Fluctuations and Odd-Even Effects in Small Superfluid Systems

Annals of Physics 275, 126 (1999) Article ID aphy.1999.5914, available online at http:www.idealibrary.com on

Fluctuations and Odd-Even Effects in Small Superfluid Systems R. Rossignoli and N. Canosa Departamento de F@ sica, Universidad Nacional de La Plata, c.c. 67, 1900 La Plata, Argentina

and P. Ring Physik-Department der Technischen Universitat Munchen, D-85748 Garching, Germany Received June 26, 1998; revised November 9, 1998

Fluctuations and odd-even effects in small superfluid systems at finite temperature are investigated by means of the static path plus RPA approximation. A general derivation of this method is presented, which allows a straightforward implementation in statistical ensembles with fixed number parity (NP). A significant smoothing of the superconducting to normal transition is obtained in systems where the gap is comparable to the level spacing, as well as a decrease of pairing correlations in the odd case. Results are shown for a schematic model, where an excellent agreement with exact canonical results is obtained with the present method, and then for a heavy nucleus. Comparison with NP projected BCS and BCS+RPA results is made. An effective BCS+RPA approach which remains smooth in transitional regions is also derived.  1999 Academic Press

I. INTRODUCTION Small correlated quantum systems exhibit important fluctuation phenomena, which lead to significant deviations from the predictions of conventional mean field approximations (MFA). In particular, in finite systems at finite temperature, the sharp phase transitions displayed by the mean field become increasingly washed out as the size of the system decreases, which can be attributed to the effects of growing fluctuations in the relevant order parameters [1, 2]. On the other hand, in small systems with fixed particle number, one can also expect important deviations from the predictions of standard grand canonical (GC) statistics due to canonical corrections. These facts have recently acquired special relevance in solid state physics with relation to the development of diverse mesoscopic structures, as for instance superconducting islands [3] and ultrasmall superconducting metallic grains [4], where odd-even effects, i.e., differences between systems with odd and even particle number, are non-negligible and play an important role [511]. Most theoretical studies at finite temperature have been based, however, on the extension of the BCS approximation to an ensemble with fixed number parity [5], rigorously derived in [11] 1 0003-491699 30.00 Copyright  1999 by Academic Press All rights of reproduction in any form reserved.

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ROSSIGNOLI, CANOSA, AND RING

from the variational principle, which, although providing the basic picture of oddeven differences, omits the important effects of large amplitude fluctuations in transitional regions. Fluctuations and odd-even effects at finite temperature are also relevant in other small systems like warm finite nuclei [12], of renewed interest due to the recent development of improved #-ray detectors, where a statistical description becomes feasible due to the high density of states involved and where number parity projection at finite temperature was considered already in [13]. A convenient microscopic framework to deal with fluctuations is provided by the path integral representation of the partition function obtained with the Hubbard Stratonovich transformation [14]. In this context, large amplitude static fluctuations around the mean field can be included by integrating exactly over the static components of the auxiliary fields, which leads to the static path approximation (SPA), introduced in [1] to describe finite size effects in small superconductors, and later introduced in nuclear physics [1518]. Small amplitude quantum fluctuations can then be incorporated by integration over the remaining components in the gaussian approximation, which leads to the SPA+RPA [1923], to be denoted for brevity as correlated (C) SPA. This approach has been originally derived in the GC ensemble [1921]. The aim of this work is first to present, in Section II and Appendix A, a general derivation of this method, which enables a straightforward exact implementation in statistical ensembles with fixed particle number parity (NP) for a general representation of a two-body interaction. We also rederive the general mean field+RPA approach to the partition function [24], in order to make it applicable in NP projected statistics. In Section III we implement these methods for a pairing interaction in NP projected ensembles. Some results were given in [25]. We first briefly review the NP projected BCS equations derived from the saddle point approximation to the SPA [5] and the concomitant phase diagram in even and odd systems [8, 11]. We then consider a schematic model, where results obtained from the exact canonical partition function are compared with those of NP projected SPA, CSPA, BCS, and BCS+RPA. We also derive from the CSPA an effective BCS+RPA approach, which contains the essential features of the full CSPA and remains smooth in transitional regions. The previous methods are then applied to the description of odd-even effects in a deformed heavy nucleus. Finally, conclusions are drawn in Section IV.

II. FORMALISM A. General CSPA Approach For a Hamiltonian of the form H=H 0 & 12 : v & Q 2& , &

(1)

3

SMALL SUPERFLUID SYSTEMS

where H 0 , Q & , are bounded hermitian operators, the HubbardStratonovich transformation allows us to express the partition function (PF) as the auxiliary field path integral [14]

{

|

|

;

Z#Tr exp[&;H]= D[x] Tr T exp &

=

d{ H[x({)] , 0

(2)

where x=[x & ] is a set of auxiliary fields, T denotes time ordering and H(x)=H 0 +: &

x&2 &x & Q & , 2v &

(3)

is a linearized Hamiltonian. The normalization  D[x] exp[& &  ;0 d{ x 2& ({)2v & ] =1 is assumed. By means of a Fourier expansion of the fields in the interval (0, ;), x &({)=x & + : x &n e &i|n {,

| n =2?n;,

(4)

n{0

Eq. (2) can be written explicitly as

|

Z= d(x) ` d(x n ) Tr T n{0

{

|

;

x &&n x &n &x &n e &i|n { Q & 2v & &, n{0

_

d{ H(x) + :

_exp &

0

&= ,

(5)

with d(x)=> & (;2?v & ) 12 dx & , d(x n )=> & ( ;2?v & i) 12 dx &n (n{0). In the CSPA [19, 21, 22], one retains the exact integration over the static coefficients x & , which represent the time average of x &({) in the interval (0, ;), while the remaining variables x &n are integrated in the gaussian approximation for each value of x & . The aim is to take into account large amplitude static fluctuations of x &({), which become relevant in small systems and critical regions, including at the same time small amplitude quantum-like fluctuations. The sharp phase transitions that arise in the ``mean field'' approximation, i.e., in a full saddle point evaluation of (2) (see Subsection II.C) become in this way smooth in a small system. After expanding the logarithm of the trace in (5) up to second order in x &n{0 , we obtain, using Eqs. (76)(79) of Appendix A for H Ä H(x), Q n Ä  & x &n Q & , n{0,

|

Z CSPA = d(x) Z(x) C RPA(x),

(6)

where Z(x)=Tr exp[&;H(x)]=: e &;EK, K

(7)

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ROSSIGNOLI, CANOSA, AND RING 

C RPA(x)=

|

{

` d(x n ) d(x &n ) exp &; : x &&n n=1

$ &&$

_ v +R

&&$

& =

(x, i| n ) x &$n

&

&, &$

(8)



