content_image/393672.jpg content_image/393673.jpg content_image/393668.jpg content_image/393667.jpg content_image/393666.jpg content_image/393670.jpg content_image/393669.jpg
assumption we obtain instead 

and with the new wavefunction Eq. (26) 

The last term in Eq. (31b) lowers the energy if 

that is, if the occupation of ( ˜ k ↑ , − ˜ k ↓ ) is different from that of ( − ˜ k ↑ , ˜ k ↓ ) . Thus, it favors a spontaneous break- ing of parity and the existence of a spin current[9]. 

In earlier work we have presented several arguments in favor of the suggestion that superconductors possess a spontaneous spin current in their ground state[5, 7– 9]. The direction of the spin current is determined by the spin-orbit interaction in the presence of an outward- pointing electric field predicted by the theory of hole superconductivity[11] together with the sample geome- try. Namely, near a surface with outward normal ˆ n the member of the Cooper pair of spin σ has larger velocity in direction ˆ n × ⃗σ [9]. Also, C.R. Hu has argued quite gen- erally that the structure of the BCS state is compatible with the existence of a spin current[12]. 

## VI. COULOMB INTERACTION IN THE PAIR WAVEFUNCTIONS 

The average of the Coulomb interaction in the BCS wavefunction Eq. (4) is given by 

<img src='content_image/393672.jpg'>

and in the new wavefunction Eq. (26) by 

<img src='content_image/393673.jpg'>

<img src='content_image/393668.jpg'>

<img src='content_image/393667.jpg'>

<img src='content_image/393666.jpg'>

For a clearer comparison we separate self-energy terms in the double sums and obtain for the BCS case 

and for the new wavefunction 

<img src='content_image/393670.jpg'>

The BCS expression Eq. (35) has the same form whether or not parity is broken. Instead, in the ex- pression Eq. (36) both the first and last term vanish identically if parity is unbroken, and the average of the Coulomb interaction (as well as of any other interaction) gives a self-energy which is identical to that of the BCS case (2nd lines in Eqs. (35) and (36)). 

Thus we conclude that there is a stark difference be- tween the BCS wavefunction and the new wavefunction. The BCS wavefunction will give rise to superconductiv- ity if the first term in Eq. (35) is negative on average[3]. This will be the case if the interaction V ( k − k $^{′}$) in addi- tion to the repulsive Coulomb interaction contains neg- ative (attractive) terms arising e.g. from the electron- phonon interaction, and if the attractive terms are suf- ficiently large that they can overwhelm the repulsive Coulomb terms. Of course the frequency dependence of the electron-phonon interaction provides additional help by reducing the effective strength of the Coulomb interaction[13]. 

Instead, with the new wavefunction an attractive inter- action (whether instantaneous or retarded) will not help if parity is unbroken, since it will cancel identically in the first term in Eq. (36). The direct Coulomb repulsion is however also eliminated in the first term in Eq. (36) in contrast to the first term in Eq. (35) if parity is unbro- ken. Furthermore, if parity is broken, the last term in Eq. (36) provides a lowering of energy in the supercon- ducting state for a repulsive interaction , and particularly if the region where u$_{k}$v$_{k}$ ̸ = 0 occurs near k = 0. 

It is natural to try to describe the superconducting state using electronic states that do not optimize the electron-ion interaction and many such attempts have 

<img src='content_image/393669.jpg'>

## VII. DISCUSSION 

