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POTENTIAL DEPENDENCE OF PAIRING
SOLUTION FOR THE BOSE SUPERFLUID
W. Evans, C. Harris
To cite this version:
JOURNAL D E PHYSIQUE
Colloque
C6,
supplement au n°
8,
Tome
39,
aout
1978,
page
C6-237
POTENTIAL DEPENDENCE OF PAIRING SOLUTION FOR THE BOSE SUPERFLUID
W.A.B. Evans and C.G. Harris
The Physios Laboratories, University of Kent at Canterbury, Canterbury, Kent, U.K.
Résumé.- On explore la dépendence qualitative de la solution "Paire de Base" avec le potential uti-lisé. Pour le cas d'un potentiel attractif, mais avec un coeur répulsif faible, la phase paire pos-sède un composant du type Bose-Einstein (Bose-Einstein condensate) avec une transition de premier ordre. Par contre, pour un coeur très répulsif la phase paire n'a pas de partie condensée et la transition est de deuxième ordre. Il est proposé que des solutions sans composant condensé, suggé-rées d'abord par Di Castro /l/ et par Coniglio et Marinaro 121, présentent les meilleurs représen-tations du comportement de l'hélium quatre superfluide.
Abstract.- We investigate the qualitative dependence of the Bose pair solution on the potential em-ployed. For attractive potentials with weak repulsive cores we find the pair phase to possess a Bose-Einstein condensate (B.E.C.) with a first order transition, whereas, with strong repulsive co-res, the pair phase is "condensate-less" and the transition of second order. Such "condensate-less" solutions of the type first proposed by Di Castro /l/ and Coniglio and Marinaro /2/ are argued to best simulate the behaviour of superfluid ''Helium.
Pairing theories of superfluidity start by assuming that the socalled "pair Hamiltonian" which is a subset of the total Hamiltonian is itself capa-ble of giving an adequate description of the super-fluid transition. The pair Hamiltonian can be writ-ten as +
Vk-o-^.k-X'-V-k^iA^
4>AIR
=K \
+i,HVk'
+V
k,k'1 --M.
+^ 2 V k -
( 1-
6k , k '
) ( 1" V k - > \ V
k,k' —'— — —where n, = a, a, , Xt = ^ T , * ! , an<* w e n a v e employed the standard notation of second quantization. The Krona-cker S-functions are there to avoid the double coun-ting of terms from the full Hamiltonian. Following a method originally due to Wentzel /3/, it can be shown that / 4 / , at least when a B.E.C. is absent in the solution, the thermodynamic potential of PpATi3 is equivalent to the thermodynamic potential of the bilinear Hamiltonian lJ
o
= A +^
{ gk
nk
+it^k
+ Ak \ 3 >
where e - e. + , £ (V, , , + v ) k' and\
J.W
Vk'X
1 ' ~ 2 \,k' (2)V
(3)\,-k'>V <
4)and A = - ^
J
{(V
k_
k,
+V
o)
(1-I 6 J ^ , +
k,k — — (5) provided the c-numbers n, and \ (or equivalently e and A, ) are the solutions of the basic coupled nonlinear equations of the boson pair theory viz.e
k
+n
|A-k'
+
V ( ' - V
1
)
\nk -2Ek, coth 6Ek, (6)\ - - i I Vk- "-V-k-^'-Vk^ ^ 7
c o where E, = / efl\i
: A k-2 Ek< c o t hK'j
2 (7) (8)The thermodynamic potentiel of the bilinear VI is easily evaluated using the Bogoliubov Valatin transformation. The difference between this and the thermodynamic potential of VL,T R may be shown /4/ to be of 0(1) pro-e, and A, are chosen as above and also that there is no B.E.C. in the solution to M
_ o (i.e. all the nk,s and XvtS are 0(1)). Thus under
these conditions the extensive parts of the thermo-dynamic potentials coincide and, in this sense, con-densate-less pair solutions (at least) are exact solutions for the thermodynamics of U pA T R.
We have investigated the solutions of the pair equations (6) and (7), numerically, for various potentials (or, more correctly, pseudo-potentials, see Evans and Imry /5/. We have neglected the
trictions imposed by the Kronacker &-functions in (6) and (7) since it can be argued that, even for pair solutions with a B.E.C., the condensate will be "smeared" over a macroscopically large number of levels near k = 0 thus making the omitted terms ne- gligibly small for these "smeared condensate" type solutions (as they also are for condensate-less ty- pe solutions). For non-trivial solutions to (7) it has long been realised /S/ that an attractive part to the potential is necessary (i.e. V(r) has to be- come negative somewhere in r-space). We have now ,.. shown / 4 / rigorously that this is also sufficient in the sense that a non-trivial solution will nu- cleate at a temperature that is definitely above Any Bose Einstein condensation temperature that may exist in the normal (Hartree-Fock) phase (i.e. the solCtion to (6) with Ak = 0) or will definitely 6 c d k at same finite temperature above 0 K in the dld'ernative case (i.e. when a B.E.C. never appears
'id C'he Hartree-Fock picture). Thus we have investi- Bated pair solutions (all of the smeared condensate
or condensate-less types) that result from poten- dials with attractive wells and a varying amount of hafd core repulsion. The equations were solved by the Newton-Raphson method as detailed elsewhere 141. Our findings are shown schematically in figure 1 .
Fig. 1 : Schematic dependence of pair solution on core repulsion
The types of solutions we obtained seem to fall into three categories. The insets show the temperature dependence of the pair amplitudes for the three cases. The point at which the pair ampli- tude vanishes denotes the "Cooper instability" in the normal phase represented by the line PQR in
figure 1. This temperature also marks the transition from normal into superfluid provided the core repul- sion is large enough i.e. as in regions (a) and (b)
.
The temperature dependence then observed is as in insets (a) and (b). In (a) we have a second order transition into a condensate-less pair phase that is denoted by a continuous line in the insets. In (b) we firstly enter a condensate-less pair phase but at some lower temperature a B.E.C. nucleates in the pair phase and below this temperature we have a "smeared-condensate" type pair solution (dotted line in insets). For weaker cores as in ( c ) , pair solu- tions can exist at a higher temperature than the Cooper instability. At the Cooper instability the pair solution must be condensate-less but as the pair amplitude increases the temperature also in- creases at first until it reaches a certain maximum after which it decreases to zero as shown in inset
(C). Also the pair solution develops a B.E.C. on the lower branch and remains a smeared B.E.C.-type so- lution from then on as the amplitude increases. However for the temperature range for which two pair solutions (ane one normal solution) are possible, the upper pair branch has the lowest Helmholtz Free Energy and so is the physical solution. Thus under conditions of constant T and p we expect a first order transition into a condensate pair phase which would remain a B.E.C. type solution as we further decrease the temperature to T = 0 K. In this case the Cooper instability would not mark the onset of superfluidity, as it is not on the lowest free ener- gy branch. Under conditions of constant T and P a Maxwell-type construction is required to determine
the nature of the first order transition which would now be accompanied by a discontinuous density change.
JOURNAL DE PHYSIQUE
Colloque
C6,
supplkment au
no
8, Tome 39, aoat 1978, page
C6-239
References
/l/ Di Castro, C., Ph.D. Thesis, University of Rome, unpublished
/ 2 / Coniglio, A. and Marinaro, M., Nuovo Cimento
48B
(1967) 249/ 3 / Wentzel, G . , Phys. Rev.
120
(1960) 1572141 Evans, W.A.B. and Harris, C.G., to be published
151 Evans, W.A.B. and Imry, Y., Nuovo Cimento (1969) 155 /6/ Mook, H.A., Phys. Rev. Letters