= ` Det[$ &&$ +v & R &&$(x, i| n )] &1,

(9)

n=1

R &&$(x, i|)= : ( K| Q & |K$)( K$| Q &$ |K) K{K$

P K &P K$ , E K &E K$ +i|

(10)

with |K) the eigenstates of the static Hamiltonian H(x), ( K| H(x) |K$) =$ KK$ E K , and P K =e &;EK Z(x)=&; &1  ln Z(x)E K

(11)

the corresponding statistical weights. Equation (6) can be applied provided the determinants in (9) are positive definite \x. The SPA is recovered if the factor C RPA(x) is omitted in (6) and is exact when the operators Q & commute with each other and with H 0 (in which case [Q & , H(x)]=0 and C RPA(x)=1). Integrals over static variables associated with repulsive terms (v & <0 in (3)) are to be performed along the imaginary axes and can be evaluated in the saddle point approximation (see Subsection II.C and Refs. [22, 23] for more details). Equations (7)(10) hold for traces taken in any finite closed representation of the operators [H 0 , Q & ]. In many fermion systems, they can be applied in a finite configuration space to any type of operators H 0 , Q & , and are valid in the GC ensemble (where H Ä H&+N, with + the chemical potential and N the particle number operator) as well as in restricted ensembles defined by projectors P which commute with H 0 and the operators Q & (see Appendix A). B. Extended One-Body Operators and Ensembles with Fixed Number Parity Let us consider now the case where H 0 and Q & are fermion operators of the form 1 20 - 02 1 c Q=: [Q 11 ij c i c j + 2 (Q ij c i c j +Q ij c i c j )]=q+ 2 (c c) Q( c - ),

(12)

i, j

q= 12 : Q 11 ii , i

Q=

\

Q 11 Q 20 , Q 02 &(Q 11 ) t

+

(13)

where (c -c)=( } } } c -i } } } c i } } } ) denotes a set of fermion creation and annihilation operators. Within a finite configuration space, any two-body Hamiltonian can be written in the (non-unique) separable form (1) in terms of operators of the general type (12) [26]. In this case, H(x)=h(x)+ 12 (c -c) H(x)

c , c-

\ +

H(x)=H0 &+N&: x & Q& , &

(14)

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SMALL SUPERFLUID SYSTEMS

where h(x)=h 0 & 12 +0+ & (x 2& 2v & )&x & q & , with 0 the dimension of the single particle space. By means of a Bogoliubov transformation [27] c

a

\c + =W(x) \a + , -

-

W -(x) W(x)=1,

determined such that H  (x)=W -(x) H(x) W(x) is diagonal, we can express H(x) and Q & in terms of quasiparticle operators a -k , a k as H(x)=h(x)+ 12 : * k(a -k a k &a k a -k ),

(15)

k

Q & =q & + 12 (a -a) Q &( aa- ),

Q & =W -(x) Q& W(x),

(16)

where * k are the eigenvalues of H(x) (which come in pairs of opposite sign) and the sum in (15) runs over the positive * k . The many-body eigenstates of H(x) are now independent quasiparticle states |K ) => k (a -k ) nk |0), and Eq. (10) can be written in terms of the elements Q &kk$ of the extended matrices Q & (see Appendix A) as R &&$(x, i|)=

1 p k &p k$ :$ Q & Q &$ , 2 k{k$ kk$ k$k * k &* k$ +i|

(17)

where the prime indicates the sum over all elements k{k$ of the extended matrices and p k =&2; &1  ln Z(x)* k

(18)

are the quasiparticle occupation probabilities. In (18) we have considered * k and * &k #&* k independent variables, such that p k =( a -k a k ) x # K P K (K | a -k a k |K), p &k =( a k a -k ) x =1& p k . The type of statistics will determine Z(x) and p k . In an ordinary GC ensemble, Z(x)=e &;h(x) ` 2 cosh k

pk=

1

_2 ;* & #Z (x), k

1 #p 0k . 1+e ;*k

0

(19)

(20)

Equation (17) is also valid in a restricted GC ensemble with fixed particle number parity, as the NP projector P _ = 12 (1+_e i?N ),

_=\1,

(21)

6

ROSSIGNOLI, CANOSA, AND RING

commutes with operators of the form (13) and hence with H(x). We have e i?N |K) =_ 0(&) NK |K ), with N K = k n k the number of quasiparticles and _ 0 the number parity of the quasiparticle vacuum [27]. Therefore, 1 Z(x)=Tr P _ exp[&;H(x)]= Z 0(x)[1+_$1], 2 1=` tanh k

pk=

1

_2 ;* & =` (1&2p ), 0 k

k

(22) (23)

k

1 1 _$1 & 1+e ;*k 1+_$1 sinh[ ;* k ]

=p 0k &

2_$1 p 0k(1& p 0k ) , 1+_$1 1&2p 0k

(24)

where _$=__ 0 . Thus, the only effect of NP projection in (17) is the replacement of the FermiDirac probabilities (20) with the modified probabilities (24). The correction factor 1 becomes small when the temperature T=1; is larger than the lowest quasiparticle energy (i.e., ;* min < <1). With the notation * : #* k &* k$ , p : #p k & p k$ , Q &: #Q &kk$ , with : labelling all pairs k{k$ of the matrices Q & , Eq. (17) becomes 12  : Q &: Q &$* : p : (* : +i|) and the determinants in (9) can be rewritten as Det[$ &&$ +v & R &&$(x, i| n )]=

Det[A ::$ +i| n $ ::$ ] | 2 +| 2n , = ` 2: Det[(* : +i| n ) $ ::$ ] :>0 * : +| 2n

1  &:$ , A ::$ =* : $ ::$ + p : : v & Q &* : Q 2 &

(25) (26)

where the | :(x) are the finite temperature quasiparticle RPA energies [28] around H(x), defined here as the eigenvalues of the matrix (26) [21, 23], which can be determined from Det[$ &&$ +v & R &&$(x, | : )]=0

(27)

for the non-trivial cases | : {* : . As R &&$(x, |)=R &$&(x, &|), the | : 's come in pairs of opposite sign and the product in (25) runs over those of a definite sign 2 2 only. Using then Euler's formula, >  n=1 (1+z n )=sinh[?z]?z, we can evaluate (9) as [21, 22] | : sinh[;* : 2] . :>0 * : sinh[;| : 2]

C RPA(x)= `

(28)

C. Mean Field+RPA (CMFA) Let us consider now the full saddle point evaluation of (2), where all variables, including the static coefficients x & , are simultaneously integrated in the gaussian

7

SMALL SUPERFLUID SYSTEMS

approximation. The (static) stationary points of Z(x) are determined by the selfconsistent equations &

v &  ln Z(x) =x & &v & ( Q & ) x =0, ; x &

(29)

where ( Q & ) x = K P K ( K| Q & |K) is the thermal average corresponding to H(x). The full gaussian approximation around a solution of (29) leads, using again Eqs. (76)(79) of Appendix A, to Z CMFA =Z(x) C 0(x) C RPA(x),

(30)

where C 0(x) is the determinant associated with the static variables, $ &&$ 1 +R 0&&$(x) x$&$ C 0(x)= d(x$) exp & ; : x$& 2 &, &$ v&

