticism is unfounded. On the contrary we argue, that the Sugawara-Tanabe interpretation of the effect is misleading.

In a recent letter by Sugawara-Tanabe and Tanabe [ 1 ] it is argued that in our publications on particle number projection [2,3 ] we have neglected a phase in the evaluation of certain overlap matrix elements and that, due to this fact their conclusion on the nuclear Meissner effect is different from ours. The purpose of this letter is twofold. First, to show that the expression in ref. [ 1 ] for the overlap of two Hartree-Fock-Bogoliubov wavefunctions rotated against each other in gauge space reduces to ours after a simple manipulation, i.e. no phase has been neglected in the calculations of ref. [ 3 ] and second to indicate that the definition of the pairing energy in ref. [ 1 ] deviates from the usual expression found in the scientific literature. This leads in ref. [ 1 ] to a misinterpretation of the results of the number projected calculation. The overlap in question, which is needed for the evaluation of number projected matrix elements is given by

(1)

(3)

where c,,, c~ are particle operators. The wave function [ ~ t ) rotated in gauge space is then given by I ~ t ) ~exp[i(nl/L)N]lq)) =c~, ...c~u [ - ) ,

(4)

with a ~ ~- E UmkC+m"~ ~'mkCm' m

(5)

and

U=exp[i(nl/L)]U,

~'=exp[-iOd/L)]V.

(6)

By means of the Bloch-Messiah theorem [4] we can express the wave functions (2) and (4) in the canonical basis k>O

Let the Hartree-Fock-Bogoliubov function I ~b) be defined by the quasiparticle operators tXk, i.e. I ~l9 ) ~ Ot l . . . OtN I - ) ,

a ~ = E UmkC+m"~ VmkCm, m

Iq~)= I-I ( u k + v k a ~ a ~ ) l - - )

x~ = ( q~lexp[i(nl/L ) ( ~ - N ) ]1 ~ ) =exp[ -i(nl/L)N] (~1 ~t) •

with

(7)

and I~)=

l-I {uk+vkexp[i(2nl/L)]a~a~}l--).

k>O

(2)

0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)

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Volume 251, number 3

PHYSICS LETTERS B

The overlap o f these two wave functions is then given by (@l@t)=

I 1 {u2+exp[i(2nl/L)]V2k}

22 November 1990

( @l@~) = (k~>oeXp[i(nl/L) l) × det(exp [ -i(rd/L) ] U +U

k>0

+ e x p [ i ( r d / L ) ] V + V)~/z, = (k~>o eXp[i(xl/L) ]) det( ~7+ U+ V+ V)l/2 .

(9) Here we have used the invariance o f a d e t e r m i n a n t u n d e r the unitary t r a n s f o r m a t i o n from the canonical basis to the quasiparticle basis o f eq. ( 3 ) . Using the orthogonality relations o f the matrices U a n d V we obtain ( @ l @ l ) =det( l - ( t V

+ V) I/2,

(10)

where

(~ = 1 - e x p [ i ( 2 n l / L ) ].

( 11 )

In the expressions ( 9 ) , ( 1 0 ) there is an undeterm i n e d sign due to the square root. I f we have the signature symmetry, however, we can use G o o d m a n ' s basis [ 5 ] a n d d e c o m p o s e the matrix V + V into two submatrices V t + ) + V ~+ ) a n d V ¢- ) + V ~- ) for different signatures. In the calculations of ref. [ 3 ] we used this s y m m e t r y and found x~=exp[ -i(xl/L)m]

det( 1 - ( i V ~-)+ V ¢ - ) ) .

(12)

This expression coincides exactly ~l with eq. (A. 14) ofref. [ 3 ], and it is used in our c o m p u t e r code. On the other h a n d the expression given by Sugawara-Tanabe et al. is given by ( @1 @~) = exp{i0/tr [ V * V r + ½( U + U - V + V) ]}

×det(cosOt-isinCt(U+U-V+V)]

~/2 ,

(13)

with (~t=r~l/L and ~¢2it is a simple exercise, using the properties o f invariance o f the trace a n d the orthonormality of the matrices U and V, to show that this expression exactly coincides with the expression

(14)

d e d u c e d in eq. ( 9 ) . Concerning the second point, the authors ofref. [ 1 ] define the pairing energy as

a X (C+C+mC.C") mn>O

= G Z ( ( C m+ C + m ) (CnCt,i) "~ ( C ~+C ~ ) ( C n+ Cm ) mn>O +

+

--(CthCn)(CmCpi) ) ,

(15)

and the corresponding expression for the projected case. Usually the second and third term (the Hartree a n d the Fock t e r m ) are neglected a n d supposed to be included as a r e n o r m a l i z a t i o n o f the single-particle energies. In the simple BCS case these terms give the well-known - G Y~k>oV 4, which obviously d e p e n d s on the size o f the configuration space a n d is always different from zero. In our calculations o f ref. [ 3 ], we also neglected these terms ( c o n t r a r y as one could infer after reading ref. [ 1 ] ) and, therefore our pairing energy can go to zero, i.e. the fact that we do not find vanishing values is not due to the inclusion o f the second and third term of eq. ( 15 ), but due to the q u a n t u m - m e c h a n i c a l fluctuations in a number-proj e c t e d wavefunction o b t a i n e d by variation after projection. Finally we would like to point out that an investigation o f a phase-transition can be considerably imp r o v e d by carrying out the variation after the projection because in case o f a vanishing pairing gap automatically the wave function is an eigenstate o f the particle n u m b e r operator and nothing is gained by projection.

References .t The factor exp[i(n/L)lN n] that appears in addition in ref. [ 3 ] is missing here because now we concentrate on unblocked states. ~2 Notice that this expression is again determined only up to a sign. In ref. [ 1] this problem is not mentioned at all. 332

[ 1] K. Sugawara-Tanabe and K. Tanabe, Phys. Lett. B 238 (1990) 15. [2] J.L. Egido and P. Ring, Nucl. Phys. A 383 (1982) 189. [3] J.L. Egido and P. Ring, Nucl. Phys. A 388 (1982) 19. [4] C. Bloch and A. Messiah, Nucl. Phys. 39 (1962) 95. [5] A.L. Goodman, Nucl. Phys. A 256 (1976) 113.