{

|

_

& =

=Det[$ &&$ +v & R 0&&$(x)] &12, R 0&&$(x)=&

(31)

 ( Q&) x P &P K$ = : ( K| Q & |K$)( K$| Q &$ |K) K x &$ E K &E K$ K{K$

_

&

&; : P K ( K| Q & |K)( K| Q &$ |K) &(Q & ) x ( Q &$ ) x . K

(32)

Here (P K &P K$ )(E K &E K$ ) Ä &;P K if E K Ä E K$ . Note that (32) differs from the limit of (10) for | Ä 0 due to the diagonal terms, which arise from the variation of the weights P K . In a basis where ( K| Q &$ |K$) =$ KK$ ( K| Q &$ |K) if E K =E K$ , R 0&&$(x)=R &&$(x, 0)&: K

P K ( K| Q & |K). x &$

(33)

For the operators (13), Eqs. (29) represent the finite temperature HartreeBogoliubov (FTHB) equations. When evaluated at the fundamental solution of (29), Z(x) becomes the FTHB PF, while Eq. (30) can be regarded as the FTHB+RPA PF. In this case (see Appendix A), ( Q & ) x =q & + R 0&&$(x)=

1 $ & pk , : Q 2 k kk

p k & p k$ 1 1 & ; :$ Q & Q &$ C kk$ , :$ Q & Q &$ 2 k{k$ kk$ k$k * k &* k$ 4 k, k$ kk k$k$

(34)

(35)

8

ROSSIGNOLI, CANOSA, AND RING

2 p k C kk$ =& =\$ k, \k$ p k(1& p k ) ; * k$ +(1&$ k, \k$ )

_

_$1 _$1 & 1+_$1 1+_$1

\

2

1

+ & sinh[ ;* ] sinh[ ;* ] , k

(36)

k$

where C kk$ is the quasiparticle correlation and the prime indicates again the sum over extended indexes. These expressions are valid in both ordinary and NP projected GC ensembles. In the standard GC case (_$=0), the second sum in (35) becomes &(;2) $k Q &kk Q &$kk p 0k (1 & p 0k ) . In general, ( p k & p k$ )  (* k & * k$ ) Ä ;[C kk$ & p k(1& p k$ )] if * k Ä * k$ in (35). The ``unperturbed'' solution x & =0 \& of Eqs. (29) is feasible if ( Q & ) 0 =0. If, moreover, for x & =0, Q &kk$ =0 for k=k$ or * k =* k$ (as occurs for pair creation or annihilation operators when [H 0 , N]=0), then R 0&&$(0)=R &&$(0, 0) and Eqs. (30)(31) become *: , | : :>0

C 0(0)=Det[$ &&$ +v & R &&$(0, 0)] &12 = ` sinh[;* : 2] , sinh[;| : 2] :>0

Z CMFA =Z(0) `

(37) (38)

where * : , | : , are here evaluated for H(x)=H 0 . As 12 [sinh( 12 ;|)] &1 is the PF of an harmonic oscillator of frequency |, the RPA correction to the mean field PF Z(0) is in this case exactly the quotient of the PF of a system of independent RPA bosons of energies | : to that of independent fermion pairs of energies * : considered as bosons. This correction is well defined only for a stable unperturbed solution, i.e., when all | : are real non zero. Equation (38) is thus normally reliable for high temperatures well above transitional regions (see the next section). As an instability is approached (i.e., as T decreases and a symmetry-breaking solution with x & {0 for some & becomes feasible at T=T c ), the lowest | : becomes zero and the factor C 0(x) diverges, making Eq. (38) highly unreliable in the vicinity of critical regions. In addition, C 0(x) will also diverge when a continuous symmetry is broken by the mean field, as in this case the determinant in (31) as well as the lowest RPA energy will vanish [29] (see Appendix B). In these phases Eq. (30) is not directly applicable. One should consider just the ``intrinsic'' variables in the static determinant (see the next section). On the other hand, due to the omission of the n=0 term (37), the factor C RPA(x), Eq. (28), remains finite and positive for zero as well as for imaginary RPA energies provided ; || : | <2?. This allows us to apply Eq. (28) for T2?, in which case the gaussian approximation for the x &n{0 is no longer valid [19, 21, 22] and large amplitude quantal fluctuations should be considered.

9

SMALL SUPERFLUID SYSTEMS

III. APPLICATION A. Pairing Hamiltonian in NP Projected Ensembles We consider now a monopole pairing interaction, H=: = k n k &GP -P,

(39)

k

n k =c -k c k +c -k c k ,

P - = : c -k c -k ,

P= : c k c k ,

k#I

(40)

k#I

where k denotes the time reversed states and I an interval of 0 twofold degenerate states around the Fermi level. As P -P= 14 (Q 2+ +Q 2& )+ 12 : (n k &1),

Q\=

k#I

1

\ i + (P\P ), -

the CSPA will lead to a two-dimensional integral, with the operator (3) given by x 2& 1 &x & Q & + G0 G 2 &=\

H(x + , x & )=: (=~ k &+) n k + : k

=: (=~ k &+) n k + k

22 1 &2(P -e &i, +Pe i, )+ G0 G 2

=e &i,N2H(2, 0) e i,N2,

(41)

where =~ k == k & 12 G for k # I, =~ k == k otherwise, 2e \i, =x + \ix & and N= k n k . Both Z(x + , x & ) and C RPA(x + , x & ) will nevertheless be independent of the gauge orientation ,, depending just on 2 2 =x 2+ +x 2& . The NP projected CSPA PF becomes then Z _CSPA =

2; G

|

 0

2 d2Z _(2) C _RPA(2),

(42)

where, writing H(2, 0) in terms of quasiparticles as H(2, 0)=

22 +: (= k &+&* k )+* k(a -k a k +a -k a k ), G k

* k =[(=~ k &+) 2 +2 2 ] 12,

k # I,

* k = |= k &+|,

(43) k  I,

(44)

1 Z 0(2)[1+_1(2)], 1+_ 2

(45)

with * k the quasiparticle energies, we obtain Z _(2)=Tr P _ exp[&;H(2, 0)]= Z 0(2)=e &;2

2 G

` e &;(=k &+)4 cosh 2 k

1

_2 ;* & , k

(46)

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ROSSIGNOLI, CANOSA, AND RING

1(2)=` tanh 2 k

1

_2 ;* & ,

(47)

k

| _k sinh[ ;* k ] , 2* k sinh[;| _k 2] k#I

C _RPA(2)= `

(48)

with _=1 (&1) for even (odd) statistics and _=0 for the ordinary GC ensemble (P 0 #1). In (48), the | _k(2) are the extended quasiparticle RPA energies, determined as the roots of Det[1+GR _(2, |)]=0,

(49)

2* k f _k : 2k 4* 2k &| 2 &i|: k * k k#I

\

R _(2, |)=& :

i|: k * k , 1

+

(50)

or equivalently, as the eigenvalues of the matrix

\

2* k $ kk$ &

G _ f (: k : k$ +1) 2 k

G _ f (: k : k$ &1) 2 k

G & f _k(: k : k$ &1) 2 G _ f (: k : k$ +1)&2* k $ kk$ 2 k

+

,

where k, k$ # I, : k =(=~ k &+)* k and f _k #1&2p _k =; &1  ln Z _(2)* k =tanh

1

2_1

1

_2 ;* & + 1+_1 sinh[ ;* ] , k

(51)

k

-

with p _k =( a -k a k ) 2_ =( a k a k ) 2_ . The thermal effects in | _k are determined entirely by f _k . Equivalent expressions for the NP projected PF Z _(2) and occupation probabilities p _k were given in [5]. The ordinary GC BCS PF (Hartree-type) is just Z 0(2) evaluated in its maximum. It provides the dominant contribution to Z CSPA in the limit of infinite particle number or volume, where the size parameter $#=T c [1], with = the average level spacing and T c the ordinary BCS critical temperature, vanishes. As $ increases, gap fluctuations, RPA correlations and NP projection effects, all taken into account in (42), become relevant and will be most important in very small systems where $r1, i.e., when T c is of the order of the level spacing. Such values are close to those reached in ultra-small metallic grains [8, 11]. In deformed heavy nuclei, typically $r0.7. B. NP Projected BCS We shall first briefly review in the present context the BCS approximation in NP projected ensembles, which has been thoroughly discussed in Refs. [5, 6, 11].

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SMALL SUPERFLUID SYSTEMS

The maximum of Z _(2) defines, for _=\1, a NP projected BCS (Hartree-type), determined by the gap equation (cf. Eq. (29)) 2=G ( P) 2_ = 12 G 2 : f _k * k ,

(52)

k#I

which for 2{0 becomes 1= 12 G : f _k * k ,

(53)

k#I

where the chemical potential + is to be adjusted by the average particle number constraint N=( N) 2_ =; &1

 ln Z _(2) =~ k &+ . =: 1&f _k + *k k

\

+

(54)

Nonetheless, note that NP projection ensures the correct number parity for any value of +. The ordinary gap equation corresponds to the GC case _=0 ( f 0k =tanh [ 12 ;* k ]). Let us recall that in large systems where $< <1, the sum in (53) can be replaced by <| D # 12 0=, with 2 0 the an integral and one obtains, for uniform spacing and 2 0 < T=0 GC BCS gap, the relations 2 0 | D r2e &1g, T c | D r1.14e &1g, with g#G=, whence T c r0.57 2 0 . Deviations from these results will arise as $ increases, especially in NP projected statistics. NP projection affects the low T behavior of p _k and f _k for those levels near the Fermi energy, especially in the odd case, as seen in Fig. 1 for a schematic example of 0 levels with uniform spacing =. In this case, Eq. (54) yields, for 2< <| D , T< <| D , += 12 (=~ 02 +=~ 02+1 )#+ 0 for ( N) =0 (even) and +r=~ 02+1 =+ 0 +=2 for ( N) =0+1 (odd), for any value of _. For T Ä 0 and 2{0, p _k Ä 0 \k for _=0, 1,

FIG. 1. Average quasiparticle occupation probabilities p _k =(a -k a k ) 2_ vs temperature for a schematic model with uniform spacing =, 0=10, and 2==. The ordinary grand canonical (GC) fermi probabilities (20) and the number parity (NP) projected probabilities (24) for the even (E, _=1) and odd (O, _=&1) cases are depicted (the largest p k corresponds to the level closest to the Fermi energy +). The left panel corresponds to +=+0 (( N ) =0), the right to +=+ 0 +=2 (( N) r0+1) with + 0 the middle energy (see text).

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ROSSIGNOLI, CANOSA, AND RING

while for _=&1, the lowest quasiparticle levels become occupied with the odd quasiparticle and p & k Ä 12n for these states, with 2n the total degeneracy of the lowest level (in Fig. 1, n=2 for +=+ 0 , n=1 for +=+ 0 +=2). This diminishes ( P) 2& and hence the odd self-consistent gap at T=0. For T>0 and fixed +, 2, 0 & p+ k p k p k so that the self-consistent gaps fulfill 2 + 2 0 2 & . Illustrative results for the BCS gap are depicted in Fig. 2 (top), for the schematic uniform model with 0=10, where comparison with results from the exact canonical PF will be shown in the next section. Results are depicted for the scaled strengths g=0.55 (a), 0.45 (b), and 0.35 (c), corresponding to effective size parameters $r 1.1, 1.6, and 3, respectively. In the even case (_=1, ( N ) =0), the final effect of NP projection is just an initial delay of thermal effects (as seen also in Fig. 1) plus a slight increase in T c for small g (case (c)). In the odd case (_=&1, (N) =0+1) the effects of NP projection are more important. For not too small strengths (case (a)), the T=0 gap is smaller than in the even case by an amount of the order of half the level spacing [11] (2 + &2 & r=2). As T increases, 2 increases first slightly, as the odd quasiparticle becomes distributed over higher levels and the sum

FIG. 2. Top left, BCS gap vs temperature in the uniform model, according to GC BCS and NP projected BCS for N=0 (E) and N=0+1 (O) particles, for g#G==0.55(a), 0.45(b), and 0.35(c). In (c), 2=0 in the odd case. Top right, the corresponding scaled gaps vs. TT c , with 2 0 the T=0 GC BCS gaps (O' is the result for +=+ 0 and _=&1). Bottom, phase diagrams in GC and NP projected (E, O) BCS in the previous model. The left panel corresponds to ( N) =0 ( +=+ 0 ) for _=0, 1 (GC, E) and ( N ) =0+1 (+r+ 0 +=2) for _=&1 (O), the right to ( N ) =0+1 (GC', E') and ( N ) =0 (O'), respectively.

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SMALL SUPERFLUID SYSTEMS

13

in (52) initially increases. As g decreases, there is first a range of g (case (b)) where 2 vanishes at T=0, but a superfluid solution arises at some T>0, due to the previous effect (the same behavior can be inferred from the phase diagram of Ref. [6] and has been discussed in the full variational BCS treatment of [11]). For still weaker strengths, a superfluid solution no longer exists at any T for N=0+1, even though it may still exist for N=0 (case (c)). Note also that in the ordinary GC BCS, 2(T )2 0 practically coincides for the cases depicted when plotted in terms of TT c , but this no longer occurs in NP projected BCS, as seen in the top right panel, which reflects an increased sensitivity to finite size effects. The previous behavior can be understood from the BCS phase diagram of the bottom left panel of Fig. 2, which depicts the critical strength g c for the onset of superfluidity as a function of T (or equivalently the critical temperature(s) T c in terms of g) in the previous model, as determined by Eq. (53) for 2 Ä 0. In the GC case, g c is almost coincident with the bulk value g c r&1ln(T1.14| D ) for T>=4. The effects of NP projection on the critical curve become relevant for T<0.75=. In the odd case, g c is larger than in the GC case, decreasing with increasing T for T<0.35=, which explains the peculiar behavior of the odd case b in the top right panel. In the even system, g c is slightly smaller for 00.28 (0.46) in the even (odd) cases. The BCS phase diagram depends nevertheless also on the choice of +, as seen in the bottom right panel of Fig. 2. If we set +=+ 0 and _=&1, we are representing a statistical mixture of odd systems, with ( N) =0 (even). In this case, g c becomes smaller for T<=2 and no longer decreases as T increases, so that the behavior of previous odd case (b) no longer occurs (see curves O' in the top right panel). For +=+ 0 +=2 and _=0, 1, a superfluid solution 2{0 exists at T=0 for all g>0, as in this case the lowest quasiparticle energy vanishes for 2 Ä 0 (in which case the sum in (53) diverges for T Ä 0 if _=0, 1). For _=1, this corresponds to a mixture of even systems, with ( N) =0+1 (the same holds for T Ä 0 in the GC case _=0). These variations of + affect the critical strength just for T<=2 for _=\1, and T<=4 for _=0, but indicate nevertheless some degree of arbitrariness of the BCS transitions in such limiting cases. These will become considerably washed out when higher order correlations are incorporated, as will be discussed below. C. Comparison with Exact and CSPA Results The actual thermodynamic behavior of a finite system is smooth, both as a function of T or g. In order to examine the accuracy of BCS and CSPA, we have calculated the exact canonical PF, by diagonalization of the Hamiltonian (39) in 5 the complete accessible space for N=0=10 (of total dimension ( 20 0 )r1.8_10 ) and N=0+1. Obviously, ( P ) =( P) =0 in the exact case due to particle number conservation, and also in the CSPA owing to , integration. We thus consider first as a measure of pairing correlation strength the quantity 2 P #G[( P -P) G &( P -P) G=0 ] 12,

(55)

14

ROSSIGNOLI, CANOSA, AND RING

where ( P -P) G denotes the thermal average for strength G, ( P -P) G =Z &1 Tr[exp(&;H) P -P]=; &1  ln ZG,

(56)

with Z=Tr exp[&;H] the (exact) PF for the Hamiltonian (39). Thus, 2 2P G= G(( P -P) G &( P -P) G=0 ) is a direct measure of the increase in the pairing correlation energy due to a non-zero strength G. In BCS, a direct evaluation as a derivative of Z _(2) at the self-consistent gap yields obviously ( P -P) G =2 2G 2 and 2 P =2. This omits exchange-like terms, which can be included by evaluation as expectation value, i.e., ( P -P) G =( P -P) 2_ , with ( P -P) 2_ #Z &1 _ (2) Tr[P _ exp[&;H(2)] P P]=

( P -P) 02 +_1(2) ( P -P) 12 , 1+_1(2) (57)

FIG. 3. Top, the effective gap parameter (55) for N=0 (left) and N=0+1 (right), according to exact canonical results and NP projected CSPA, SPA, and BCS results. Curves (a), (b), (c), correspond to the strengths of Fig. 2 and d to g=0.2. CSPA and exact results almost overlap above the CSPA breakdown. BCS results are obtained using Eq. (57). Bottom left, comparison between the exact results for 2 P for N=0 (E) and N=0+1 (O). The even BCS results are indicated (dotted lines). Right, exact and NP projected BCS scaled values of 2 P for N=0 (2 0 is the T=0 GC BCS gap).

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15

SMALL SUPERFLUID SYSTEMS

where, by application of Wick's theorem, ( P -P) &2 #

Tr e &;H(2)+i&?Nq P -P =( P - ) &2 ( P) &2 + : ( c -k c k ) &2 ( c -k c k ) &2 , Tr e &;H(2)+i&?Nq k#I

1 1 ( ;* k +i&?) ( P - ) &2 =( P) &2 = 2 : tanh 2 k#I 2

_

&<* , 1 =~ &+ 1 1 ( c c ) =( c c ) = ( n ) = 1&tanh ( ;* +i&?) _2 & * +, 2 2\ k k

& 2

k k

& 2

k

& 2

k

k

k

(58) (59) (60)

k

for &=0, 1. Thus, in the GC case _=0, ( P -P) 20 =2 2G 2 + 14  k # I ( n k ) 220 at the self-consistent gap, so that 2 P r2 if the variation of the exchange term with 2 is neglected. In NP projected BCS, 2&2 P is also small in the cases considered (as seen by comparing the BCS results for 2 P depicted in Fig. 3 with the gaps of Fig. 2) and clearly 2 P =0 if 2=0. As seen in Fig. 3, the exact result for 2 P is smooth and non-vanishing for T>T c , in contrast with all BCS approximations, decreasing as (g 3T ) 12 for high T (see the next subsection). Nonetheless, there is a slight initial increase of 2 P in the odd case, in qualitative agreement with the NP projected BCS result in case (a), although no other signature of the peculiar BCS behavior in odd case (b) remains. Exact results for 2 P are actually closer for small T to those for _=&1 and +=+ 0 in BCS (see curves O' in top left panel of Fig. 2). Note, however, that odd-even differences in >T c , although they are non-negligible for T around T c . the exact 2 P vanish for T> When scaled and plotted in terms of TT c , the exact 2 P becomes increasingly flat as g decreases, and the deviation from BCS increases. Case (d) in Fig. 3 corresponds to g=0.2, where, for 0=10, 2=2P =0 \T>0 in both GC and NP projected BCS. Nonetheless, there is a non-negligible value of 2 P in the exact and CSPA results, as well as a non-vanishing odd-even difference in 2 P for small T. Moreover, the scaled value 2 P 2 0 , with 2 0 =2| D e &1g the bulk T=0 gap, becomes rather large for small strengths (for case d, 2 P 2 0 r3.7 at T=0, beyond the range of bottom right panel in Fig. 3). On the other hand, NP projected CSPA results, obtained from the PF (42), practically overlap with the exact ones in both the even and odd systems, for T above the CSPA breakdown (which occurs for T< 12 T c in the cases considered; results diverge at the breakdown and are shown for T> 12 T c ). In the CSPA, NP projection is essential in the odd case, but is also required in the even case for an accurate agreement in the region below T c [25]. NP projected SPA results, obtained as the SPA average of Eq. (57) (see below) give the correct trend for high T but approach just the NP projected BCS results for T Ä 0, underestimating 2 P for low T (the result obtained by derivation of the SPA PF (Eq. (42) without C _RPA ) omits the exchange terms for low T and is less accurate [17]). Figure 4 (top) depicts the CSPA probability distributions P &(2)#N &_ Z _(2) C _RPA(2) 2 &,

(61)

16

ROSSIGNOLI, CANOSA, AND RING

& for &=0, 1, where N &_ is a normalization constant (  0 P (2) d2=1). The SPA _ distributions are obtained if C RPA(2) is omitted in (61), in which case P 0(2) corresponds to the ``BCS'' probability whose maximum determines the BCS solution, while P 1(2) to the final effective distribution determining the SPA averages. In a small system, P 0(2) becomes quite flat for T close to T c , but the measure 2 has nonetheless an important effect and P 1(2) exhibits a well defined peak at 2{0, even for T>T c (see Ref. [30] and Subsection E]). On the other hand, the factor C _RPA(2) has a small effect on the distribution shape. Due to the attractive character of the pairing interaction, | _k 2 <(2* k ) 2 and C _RPA(2)>1 \2 [22]. NP projection effects on C _RPA(2) are rather small, affecting mainly the lowest RPA energy, which lies below (above) the corresponding GC result in the even (odd) case as seen in the left (right) bottom panels of Fig. 4. For TT c , all RPA energies are real \2 but the collective mode still deviates considerable from the lowest quasiparticle pair energy for large 2). For uniform spacing, the quasiparticle energies * k are degenerate

FIG. 4. Top, normalized NP projected CSPA distributions (61) (dashed lines) for g=0.55 and T=0.8T c , in the even (E, N=0) and odd (O, N=0+1) systems for &=0 and &=1. The solid lines depict the corresponding SPA distributions (Eqs. (61) with C _RPA(2) omitted). Bottom, the lowest quasiparticle RPA energies (squared) vs. 2 for the same values of g and T, in the even (left) ond odd (right) systems, according to NP projected (solid lines) and GC (dashed lines) treatments (almost coincident except for the lowest mode). The dotted lines indicate the (squared) quasiparticle pair energies 4* 2k .

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SMALL SUPERFLUID SYSTEMS

17

for states symmetrical with respect to the Fermi level. The RPA energies remain almost degenerate in the relevant region of 2, except for the lowest mode. In the CSPA, the chemical potential + can be obtained after integration by the standard constraint ; &1  ln Z _CSPA +=N, or also adjusted at each point 2 by Eq. (54). This corresponds to particle number projection in the saddle point approximation applied before integration [1, 25, 30], which leads to the canonical CSPA PF Z NCSPA r

4; G

|

2d2Z _(2)

e &;+(2) N C _ (2), [2?_ 2N(2)] 12 RPA

(62)

with _ 2N =; &2  2 ln Z _(2)+ 2 the number fluctuation (we have here neglected small RPA effects on ( N) 2 and _ 2N ; Eq. (62) corresponds to a simultaneous saddle point approximation in the variables x &n and ;+ in the path integral). When applied after NP projection (with _=(&1) N ), Eq. (62) provides an accurate evaluation of the true canonical PF for T>=2. In the present case, the variation of + with 2 is negligible and differences with results obtained directly from Eq. (42) are small in the magnitudes so far depicted. We are now able to consider the odd-even free energy difference 2 F (N)# 12 [F N+1 +F N&1 &2F N ],

(63)

where F N =&T ln Z N with Z N the canonical PF. In the standard GC approximation, Z N rZ 0 e &;+N and F N r&T ln Z 0 ++N, with Z 0 the GC PF, whence 2 F (N) r 12  2 F N N 2 r 12 +N. The deviation of 2 F (N) from this value is indicative of canonical corrections. These are strongly enhanced by pairing effects, in which case number projection, or at least NP projection (i.e., Z N rZ _ e &;+N with _=(&) N ) is essential for an adequate evaluation. In the absence of pairing (G=0), at T=0 we have 2 F (N)==2 (0) for N even (odd) in the present uniform model, and the GC result 2 F (N)r 12 +Nr=4 is approached in both cases for T>=2. In the presence of pairing, for large superconductors ($< <1), 2 F (N)r(&) N2 at T=0 (as FN\1 &F N r2\+) and the GC limit 2 F (N)r=4 is approached only for T>T c . Corrections to this picture will arise as the size decreases. Let us consider first 2 F (N) for N=0 (even, half filled). In the uniform model, for energies measured from + 0 , F N+1 =F N&1 and 2 F (N)=F N\1 &F N . In NP projected BCS, Z N rZ _(2)e &;+N, with 2 the self-consistent gap. As seen in Fig. 5, the ensuing BCS result practically coincides with the even BCS gap at T=0 and approaches the GC limit r=4 for T>T c . However, BCS results underestimate the exact value of 2 F (N), particularly for small g and in the critical region. The exact result exhibits a smooth behavior and approaches the GC limit only for T>2T c in the cases considered. On the other hand, NP projected CSPA results, obtained with Eq. (62), are practically coincident with the exact ones above the breakdown, while those from SPA give the correct trend for high T. Let us consider now 2 F (N+1), for N=0. For high T, the GC limit holds and 2 F (N+1)r2 F (N), while for T Ä 0, we obtain 2 F (N+1)r&2 F (N)+=2, in agreement with Ref. [8], both in the exact

18

ROSSIGNOLI, CANOSA, AND RING

FIG. 5. Left, odd-even free energy difference, Eq. (63), for N=0, according to exact and NP projected CSPA, SPA, and BCS results, for cases (a) and (c) of Fig. 3. Right, the exact free energy difference for N=0 in the cases of Fig. 3, with e the result for g=0. The shifted result for N=0+1 is also plotted (see text).

case and in BCS. This relation holds approximately also at finite temperature in the range considered, as seen in Fig. 5 (right). Note also that in case (d) (where 2=0 in BCS) there is still an appreciable value of 2 F (N) above the result for G=0 (curve (e)). D. BCS+RPA (CBCS) Let us consider now the gaussian approximation (30). For T>T c , we obtain (Eq. (38)) sinh[ ;* k ] , sinh[;| _k 2] k#I

_ =Z _(0) C _0 C _RPA(0)=Z _(0) ` Z CBCS

(64)

where | _k are the thermal quasiparticle RPA energies for 2=0, with * k = |=~ k &+| and 1 2* k =Det[1+GR _(0, 0)] &12 = 1& G : f _k * k |k 2 k#I k#I

C _0 = `

_

&

&1

,

(65)

with R _(0, 0) the limit of (50) for 2 Ä 0, | Ä 0. Equation (64) is reliable only for T well above T c in the small systems here considered. As T Ä T c , the lowest RPA energy vanishes and C _0 diverges. Nevertheless, Eq. (64) yields exact asymptotic expressions for high T. In this limit, within a finite configuration space, C _RPA(0) _ rZ _(0)C _0 , whence Ä 1 and Z CBCS ( P -P) G r; &1

_  ln Z CBCS  k # I f _k * k 1 , r ; &1 G 2 1&(G2) : k # I f _k * k

so that Eq. (55) behaves as 2 P r 14 G0(G;) 12 for T Ä  in a finite space.

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(66)

19

SMALL SUPERFLUID SYSTEMS

In the superconducting phase, one can attempt a saddle point evaluation by considering just the ``intrinsic'' coordinate 2 in the static variables. This leads to _ Z CBCS =[4?;2 2G] 12 Z _(2) C _0 (2) C _RPA(2),

(67)

where 2 is the self-consistent gap and

_

C _0 (2)= &

G  2 ln Z _(2) 2; 2 2

1 R 0_(2)=& : 2 k#I

_

f _k

&

&12

=[1+GR 0_(2)] &12,

(=~ k &+) 2 22 +; : C _kk$ , 3 *k * k * k$ k$ # I

&

(68) (69)

with C _kk$ =; &1 f _k * k$ . Note that at the superfluid solution of (52), the lowest RPA energy and the full static determinant (31) in both variables x \ vanish due to broken U(1) gauge symmetry (Appendix B). Equation (67) is reliable well below T c , as C _0 (2) diverges for T Ä T c . As seen in Fig. 6 (top left) the temperature range

FIG. 6. Top, the effective gap parameter (55) in the even system (N=0) for the same cases of Fig. 3, according to NP projected BCS+RPA (CBCS, left), Eqs. (64)(67), and effective BCS+RPA (CEBCS, right), Eq. 71). Exact and NP projected BCS (right) results are also depicted. Bottom left, exact and NP projected CBCS, CEBCS, and BCS results for the odd case (N=0+1), with the same convention as the top panels. Right, the solutions of the effective BCS Eq. (70), in the GC (solid lines), even (dashed lines), and odd (dotted lines) cases for the same strengths of Fig. 3.

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20

ROSSIGNOLI, CANOSA, AND RING

where (64)(67) is inaccurate is rather wide in cases (a), (b), and (c), particularly for T>T c , although it does provide a good evaluation of 2 P in the weak coupling case (d), where 2=0 in BCS and no transition takes place. For odd case (c) (bottom left) there is no transition either but, being the system closer to the onset of superfluidity, the CBCS result is much less accurate. E. Effective BCS+RPA (CEBCS) It is remarkable that a significantly improved gaussian approximation to the partition function can be obtained from the CSPA in a very simple way, just by including the measure 2 in (42) when determining the expansion point [31]. The maximum of Z _(2)2 is determined by the modified gap equation 2=G[( P) 2_ +1(2;2)] = 12 G2

_ : f * +1( ;2 )& , _ k

2

k

(70)

k#I

whose solutions are smooth and non-zero \g, T>0 (see bottom right panel in Fig. 6) and represent approximately the CSPA average of 2 [31]. They approach the corresponding BCS gaps for T Ä 0 but are larger for T>0, behaving as (G2;) 12 for T Ä  in a finite space. Note that Z _(2) 2 exhibits an approximate gaussian shape \T>0, including the region around T c , as seen in the upper right panel of Fig. 4. The gaussian approximation around the solution of (70) leads to _ Z CEBCS =(4?;2 2G) 12 Z _(2) C _0 (2)C _RPA(2),

_

C _0 (2)= &

G  2 ln[Z _(2) 2] 2; 2 2

&

(71)

&12

=[1+G[R 0_(2)+1(2;2 2 )]] &12. (72)

Now C _0 (2) and Eq. (71) remain finite and positive \T>0. For T Ä 0 the BCS+RPA results are approached. As shown in Fig. 6, results for Eq. (55) are remarkably accurate both in the even and odd systems for all strengths considered, except for a narrow region around the critical temperature, where, nevertheless, they remain finite and close to the exact results. Similar accuracy is obtained for 2 F (N). The exact asymptotic expressions for high T can also be obtained from Eq. (71). F. Application to Heavy Nuclei Having verified the accuracy of the CSPA for a pairing Hamiltonian in small systems, we now consider an illustrative example in a well deformed rare earth nucleus. We employ the BarangerKumar configuration space [32] for neutrons (0=49 twofold degenerate levels for the N=5, 6 major shells), with a pairing strength G n =22A MeV, where A is the mass number. We choose for H 0 a deformed single particle Hamiltonian (with non-uniform spectrum) of the form H 0 =H s & ; d | 0 Q 0 , where H s is a spherical part, Q 0 a quadrupole operator, taken both from [32], ; d the T=0 quadrupole deformation and | 0 the oscillator constant. We have depicted in Fig. 7 the effective gap parameter (55) for neutrons in 164Er and 165Er,

SMALL SUPERFLUID SYSTEMS

21

FIG. 7. The effective gap parameter (55) for neutrons (left) in 164Er(E) and 165Er(O), and the neutron odd-even free energy difference (63) (right) for N=164, according to NP projected CSPA, SPA, BCS, and CEBCS results. The spaced dotted line indicates the result for G=0.

and the free energy difference (63) for 164Er, both for T<1 MeV. We observe the same trends as in the previous model. The average level spacing at the fermi level is =r0.35 MeV and T c r0.45 MeV. At T=0, the even-odd difference in the NP projected BCS gap is 0.26 MeV. The CSPA results can be considered practically exact for T>0.25 MeV and yield a substantial value of 2 P for T>T c , which is in agreement with Eq. (66) for high T, although for 2 F the deviation from the BCS estimate is smaller. The even-odd difference in 2 P and the deviation of 2 F from the GC value 12 +N vanish for T>0.6 MeV. Effective BCS+RPA results are again remarkably accurate, particularly for 2 F , and allow us to extrapolate CSPA results up to T Ä 0 at the NP projected BCS+RPA level. SPA results provide as well a very good description of 2 F and the correct trend for 2 P .

IV. CONCLUSIONS We have presented a general derivation of the CSPA formalism, suitable for dealing with small superfluid systems at finite temperature in NP projected ensembles. We have also derived general expressions for the partition function in the mean field+RPA approximation, valid for these ensembles. Results for a pairing interaction indicate that NP projected CSPA is able to provide an accurate description of small systems in which the critical temperature is of the order of the single particle level spacing, significantly improving NP projected BCS and BCS+RPA results in critical regions, as well as NP projected SPA results at low temperatures above the CSPA breakdown. The NP projected BCS transitions become increasingly smoothed out in the exact and CSPA results as the effective size of the system decreases, as indicated for instance by the behavior of 2 P , although odd-even effects beyond the predictions of standard GC statistics become nevertheless small for T above the transitional region. Finally, we have derived from the CSPA a simple effective BCS+RPA approach which remains smooth in critical regions and avoids

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22

ROSSIGNOLI, CANOSA, AND RING

the main shortcomings of conventional BCS and BCS+RPA at finite temperature. Alltogether, the CSPA approach constitutes, at least for effective interactions containing a few separable terms, a simple yet reliable alternative at finite temperature to more involved treatments like Monte-Carlo evaluations of Eq. (2) [26] or improved mean field approximations with full symmetry restoration before variation [11, 33, 31], which in principle are also able to yield a correct description of small systems.

APPENDIX A: EXPANSION OF IMAGINARY TIME EVOLUTION OPERATOR Let us consider the imaginary time evolution operator corresponding to a static Hamiltonian H plus a time dependent perturbation Q({),

{

|

;

U(;)#T exp &

=

_

|

;

d{[H+Q({)] =exp[&;H] T exp & 0

&

d{ Q H ({) , 0

(73)

where in the last expression we used the interaction representation in imaginary time, with Q H ({)#e {HQ({) e &{H. We can expand (73) and its trace, up to second order in Q({), as ;

_ | _1&| d{Q({)

U( ;)rexp[&;H] 1&

d{ Q H ({)+

0

{

Tr[PU( ;)]rTr Pe &;H

|

;

d{ Q H ({)

0

|

{

&

d{$ Q H ({$) ,

0

;

(74)

0

+

|

;

d{ 0

|

{

d{$ e ({&{$) H Q({) e &({&{$) HQ({$)

0

&= ,

where, in a many-particle system, Tr denotes the trace in the GC ensemble (full Fock space) and P a projector onto a restricted ensemble, with [P, H]=0. Expanding now Q({) in the interval (0, ;) as 

Q({)= :

Q n e &i|n {,

| n =2?n;,

n=&

where Q n are {-independent operators and | n the Matsubara frequencies, Eq. (74) yields, for [P, Q n ]=0,

_

\

Tr[PU( ;)]rZ 1&; (Q 0 ) + 12 R 0 + : R n

{ _

n>0

+&

(75)

rZ exp &; ( Q 0 ) + 12 (R 0 +; ( Q 0 ) 2 )+ : R n n>0

&= ,

(76)

23

SMALL SUPERFLUID SYSTEMS

where Z=Tr[Pe &;H ]= K e &;EK, (Q 0 ) =Z &1 Tr[Pe &;HQ 0 ]= K P K( K| Q |K), and R n #&Z &1

|

;

e i|n { d{ Tr[Pe &( ;&{) H Q &n e &{H Q n ]

(77)

0

= : ( K| Q &n |K$)(K$| Q n |K) K{K$

P K &P K$ &$ n0 ; : P K (K| Q 0 |K) 2, E K &E K$ +i| n K (78)

with PK =Z &1e &;EK and |K) the many-body eigenstates of H in the subspace defined by P ((K$| PH |K) =EK $ KK$ ). Equation (76) is correct up to second order in Q({) and corresponds to the expansion of ln Tr[PU(;)]. If Q n Ä * n Q n , (76) implies  ln Tr[PU(;)]  2 ln Tr[PU(;)] =&$ n0 ; (Q 0 ), =&$ n, &n$ ;(R n +$ n0 ; ( Q 0 ) 2 ), * n * n * n$ (79) for derivatives evaluated at * n =0. Note that for n=0 and E K =E K$ in (78), (P K &P K$ )(E K &E K$ ) is to be replaced by the limit &;P K , as can be obtained directly from (77). Case of One-Body Operators If H and Q({) are one-body fermion operators of the standard form H=: * k a -k a k , k

Q n = : Q nkk$ a -k a k$ ,

(80)

k, k$

the eigenstates of H are independent particle states |K) => k (a -k) nk |0) and Eq. (78) becomes R n = : Q &n  nk$k kk$ Q k{k$

p k & p k$ &$ n0 ; |( Q 0 ) | 2 + : Q 0kk Q 0k$k$ C kk$ , * k &* k$ +i| n k, k$

_

&

(81)

where p k =(a -k a k ) = K P K (K| a -k a k |K), (Q 0 ) = k Q 0kk p k and C kk$ #(a -k a k a -k$ a k$ ) & p k p k$ =&; &1

p k , * k$

the correlation. These general expressions are valid, for instance, both in the GC and canonical ensembles. In the GC case, H stands for H&+N, p k becomes the Fermi probability (20), and C kk$ =$ kk$ p k(1& p k ). For n=0 in (81), ( p k & p k$ )(* k &* k$ ) Ä &;[ p k(1& p k$ )&C kk$ ] if * k Ä * k$ .

24

ROSSIGNOLI, CANOSA, AND RING

If Q({) is of the more general form (16), Eq. (78) becomes Rn= : k{k$

{

11 Q &n Q nk$k11 kk$

02 +Q &n Q nk$k20 kk$

=

p k & p k$ p k + p k$ &1 1 20 + Q &n Q nk$k02 kk$ * k &* k$ +i| n 2 * k +* k$ +i| n

_

1& p k & p k$ &* k &* k$ +i| n

1 :$ Q &n Q n 2 k{k$ kk$ k$k

&= &$ ; _ |( Q ) | + : Q p &p 1 &$ ; |( Q ) | + :$ Q Q _ * &* +i| 4 k

2

n0

k$

k$

11 Q 0k$k$ C kk$

k, k$

2

n0

k

0 11 kk

0

0 kk

0

n

k, k$

0 k$k$

&

C kk$ ,

& (82)

where the prime in (82) indicates the sum over all labels of the extended matrices (13), with p k =( n k ) and n k =a -k a k for * k >0, n &k =a k a -k =1&n k for * &k =&* k <0, while C kk$ =( n k n k$ ) & p k p k$ is the extended correlation (36). These expressions are valid both in standard and NP projected GC ensembles.

APPENDIX B: ZERO ENERGY MODES We consider here the case where the linearized Hamiltonian (3) breaks a continuous symmetry of H. We assume there is a unitary transformation U(,) such that U -(,) Q & U(,)=: U&&$(,) Q &$ ,

(83)

&$

which leaves the Hamiltonian (1) invariant, i.e., [U(,), H]=0, [U(,), H 0 ]=0, with U -(,) H(x) U(,)=H(x$),

x$& =: U&$&(,) x &$ .

(84)

&$

The eigenvalues of H(x), and hence Z(x), remain invariant under the transformation, i.e., Z(x$)=Z(x). In this case a symmetry-breaking solution x & {0 of Eqs. (29) will exhibit continuous degeneracy and can be represented as x &(,)=  &$ U&$&(,) x &$(0). Deriving Eqs. (29) (satisfied \,) with respect to ,, and taking into account that P K ,=0, we obtain, from Eqs. (31)(32) and (33), : &$

$ &&$

_ v +R &

(x, 0)

&&$

x &$

& , =0,

(85)

whence (assuming x &$ , is a non-zero vector) Det[$ &&$ +v & R &&$(x, 0)]=0.

(86)

This implies that |=0 is a root of Eq. (27) when x is a solution of Eqs. (29) in the ensemble considered. At other values of x the determinant need not vanish.

SMALL SUPERFLUID SYSTEMS

25

Equation (85) also implies the vanishing of the determinant in (31). In the BCS Hamiltonian, the gauge transformation is U(,)=e i,N2, with U -(,) P -U(,)= e &i,P -.

ACKNOWLEDGMENTS R.R. and N.C. are members of CICPBA and CONICET, respectively, of Argentina. Work supported in part by EEC (Contract CI1*-CT93-0352).

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